Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory

The Eleventh IMACS International Conference on NONLINEAR EVOLUTION EQUATIONS AND WAVE PHENOMENA: COMPUTATION AND THEORY

April 17–19, 2019

Georgia Center for Continuing Education University of Georgia, Athens, GA, USA

http://waves2019.uga.edu

Edited by Gino Biondini and Thiab Taha

Book of Abstracts

The Eleventh IMACS International Conference On

Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory

Athens, Georgia April 17—19, 2019

Sponsors

International Association for Mathematics and Computers in Simulation (IMACS) Keynote Speakers Computer Science Department at UGA David Ambrose: "Vortex sheets, Boussinesq equations, and other problems in the Wiener algebra" Organization Alex Himonas:"Initial and boundary value problems for T. Taha (USA), General Chair & Conference Coordinator evolution equations” G. Biondini (USA), Co-chair J. Bona (USA), Co-chair Stefano Trillo:"Nonlinear PDEs describing real R. Vichnevetsky (USA), experiments: recurrences, solitons, and shock waves" Honorary President of IMACS, Honorary Chair

Scientific program committee

Bedros Afeyan (USA) Yuri Latushkin (USA) David Amrbrose (USA) Jonatan Lenells (USA) Stephen Anco (Canada) Changpin Li (China) Andrea Barreiro (USA) Andrei Ludu (USA) Gino Biondini (USA) Pavel Lushnikov (USA) Lorena Bociu (USA) Dionyssis Mantzavions (USA) Jerry Bona (USA) Peter Miller (USA) Jared Bronski(USA) Dimitrios Mitsotakis (USA) Robert Buckingham (USA) Nobutaka Nakazono (Japan) Annalisa Calini (USA) Alan Newell (USA) Ricardo Carretero (USA) Katie Newhall (USA) John Carter (USA) Beatrice Pelloni (UK) Efstathios G. Charalampidis (USA) Virgil Pierce (USA) Min Chen (USA) Barbara Prinari (USA) Guangye Chen (USA) Pamela Pyzza (USA) Wooyoung Choi (USA) Zhijun (George) Qiao (USA) Antoine Cerfon (USA) Vassilios Rothos (Greece) Anton Dzhamay (USA) Xu Runzhang (China) Anna Ghazaryan (USA) Constance Schober (USA) Alex Himonas (USA) Brad Shadwick (USA) Curtis Holliman (USA) Israel Michael Sigal (Canada) Pedro Jordan (USA) Avraham Soffer (USA) Nalini Joshi (Australia) Martin Ostoja Starzewski (USA) Kenji Kajiwara (USA) Thiab Taha (USA) Henrik Kalisch (Norway) Michail Todorov (Bulgaria) David Kaup (USA) Muhammad Usman (USA) Panayotis Kevrekidis (USA) Samuel Walsh (USA) Alexander Korotkevic (USA) Jianke Yang (USA) Gregor Kovacic (USA) Vladimir Zakharov (USA) Stephane Lafortune (USA) Organized sessions

1. Jerry Bona, Min Chen,Shuming Sun, Bingyu Zhang, 21. Katie Newhall: "Stochastic dynamics in nonlinear "Nonlinear waves" systems" 2. Barbara Prinari, Alyssa K. Ortiz "Novel challenges in 22. Robert Buckkingham, Peter Miller: "Modern methods for nonlinear waves and integrable systems" dispersive wave equations" 3. John Carter, "Recent developments in mathematical 23. Sergey Dyachenko, Katelyn Leisman, Denis Silantyev: models of water waves " "Nonlinear waves in optics, fluids and plasma" 4. Andrei Ludu, Changpin Li, Thiab Taha, "Fractional 24. Michael Sigal, Jianfeng Lu: "Mathematical perspectives diferential equations" in quantum mechanics and quantum chemistry" 5. Alex Himonas, Curtis Holliman, Dionyssis 25. Alexander O. Korotkevich and Pavel Lushnikov: Mantzavinos:"Evolution equations and integrable systems" "Nonlinear waves, singularities,vortices, and turbulence in 6. Vladimir Dragovic, Anton Dzhamay, Virgil Pierce: hydrodynamics, physcal, and biological systems" "Random matrices, Painleve equations, and integrable 26. Ziad Musslimani, Matthew Russo: "Physical applied systems" mathematics" 7. Bernard Deconinck, Anna Ghazaryan, Mat Johnson, 27. Cancelled Stephane Lafortune, Yuri Latushkin, Jeremy Upsal, Samuel Walsh: "Stability and traveling waves" 28. Chaudry Masood Khalique, Muhammad Usman: "Recent advances in analytical and computational methods for 8.Avraham Soffer, Gang Zhao, S. Gustafson: "Dispersive nonlinear partial differential equations" wave equations and their soliton interactions: Theory and applications" 9. Efstathios Charalampidis, Fotini Tsitoura: "Nonlinear evolutionary equations: Theory, numerics and experiments" 10. Robin Ming Chen, Runzhang Xu: "Recent advances in PDEs from fluid dynamics and other dynamical models" 11. Cancelled 12. Gino Biondini: “Dispersive shocks, semiclassical limits and applications" 13. Qi Wang and Xueping Zhao:"Recent advances in numerical methods of PDEs and applications in life science, material science" 14. Bedros Afeyan, Brad Shadwicn, Jon Wilkening: "Nonlinear kinetic self-organized plasma dynamics driven by coherent, intense electromagnetic fields session" 15. Yi Zhu, Xu Yang, Hailong Guo: "Waves in topological materials" 16. Dmitry Pelinovsky and Anna Geyer: "Existence and stability of peaked waves in nonlinear evolution equations" 17. Pamela B. Pyzza: "Nonlinear dynamics of mathematical models in neuroscience" 18. Stephen Anco, Stephane Lafortune, Zhijun (George) Qiao: “Negative flows, peakons, integrable systems,and their applications" 19. Thomas Carty: "Network dynamics" 20. Nalini Joshi, Giorgio Gubbiotti, Nobutaka Nakazono, Milena Radnovic, Yang Shi, Dinh Tran: "Dynamical systems and integrability" PROGRAM AT A GLANCE

Wednesday, April 17, 2019

Mahler auditoriumRoom F/GRoom Y/ZRoom ERoom JRoom V/WRoom BRoom CRoom D 8.00–8.30amWelcome 8.30–9.30amKeynote lecture I: David Ambrose 9.30–10.00am Coffee break 10.00–10.50amS7 - I/IXS3 - I/IIIS24 - I/IIIS25 - I/VIIS20 - I/IIS21 - I/IIS15 - I/IIIS19 - I/II 10.55am–12.10pmS7 - II/IXS3 - II/IIIS24 - II/IIIS18 - I/IVS6 - I/IIIS5 - I/VS15 - II/III S19 - II/II 12.10–1.40pm Lunch (attendees on their own) 1.40–3.20pmS7 - III/IXS5 - II/VS9 - I/IIIS18 - II/IVS6 - II/IIIS28 - I/IS15 - III/IIIPapers 3.20–3.50pm Coffee break 3.50–5.55pmS7 - IV/IXS5 - III/VS9 - II/IIIS25 - II/VIIS6 - III/IIIS16 - I/IIIS21 - II/IIS24 - III/IIIS10 - I/I

Thursday, April 18, 2019

Masters HallRoom F/GRoom Y/ZRoom ERoom JRoom V/WRoom KRoom LRoom D 8:00–9:00am : Alex Himonas 9:10–10:00amS4 - I/IIIS8 - I/VS25 - III/VIIS12 - I/IIIS16 - II/IIIPapersS17 - I/II 10:00–10:30am Coffee break 10:30–12:10pmS18 - III/IVS9 - III/IIIS22 - I/IIS25 - IV/VIIS12 - II/IIIS16 - III/III S3 - III/IIIS2 - I/IIS17 - II/II 12:10–1:40pm Lunch (attendees on their own) 1:40–3:20pmS7 - V/IXS5 - IV/VS8 - II/VS25 - V/VIIS1 - I/IIS12 - III/IIIS26 - I/IIS2 - II/IIS20 - II/II 3:20–3:50pm Coffee break 3:50–5:55pmS7 - VI/IXS5 - V/VS8 - III/VS18 - IV/IVS1 - II/IIS22 - II/IIS26 - II/IIS4 - II/IIIS23 - I/II 5:00–7:00pmPosters, Hill Atrium (outside Mahler auditorium) 7:00–9:00pm Conference banquet (including student papers award)

Friday, April 19, 2019

Masters HallRoom F/GRoom Y/ZRoom KRoom V/W 8:00–9:00amKeynote lecture 3: Stefano Trillo 9:10–10:00amS7 - VII/IXS13 - I/IIS14 - I/IIS25 - VI/VII 10:00–10:30am Coffee break 10:30–12:10pmS7 - VIII/IXS13 - II/IIS8 - IV/VS4 - III/IIIS25 - VII/VII 12:10–1:40pm Lunch (attendees on their own) 1:40–3:20pmS7 - IX/IXS23 - II/IIS8 - V/VS14 - II/II 3:20–3:50pm Coffee break CONFERENCE PROGRAM

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TUESDAY, APRIL 16, 2019

5:00–6:00 REGISTRATION (in front of Mahler Hall) 5:00–7:00 RECEPTION

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WEDNESDAY, APRIL 17, 2019

7:30–9:30 REGISTRATION 8:00–8:30 WELCOME Thiab Taha, Program Chair and Conference Coordinator Alan Dorsey, Dean of the Franklin College of Arts and Sciences, UGA 8:30–9:30 KEYNOTE LECTURE I, Mahler Hall David Ambrose: Vortex sheets, Boussinesq equations, and other problems in the Wiener algebra Chair: Thiab Taha

9:30–10:00 COFFEE BREAK

10:00–10:50 SESSION 7, Mahler Hall: Stability and traveling waves – Part I/IX Chairs: Bernard Deconinck, Anna Ghazaryan, Mat Johnson, Stephane Lafortune, Yuri Latushkin, Jeremy Upsal, Samuel Walsh 10:00–10:25 Stephane Lafortune: Study of a model of a liquid in presence of a surfactant 10:25–10:50 Panayotis Kevrekidis: On some Select Klein-Gordon problems: internal modes, fat tails, wave collisions and beyond

10:00–10:50 SESSION 3, F/G: Recent developments in mathematical studies of water waves – Part I/III Chair: John Carter 10:00–10:25 John Carter: Particle paths and transport properties of NLS and its generalizations 10:25–10:50 Ben Akers: Asymptotics and numerics for modulational instabilities of traveling waves

10:00–10:50 SESSION 24, Room Y/Z: Mathematical perspectives in quantum mechanics and quantum chemistry – Part I/III Chairs: Jianfeng Lu and Israel Michael Sigal 10:00–10:25 Christof Melcher: Spinning Landau-Lifschitz solitons - a quantum mechanical analogy 10:25–10:50 Benjamin Stamm: A perturbation-method-based post-processing of plane wave approximations for nonlinear Schoedinger operators

10:00–10:50 SESSION 25, Room E: Nonlinear waves, singularities, vortices, and turbulence in hydrodynamics, physical, and biological systems – Part I/VII Chairs: Alexander O. Korotkevich and Pavel Lushnikov 10:00–10:25 David Kaup: Optical phase-modulated nonlinear waves in a graphene waveguide 10:25–10:50 Bo Yang and Jianke Yang: Rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger equation

10:00 - 10:50 SESSION 20, Room J: Dynamical systems and integrability – Part I/II Chairs: Nalini Joshi and Nobutaka Nakazono 10:00–10:25 Vladimir Dragovic and Milena Radnovic: Ellipsoidal Billiards and Chebyshev-type polynomials 10:25–10:50 Nalini Joshi, Christopher Lustri and Steven Luu: Hidden solutions of discrete systems 10:00–10:50 SESSION 21, Room V/W: Stochastic Dynamics in Nonlinear Systems – Part I/II Chair: Katie Newhall 10:00–10:25 Katie Newhall: A network of transition pathways in a model granular system 10:25–10:50 Jay Newby: The effect of moderate noise on a limit cycle oscillator: counterrotation and bistability

10:00–10:50 SESSION 15, Room B: Waves in topological materials – Part I/III Chairs: Yi Zhu, Xu Yang, Hailong Guo 10:00–10:25 Alexander Watson: Computing edge spectrum in the presence of disorder without spectral pollution 10:25–10:50 Justin Cole: Topologically Protected Edge Modes in Longitudinally Driven Waveguides

10:00–10:50 SESSION 19, Room C: Network Dynamics – Part I/II Chair: Tom Carty 10:00–10:25 Mamoon Ahmed: The universal covariant representation and amenability 10:25–10:50 Dashiell Fryer: Adaptive zero determinant strategies in the iterated prisoner’s dilemma tournament

10:55–12:10 SESSION 7, Mahler Hall: Stability and traveling waves – Part II/IX Chairs: Bernard Deconinck, Anna Ghazaryan, Mat Johnson, Stephane Lafortune, Yuri Latushkin, Jeremy Upsal, Samuel Walsh 10:55–11:20 Milena Stanislavova: Asymptotic stability for spectrally stable Lugiato-Lefever solutions in periodic waveguides 11:20–11:45 Efstathios Charalampidis: Formation of extreme events in NLS systems 11:45–12:10 Todd Kapitula: Viewing spectral problems through the lens of the Krein matrix

10:55–12:10 SESSION 3, F/G: Recent Developments in Mathematical Studies of Water Waves – Part II/III Chair: John Carter 10:55–11:20 Chris Curtis: Nonlinear waves over patches of vorticity 11:20–11:45 Henrik Kalisch: Fully dispersive model equations for hydroelastic waves 11:45–12:10 Harvey Segur: Tsunami

10:55–12:10 SESSION 24, Room Y/Z: Mathematical perspectives in quantum mechanics and quantum chemistry – Part II/III Chairs: Michael Sigal and Jianfeng Lu 10:55–11:20 Michael Weinstein: Edge states in honeycomb structures 11:20–11:45 Fabio Pusateri: Nonlinear Schroedinger equations with a potential in dimension 3 11:45–12:10 Artur Izmaylov: New developments in quantum chemistry on a quantum computer

10:55–12:10 SESSION 18, Room E: Negative flows, peakons, integrable systems, and their applications – Part I/IV Chair: Zhijun (George) Qiao 10:55–11:20 Jing Kang: Liouville correspondences between multi-component integrable hierarchies 11:20–11:45 Huafei Di: Global well-posedness for a nonlocal semilinear pseudo-parabolic equation with conical degeneration

10:55–12:10 SESSION 6, Room J: Random matrices, Painleve equations, and integrable systems – Part I/III Chair: Vladimir Dragovic 10:55–11:20 Anton Dzhamay: Discrete Painlevé equations in tiling problems 11:20–11:45 Tomoyuki Takenawa: The space of initial conditions for some 4D Painlevé systems 11:45–12:10 Nobutaka Nakazono: Classification of quad-equations on a cuboctahedron

10:55–12:10 SESSION 5, Room V/W: Evolution equations and integrable systems – Part I/IV Chairs: Alex Himonas, Curtis Holliman & Dionyssis Mantzavinos 10:55–11:20 Gino Biondini: Riemann problems, solitons and dispersive shocks in modulationally unstable media 11:20–11:45 Barbara Prinari: Inverse scattering transform for the defocusing Ablowitz-Ladik equation with arbitrary nonzero background 11:45–12:10 Satbir Malhi: Energy decay for the linear damped Klein Gordon equation on unbounded domain 10:55–12: 10 SESSION 15, Room B: Waves in topological materials – Part II/III Chairs: Yi Zhu, Xu Yang, Hailong Guo 10:55–11:20 Junshan Lin: Embedded eigenvalues and Fano resonance for metallic structures with small holes 11:20–11:45 Alexis Drouo: Edge states in near-honeycomb structures 11:45–12:10 Hailong Guo: Unfitted Nitsche's method for computing edge modes n iphotonic graphene

10:55–12: 10 SESSION 19, Room C: Network dynamics – Part II/II Chair: Tom Carty 10:55–11:20 Timothy Ferguson: Bistability in the Kuramoto model 11:20–11:45 Tom Carty: Configurational stability for the Kuramoto-Sakaguchi modelH 11:45–12:10 Sarah Simpson: A Matrix Valued Kuramoto Model

12:10–1:40 LUNCH (attendees on their own)

1:40–3:20 SESSION 7, Mahler Hall: Stability and traveling waves – Part III/IX Chairs: Bernard Deconinck, Anna Ghazaryan, Mat Johnson, Stephane Lafortune, Yuri Latushkin, Jeremy Upsal, Samuel Walsh 1:40–2:05 Ross Parker: Spectral stability of multi-pulses via the Krein matrix 2:05–2:30 Anna Ghazaryan: Stability of planar fronts in a class of reaction-diffusion systems 2:30–2:55 Yuri Latushkin: Recent results on application of the Maslov index in spectral theory of differential operators 2:55–3:20 Alim Sukhtayev: Spectral stability of hydraulic shock profiles

1:40–3:20 SESSION 5, Room F/G: Evolution equations and integrable systems – Part II/V Chairs: Alex Himonas, Curtis Holliman & Dionyssis Mantzavinos 1:40–2:05 David Nicholls: Well-posedness and analyticity of solutions to a water wave problem with viscosity 2:05–2:30 John Gemmer: Isometric immersions and self-similar buckling in non-Euclidean elastic sheets 2:30–2:55 Curtis Holliman: Non-uniqueness and norm-inflation for Camassa-Holm-type equations 2:55–3:20 Fredrik Hildrum: Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity

1:40–3:20 SESSION 9, Room Y/Z: Nonlinear evolutionary equations: Theory, numerics and experiments – Part I/III Chairs: Efstathios Charalampidis and Fotini Tsitoura 1:40–2:05 Roy Goodman: Bifurcations on a dumbbell quantum graph 2:05–2:30 Patrick Sprenger and Mark Hoefer: Traveling waves in the fifth order KdV equation anddiscontinuous shock solutions of the Whitham modulation equations 2:30–2:55 Adilbek Kairzhan, Dmitry Pelinovsky & Roy Goodman: Nonlinear instability of spectrally stable shifted states on star graphs 2:55–3:20 Yuan Chen and Keith Promislow: Curve Lengthening and shortening in Stong FCH

1:40–3:20 SESSION 18, Room E: Negative flows, peakons, integrable systems, and their applications – Part II/IV Chair: Stephen Anco 1:40–2:05 Anna Geyer: Instability and uniqueness of the peaked periodic traveling wave in the reduced Ostrovsky equation 2:05–2:30 Huijun He: Some analysis results for the U(1)-invariant equation 2:30–2:55 Stephen Anco and Elena Recio: Accelerating dynamical peakons and their behaviour 2:55–3:20 Xiao-Jun Yang: A new perspective in anomalous viscoelasticity from the derivative with respect to another function view point

1:40–3:20 SESSION 6, Room J: Random matrices, Painleve equations, and integrable systems – Part II/III Chair: Virgil Pierce 1:40–2:05 Robert Buckingham: Representation of joint moments of CUE characteristic polynomials in terms of a Painlevé-V solution 2:05–2:30 Peter Miller: Rational solutions of Painlevé equations 2:30–2:55 Andrei Prokhorov: Asymptotic of solutions of three-component Painlevé-II equation 2:55–3:20 Sevak Mkrtchyan: Entropy of Beta Random Matrix Ensembles

1:40–3:20 SESSION 28, Room V/W: Recent advances in analytical and computational methods for nonlinear PDEs Chairs: Chaudry Masood Khalique and Muhammad Usman 1:40–2:05 Muhammad Usman: A collocation method for a class of a nonlinear partial differential equations 2:05–2:30 Arshad Muhammad: Applications of fixed point theorems to integral and differential equations 2:30–2:55 Kinza Mumtaz & Mudassar Imran: The optimal control of HPV infection and cervical cancer with HPV vaccine

1:40–3:20 SESSION 15, Room B: Waves in topological materials – Part III/III Chairs: Hailong Guo, Xu Yang, Yi Zhu 1:40–2:05 Lihui Chai: Frozen Gaussian Approximation for the Dirac equation in semi-classical regime 2:05–2:30 Yi Zhu: Linear and nonlinear waves in honeycomb photonic materials 2:30–2:55 Peng Xie and Yi Zhu: Wave-packet dynamics in slowly modulated photonic graphene

1:40–3:20 PAPERS, Room C Chairs: Gennady El 1:40–2:05 Giacomo Roberti, Gennady El, Pierre Suret and Stéphane Randoux: Early stage of integrable turbulence in 1D NLS equation: the semi-classical approach to statistics 2:05–2:30 Bryn Balls-Barker, Blake Barker & Olivier Lafitte: Spectral stability of ideal-gas shock layers in the strong shock limit 2:30–2:55 Camille R. Zaug and John D. Carter: Frequency Downshift in the Ocean 2:55–3:20 Ali Eshaghian Dorche, Ali Asghar Eftekhar and Ali Adibi: Advanced dispersion enginedgeeering for wideband on-chip optical frequency comb generation

3:20–3:50 COFFEE BREAK

3:50–5:55 SESSION 7, Mahler Hall: Stability and traveling waves – Part IV/IX Chairs: Bernard Deconinck, Anna Ghazaryan, Mat Johnson, Stephane Lafortune, Yuri Latushkin, Jeremy Upsal, Samuel Walsh 3:50–4:15 Blake Barker: Rigorous verification of wave stability 4:15–4:40 Alin Pogan: Nonlinear stability of layers in precipitation models 4:40–5:05 Vahagn Manukian: Fisher-KPP dynamics in diffusive Rosenzweig-MacArthur and Holling-Tanner models 5:05–5:30 Zhiwu Lin: Turning point principle for the stability of stellar models 5:30–5:55 Robert Marangell: Stability of travelling waves in a haptotaxis model

3:50–5:55 SESSION 5, Room F/G: Evolution Equations and Integrable Systems – Part III/V Chairs: Alex Himonas, Curtis Holliman & Dionyssis Mantzavinos 3:50–4:15 Sarah Raynor: Low regularity stability for the KdV equation 4:15–4:40 John Holmes: Existence of solutions for conservation laws 4:40–5:05 Ryan Thompson: On the evolution of dark matter 5:05–5:30 Yuexun Wang: Enhanced existence time of solutions to the fractional KdV equation 5:30–5:55 Jose Pastrana Chiclana: Non-uniform continuous dependence for Euler equations in Besov spaces

3:50–5:55 SESSION 9, Room Y/Z: Nonlinear evolutionary equations: Theory, numerics and experiments – Part II/III Chairs: Efstathios Charalampidis and Fotini Tsitoura 3:50–4:15 Foteini Tsitoura: Observation of phase domain walls in deep water surface gravity waves 4:15–4:40 Hang Yang: Models for 3D Euler Equations 4:40–5:05 Igor Barashenkov: New PT-symmetric systems with solitons: nonlinear Dirac and Landau-Lifshitz equations 5:05–5:30 Demetrios Christodoulides: Parity-Time and other symmetries in optics and photonics 5:30–55:5 Guo Deng, Gino Biondini and Surajit Sen: Generation, propagation and interaction of solitary waves in integrable versus non-integrable lattices

3:50–5:55 SESSION 25, Room E: Nonlinear waves, singularities, vortices, and turbulence in hydrodynamics, physical, and biological systems – Part II/VII Chairs: Alexander O. Korotkevich and Pavel Lushnikov 3:50–4:15 Fabio Pusateri, Massimiliano Berti, and Roberto Feola: The Zakharov-Dyachenko conjecture on the integrability of gravity water waves 4:15–4:40 Joseph Zaleski, Miguel Onorato and Yuri Lvov: Anomalous correlators, “ghost” waves and nonlinear standing waves in the beta-FPUT system 4:40–5:05 Denis Silantyev and Pavel Lushnikov: Powerful conformal maps for adaptive resolving of the complex singularities of the Stokes wave 5:05–5:30 Amir Sagiv, Adi Ditkowski and Gadi Fibich: Efficient numerical methods for nonlinear dynamics thwi random parameters

3:50–5:55 SESSION 6, Room J: Random Matrices, Painleve Equations, and Integrable Systems – Part III/III Chair: Anton Dzhamay 3:50–4:15 Vasilisa Shramchenko: Algebro-geometric solutions to Schlesinger and Painlevé-VI equations 4:15–4:40 Leonid Chekhov: SLk character varieties and quantum cluster algebras 4:40–5:05 Alessandro Arsie: A survey of bi-flat F-manifolds 5:05–5:30 Nicholas Ercolani: Integrable mappings and random walks in random environments 5:30–5:55 Virgil Pierce: Skew-orthogonal polynomials and continuum limits of the Pfaff lattice

3:50–5:55 SESSION 16, Room V/W: Existence and stability of peaked waves in nonlinear evolution equations – Part I/III Chair: Anna Geyer 3:50–4:15 Mariana Haragus: Regular patterns and defects for the Rayleigh-Bénard convection. 4:15–4:40 Richard Kollar: Krein signature without eigenfunctions and without eigenvalues. What is Krein signature and what does it measure? 4:40–5:05 Fabio Natali: Periodic Traveling-wave solutions for regularized dispersive equations: Sufficient conditions for orbital stability with applications 5:05–5:30 Uyen Le: Convergence of Petviashvili's method near periodic waves in the fractional KdV equation 5:30–5:55 Elek Csobo: Stability of standing waves for a nonlinear Klein-Gordon equation with delta potentials

3:50–5:55 SESSION 21, Room B: Stochastic dynamics in nonlinear systems - PART II/II Chair: Katie Newhall 3:50–4:15 Joe Klobusicky: Averaging for systems of nonidentical molecular motors 4:15–4:40 Ilya Timofeyev: Stochastic parameterization of subgrid-scales in one-dimensional shallow water equations 4:40–5:05 Nawaf Bou-Rabee: Coupling for Hamiltonian Monte Carlo 5:05–5:30 Yuan Gao: Limiting behaviors of high dimensional stochastic spin ensemble 5:30–5:55 Molei Tao: Improving sampling accuracy of SG-MCMC methods via non-uniform subsampling of gradients

3:50–5:55 SESSION 24, Room C: Mathematical perspectives in quantum mechanics and quantum chemistry – Part III/III Chairs: Jianfeng Lu and Israel Michael Sigal 3:50–4:15 Dionisios Margetis: On the excited state of the interacting boson system: a non-Hermitian view 4:15–4:40 Akos Nagy: Concentration properties of Majorana spinors in the Jackiw-Rossi theory 4:40–5:05 Marius Lemm: A central limit theorem for integrals of random waves 5:05–5:30 Christof Sparber: Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures 5:30–5:55 Thomas Chen: Boltzmann equations via Wigner transform and dispersive methods

3:50–5:55 SESSION 10, Room D: Recent advances in PDEs from fluid dynamics and other dynamical models – Part I/I Chairs: Robin Ming Chen, Runzhang Xu 3:50–4:15 Gary Webb, Qiang Hu, Avijeet Prasad and Stephen Anco: Godbillon-Vey helicity in magnetohydrodynamics and fluid dynamics 4:15–4:40 Hua Chen, Robert Gilbert and Philippe Guyenne: Dispersion and attenuation in a poroelastic model for gravity waves on an ice-covered ocean 4:40–5:05 Qingtian Zhang: Global solution of SQG front equation 5:05–5:30 Dongfen Bian and Jinkai Li: Finite time blow up of compressible Navier-Stokes equations on half space or outside a fixed ball 5:30–5:55 Wei Lian, Runzhang Xu and Yi Niu: Global well-posedness of coupled parabolic systems ======THURSDAY, APRIL 18, 2019

7:30–9:30 REGISTRATION 8:00–9:00 KEYNOTE LECTURE 2, Masters Hall Alex Himonas: Initial and boundary value problems for evolution equations Chair: Jerry Bona

9:10–10:00 SESSION 4, Room F/G: Fractional Diferential Equations – Part I/III Chair: Harihar Khanal 9:10–9:35 Dumitru Baleanu: On fractional calculus and nonlinear wave phenomena 9:35–1:00 Andrei Ludu: Time dependent order differential equations

9:10–10:00 SESSION 8, Room Y/Z: Dispersive Wave Equations and their Soliton Interactions: Theory and Applications – Part I/V Chairs: Avraham Soffer, Gang Zhao, . SGustafson 9:10–9:35 Peter Pickl: Higher Order Corrections to Mean Field Dynamics of Bose Cold Gases 9:35–10:00 Thomas Chen and Avy Soffer:Dynamics of a heavy quantum tracer particle in a Bose gas

9:10–10:00 SESSION 25, Room E: Nonlinear waves, singularities, vortices, and turbulence in hydrodynamics, physical, and biological systems – Part III/VII Chairs: Alexander O. Korotkevich and Pavel Lushnikov 9:10–9:35 Svetlana Roudenko, Kai Yang and Yanxiang Zhao: Stable blow-up dynamics in the critical and supercritical NLS and Hartree equations 9:35–1:00 Anastassiya Semenova, Alexander Korotkevich, and Pavel Lushnikov: Appearance of stokes waves in deep water

9:10–10:00 SESSION 12, Room J: Dispersive shocks, semiclassical limits and applications – Part I/III Chair: Gino Biondini 9:10–9:35 Stephane Randoux: Modulational instability of a plane wave in the presence of localized perturbations: some experimental results in nonlinear fiber optics 9:35–1:00 Gennady El: Wave-mean flow interactions in dispersive hydrodynamics

9:10–10:00 SESSION 16, Room V/W: Existence and stability of peaked waves in nonlinear evolution equations – Part II/III Chair: Dmitry Pelinovsky 9:10–9:35 Stephen Anco: Evolution equations with distinct sectors of peakon-type solutions 9:35–10:00 Zhijun Qiao: High order peakon models

9:10–10:00 PAPERS, Room K Chairs: Otis wright 9:10–9:35 Alessandro Barone, Alessandro Veneziani, Flavio Fenton and Alessio Gizzi: Cardiac conductivity estimation by a variational data assimilation procedure: analysis and validation 9:35–10:00 Otis Wright: Effective Integration of Some Integrable NLS Equations

9:10–10:00 SESSION 17, Room L: Nonlinear dynamics of mathematical models in neuroscience – Part I/II Chair: Pamela Pyzza 9:10–9:35 Shelby Wilson: On the dynamics of coupled Morris-Lecar neurons

10:00–10:30 COFFEE BREAK

10:30–12:10 SESSION 18, Masters Hall: Negative flows, peakons, integrablesystems, and their applications – Part III/IV Chair: Stephane Lafortune 10:30–10:55 Qilao Zha, Qiaoyi Hu and Zhijun Qiao: Short pulse systems produced through the negative WKI hierarchy 10:55–11:20 Evans Boadi, Sicheng Zhao and Stephen Anco: New integrable peakon equations from a modified AKNS scheme 11:20–11:45 Shuxia Li and Zhijun Qiao: Lax algebraic representation for an integrable hierarchy

10:30–12:10 SESSION 9, Room F/G: Nonlinear Evolutionary Equations: Theory, Numerics and Experiments – Part III/III Chair: Efstathios Charalampidis and Fotini Tsitoura 10:30–10:55 Jason Bramburger: Snakes and lattices: Understanding the bifurcation structure of localized solutions to lattice dynamical systems 10:55–11:20 Ryan Goh: Growing stripes, with and without wrinkles 11:20–11:45 Zoi Rapti, Jared Bronski & Andrea Barreiro: Nonlinear eigenvalue problems in biologically motivated PDEs 11:45–12:10 Joceline Lega: Grain boundaries of the Swift-Hohenberg equation: simulations and analysis

10:30–12:10 SESSION 22, Room Y/Z: Modern Methods for Dispersive Wave Equations – Part I/II Chairs: Robert Buckingham and Peter Miller 10:30–10:55 Peter Perry: Soliton Resolution for the Derivative Nonlinear Schrödinger Equation 10:55–11:20 Aaron Saalmann: Long-time asymptotics for the massive Thirring model 11:20–11:45 Elliot Blackstone: Singular limits of certain Hilbert-Schmidt integral operators and applications to tomography 11:45–12:10 Tom Trogdon: The computation of linear and nonlinear dispersive shocks

10:30–12:10 SESSION 25, Room E: Nonlinear waves, singularities, vortices, and turbulence in hydrodynamics, physical, and biological systems – Part IV/VII Chair: Alexander O. Korotkevich and Pavel Lushnikov 10:30–10:55 Jerry Bona: Dynamical problems arising in blood flow: nonlinearwaves on trees 10:55–11:20 Curtis Menyuk, Zhen Qi, Shaokang Wang: Stability and noise in frequency combs: harnessing the music of the spheres 11:20–11:45 Tobias Schaefer: Instantons and fluctuations in complex systems 11:45–12:10 Katelyn Plaisier Leisman and Gregor Kovacic: Nonlinear waves acting like linear waves in NLS

10:30–12:10 SESSION 12, Room J: Dispersive shocks, semiclassical limits and applications – Part II/III Chair: Gino Biondini 10:30–10:55 Alexander Tovbis: Towards kinetic equation for soliton and breather gases for the focusing NLS equation 10:55–11:20 Sitai Li: Universal behavior of modulationally unstable media with non-zero boundary conditions 1:20 –11:45 Jonathan Lottes: Nonlinear interactions between solitons and dispersive shocks in focusing media 11:45–12:10 Thibault Congy: Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure

10:30–12:10 SESSION 16, Room V/W: Existence and stability of peaked waves in nonlinear evolution equations – Part III/III Chair: Dmitry Pelinovsky 10:30–10:55 Mathias Arnesen: A nonlocal approach to waves of maximal height to the Degasperis-Procesi equation 10:55–11:20 Raj Dhara: Waves of maximal height for a class nonlocal equations with homogeneous symbol 11:20–11:45 Tien Truong: Large-amplitude solitary water waves for the Whitham equation 11:45–12:10 Bruno Vergara: Convexity of Whitham's highest cusped wave

10:30–12:10 SESSION 3, K: Recent Developments in Mathematical Studies of Water Waves – Part III/III Chair: John Carter 10:30–10:55 Diane Henderson: Faraday waves with bathymetry 10:55–11:20 Olga Trichtchenko: Water waves under ice 11:20–11:45 Bernard Deconinck: The stability of stationary solutions of the focusing NLS equation 11:45–12:10 Debbie Eeltink: Effect of viscosity and sharp wind increase on ocean wave statistics

10:30–12:10 SESSION 2, Room L: Novel challenges in nonlinear waves and integrable systems – Part I/II Chairs: Barbara Prinari, Alyssa K. Ortiz 10:30–10:55 Martin Klaus: Spectral properties of matrix-valued AKNS systems with steplike potentials 10:55–11:20 Alexei Rybkin: The effect of a positive bound state on the KdV solution. A case study 11:20–11:45 C van der Mee: Exact solutions of the focusing NLS equation with symmetric nonvanishing boundary conditions 11:45–12:10 Jeremy Upsal: Real Lax spectrum implies stability

10:30–12:10 SESSION 17, Room D: Nonlinear dynamics of mathematical models in neuroscience – Part II/II Chair: Pamela B. Pyzza 10:30–10:55 Paulina Volosov and Gregor Kovacic: Network reconstruction: architectural and functional connectivity in the cerebral cortex 10:55–11:20 Duane Nykamp and Brittany Baker: Network microstructure dominates global network connectivity in synchronous event initiation 11:20–11:45 Pamela Pyzza, Katie Newhall, Douglas Zhou, Gregor Kovacic and David Cai: Idealized models of insect olfaction 11:45–12:10 Alexei Cheviakov and Jason Gilbert: The narrow-capture problem in a unit sphere: global optimization of volume trap arrangements

12:10–1:40 LUNCH (attendees on their own)

1:40–3:20 SESSION 7, Masters Hall: Stability and traveling waves – Part V/IX Chairs: Bernard Deconinck, Anna Ghazaryan, Mat Johnson, Stephane Lafortune, Yuri Latushkin, Jeremy Upsal, Samuel Walsh 1:40–2:05 Graham Cox: A Maslov index for non-Hamiltonian systems 2:05–2:30 Claire Kiers: A bifurcation analysis of standing pulses and the Maslov index 2:30–2:55 Selim Sukhtaie: Localization for Anderson models on tree graphs 2:55–3:20 Mariana Haragus: Dynamics of frequency combs modeled by the Lugiato-Lefever equation

1:40–3:20 SESSION 5, Room F/G: Evolution Equations and Integrable Systems – Part IV/V Chair: Alex Himonas, Curtis Holliman & Dionyssis Mantzavinos 1:40–2:05 Natalie Sheils: Revivals and fractalisation in the linear free space Schrodinger equation 2:05–2:30 David Smith: Unified transform method with moving interfaces 2:30–2:55 Fangchi Yan: Well-posedness of initial-boundary value problems for dispersive equations via the Fokas method 2:55–3:20 Maria Christina van der Weele: Integrable systems in 4+2 dimensions and their reduction to 3+1 dimensions

1:40–3:20 SESSION 8, Room Y/Z: Dispersive wave equations and their soliton interactions: Theory and applications – Part II/V Chairs: Avraham Soffer, Gang Zhao, . SGustafson 1:40–2:05 Marius Beceanu, Juerg Froehlich and Avy Soffer: Semi-linear Schroedinger's equation with random time-dependent potentials 2:05–2:30 Minh Binh & Avy Soffer: On the energy cascade of acoustic wave turbulence: Beyond Kolmogorov-Zakharov solutions 2:30–2:55 Matthew Rosenzweig: Global well-posedness and scattering for the Davey-Stewartson system at critical regularity

1:40–3:20 SESSION 25, Room E: Nonlinear waves, singularities, vortices, and turbulence in hydrodynamics, physical, and biological systems – Part V/VII Chairs: Alexander O. Korotkevich and Pavel Lushnikov 1:40–2:05 Evgeny Kuznetsov, Maxim Kagan and Andrey Turlapov: Expansion of the strongly interacting superfluid Fermi gas: symmetry and self-similar regimes 2:05–2:30 Joseph Zaleski, Philip Zaleski, and Yuri Lvov: Excitation of interfacial waves via near-resonant surface-interfacial wave interactions 2:30–2:55 Sergey Dyachenko, Alexander Dyachenko, Pavel Lushnikov & Vladimir Zakharov: Singularities in 2D fluids with free surface 2:55–3:20 Israel Michael Sigal: On density functional theory

1:40–3:20 SESSION 1, Room J: Nonlinear Waves – Part I/II Chair: Jerry Bona 1:40–2:05 Guillaume Fenger: Strong error order of time-discretization of the stochastic gBBM equation 2:05–2:30 Min Chen: Mathematical analysis of Bump to Bucket problem 2:30–2:55 Olivier Goubet: Wave equations with infinite memory 2:55–3:20 Bongsuk Kwon: Small Debye length limit for Euler-Poisson system

1:40–3:20 SESSION 12, Room V/W: Dispersive shocks, semiclassical limits and applications – Part III/III Chair: Gino Biondini 1:40–2:05 Mark Hoefer: Evolution of broad initial profiles—solitary wave fission and solitary wave phase shift 2:05–2:30 Antonio Moro: Dispersive shocks dynamics of phase diagrams 2:30–2:55 Jeffrey Oregero: Semiclassical Lax spectrum of Zakharov-Shabat systems with periodic potentials 2:55–3:20 Bingying Lu: The universality of the semi-classical sine-Gordon equation at the gradient catastrophe

1:40–3:20 SESSION 26, Room K: Physical Applied Mathematics – Part I/II Chairs: Ziad Musslimani, Matthew Russo 1:40–2:05 Nick Moore: Anomalous waves induced by abrupt changes in topography 2:05–2:30 Adam Binswanger: Oblique dispersive shock waves in steady shallow water flows 2:30–2:55 Justin Cole: Solitons and Psuedo-solitons in the Korteweg-de-Vries equation with step-up boundary conditions 2:55–3:20 Sathyanarayanan Chandramouli: Spectral Renormalization algorithm applied to solving initial-boundary value problems

1:40–2:05 SESSION 2, Room L: Novel Challenges in Nonlinear Waves and Integrable Systems – Part II/II Chairs: Barbara Prinari, Alyssa K. Ortiz 1:40–2:05 Annalisa Calini: Integrable evolutions of twisted polygons in centro-affine space 2:05–2:30 Brenton LeMesurier: Studying DNA transcription pulses with refinements f oa [discrete] sine-Gordon approximation 2:30–2:55 Deniz Bilman: Extreme superposition: rogue waves of infinite order and the Painlevé-III hierarchy 2:55–3:20 Alyssa K. Ortiz: Soliton solutions of certain reductions of the matrix NLS equation with non-zero boundary conditions

1:40–3:20 SESSION 20, V/W: Dynamical Systems and integrability – Part II/II Chairs: Nalini Joshi and Nobutaka Nakazono 1:40–2:05 Y. Ohta: Two dimensional stationary vorticity distribution and integrable system 2:05–2:30 Claire Gilson: Quasi-Pfaffians and noncommutative integrable systems 2:30–2:55 Masato Shinjo and Koichi Kondo: A discrete analogue of the Toda hierarchy and its some properties 2:55–3:20 Giorgio Gubbiotti: On the inverse problem of the discrete calculus of variations

3:20–3:50 COFFEE BREAK

3:50–5:55 SESSION 7, Masters Hall: Stability and traveling waves – Part VI/IX Chairs: Bernard Deconinck, Anna Ghazaryan, Mat Johnson, Stephane Lafortune, Yuri Latushkin, Jeremy Upsal, Samuel Walsh 3:50–4:15 Mat Johnson: Modulational dynamics of spectrally stable Lugiato-Lefever periodic waves 4:15–4:40 Chongchun Zeng: Steady concentrated vorticity and its stability of the 2-dim Euler equation on bounded domains 4:40–5:05 Dmitry Pelinovsky: Double-periodic waves of the focusing NLS equation and rogue waves on the periodic background 5:05–5:30 Keith Promislow: Bulk verses Surface Diffusion in Highly AmphiphilicPolymer Networks 5:30–5:55 Doug Wright: Generalized solitary wave solutions of the capillary-gravity Whitham equation

3:50–5:55 SESSION 5, Room F/G: Evolution equations and integrable systems – Part V/V Chairs: Alex Himonas, Curtis Holliman & Dionyssis Mantzavinos 3:50–4:15 Dionyssios Mantzavinos: Analysis of nonlinear evolution equations in domains with a boundary 4:15–4:40 Feride Tiglay: Non-uniform dependence of the data-to-solution map for the Hunter--Saxton equation in Besov spaces 4:40–5:05 Rafael Barostichi: The Cauchy problem for the "good" Boussinesq equation with analytic and Gevrey initial data 5:05–5:30 Renata Figueira: Gevrey regularity in time variable for solutions to the "good" Boussinesq equation. 5:30–5:55 Alex Himonas: The Cauchy problems for evolution equations with analytic data 3:50–5:55 SESSION 8, Room Y/Z: Dispersive wave equations and their soliton interactions: Theory and applications – Part III/V Chairs: Avraham Soffer, Gang Zhao,S. Gustafson 3:50–4:15 Stefanos Aretakis: Conservation laws and asymptotics for the wave equation 4:15–4:40 Jonas Luhrmann: Local smoothing estimates for Schrodinger equations on hyperbolic space and applications 4:40–5:05 Hao Jia: Quantization of energy of blow up for wave maps 5:05–5:30 Baoping Liu: Long time dynamics for nonlinear dispersive equations 5:30–5:55 Qingquan Deng: Soliton Potential Interaction of NLS in R^{3}

3:50–5:55 SESSION 18, Room E: Negative flows, peakons, integrable systems, and their applications – Part IV/IV Chairs: Stephen Anco, Stephane Lafortune 3:50–4:15 Daniel Kraus: Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation 4:15–4:40 Wenhao Liu: Some new exact solutions for the extended (3+1)-dimensional Jimbo-Miwa equation 4:40–5:05 Vesselin Vatchev: Some Properties of Wronskian Solutions of Nonlinear Differential Equations

3:50–5:55 SESSION 1, Room J: Nonlinear waves – Part II/II Chairs: Min Chen 3:50–4:15 Douglas Svensson Seth: Three-dimensional steady water waves with vorticity 4:15–4:40 Shenghao Li: Lower regularity solutions of non-homogeneous boundary value problems of the sixth order Boussinesq equation in a quarter plane 4:40–5:05 Hongqiu Chen: Well-posedness for a higher-order, nonlinear, dispersive equation: new approach 5:05–5:30 Shu-Ming Sun: Solitary-wave solutions for some BBM-type of equations with inhomogeneous nonlinearity

3:50–5:55 SESSION 22, Room V/W: Modern methods for dispersive wave equations – Part II/II Chairs: Robert Buckingham and Peter Miller 3:50–4:15 Rowan Killip: KdV is well-posed in H^{-1} 4:15–4:40 Jiaqi Liu: Long time asymptotics of the defocussing Manakov system in weighted Sobolev space 4:40–5:05 Donatius DeMarco: Asymptotics of rational solutions of the defocusing nonlinear Schrodinger equation 5:05–5:30 Bob Jenkins: Semiclassical soliton ensembles and the three-wave resonant interaction (TWRI) equations

3:50–5:55 SESSION 26, Room K: Physical applied mathematics – Part II/II Chairs: Ziad Musslimani, Matthew Russo 3:50–4:15 Abdullah Aurko: Time-dependent spectral renormalization method applied to conservative PDEs 4:15–4:40 Constance Schober: Linear instability of the Peregrine breather: Numerical and analytical investigation 4:40–5:05 Ryan Roopnarain: Various dynamical regimes, and transitions from homogeneous to inhomogeneous steady states in oscillators with delays and diverse couplings 5:05–5:30 Michail Todorov and Vladimir Gerdjikov: On N-soliton interactions: Effects of local and non-local potentials

3:50–4:15 SESSION 4, Room L: Fractional Diferential Equations – Part II/III Chair: Andrei Ludu 3:50–4:15 Gavriil Shchedrin, Nathanael Smith, Anastasia Gladkina and Lincoln Carr: Generalized Euler's integral transform 4:15–4:40 Aghalaya Vatsala: One dimensional sub-hyperbolic equation via sequential Caputo fractional derivative 4:40–5:05 Christina Nevshehir: The gravity of light travel: riding the fractional wave of a visible universe from h to c-squared 5:05–5:30 Haret Rosu and Stefan Mancas: The factorization method for fractional quantum oscillators 5:30–5:55 Timothy Burns and Bert Rust: Closed-form projection method for regularizing a function defined by a discrete set of noisy data and for estimating its derivative and fractional derivative

3:50–5:55 SESSION 23, Room D: Nonlinear waves in optics, fluids and plasma – Part I/II Chairs: Sergey Dyachenko, Katelyn Leisman, Denis Silantyev 3:50–4:15 Jeffrey Banks & Andre Gianesini Odu: High-order accurate conservative finite differencesor f Vlasov equations in 2D+2V 4:15–4:40 Pavel M Lushnikov, Vladimir E Zakharov and Nikolay M. Zubarev: Non-canonical Hamiltonian structure and integrability for 2D fluid surface dynamics 4:40–5:05 Jolene Britton and Yulong Xing: Well-balanced discontinuous Galerkin methods for blood flow simulation with moving equilibrium 5:05–5:30 Yulong Xing: Invariant conserving local discontinuous Galerkin methods for the modified Camassa-Holm equation

5:00–7:00 POSTERS: Pecan Tree Galleria Lucas Schauer and Geng Chen: Shock formation in finite time for the 1-d compressible Euler equations Taylor Paskett and Blake Barker: Stability of traveling waves in compressible Navier-Stokes Alexei Cheviakov and Caylin Lee: Nonlinear wave equations of shear radial wave propagation in fiber-reinforced cylindrically symmetric media Ryan Marizza, Jessica Harris, Michelle Maiden and Mark Hoefer: Theory and observation of interacting linear waves and nonlinear mean flows in a viscous fluid conduit

7:00- 9:00 BANQUET Speaker: TBA Thiab Taha: Presentation of best Student Paper Awards

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FRIDAY, APRIL 19, 2019

7:30–9:30 REGISTRATION 8:00–9:00 KEYNOTE LECTURE 3, Masters Hall Stefano Trillo: Nonlinear PDEs describing real experiments: recurrences, solitons, and shock waves Chair: Gino Biondini

9:10–10:00 SESSION 7, Masters Hall: Stability and traveling waves – Part VII/IX Chairs: Bernard Deconinck, Anna Ghazaryan, Mat Johnson, Stephane Lafortune, Yuri Latushkin, Jeremy Upsal, Samuel Walsh 9:10–9:35 Dag Nilsson: Solitary wave solutions of a Whitham-Bousinessq system 9:35–10:00 Ola Maehlen: Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities

9:10–10:00 SESSION 13, Room F/G: Recent advances in numerical methods of PDEs and applications in life science, material science – Part I/II Chairs: Qi Wang and Xueping Zhao 9:10–9:35 Thomas Lewis: Approximating nonlinear reaction-diffusion problemswith multiple solutions 9:35–10:00 Shuang Liu and Xinfeng Liu: Efficient and stable numerical methods for a class stiffof reaction-diffusion systems with free boundaries

9:10–10:00 SESSION 14, Room K: Nonlinear kinetic self-organized plasma dynamics driven by coherent, intense electromagnetic fields – Part I/II Chairs: Bedros Afeyan, Shadwick Brad,Wilkening Jon 9:10–9:35 Jon Wilkening, Bedros Afeyan and Rockford Sison: Spectrally accurate methods for kinetic electron plasma wave dynamics 9:35–10:00 Bedros Afeyan and Richard Sydora: Improving the performance of plasma kinetic simulations by iteratively learned phase space tiling: Variational constrained optimization meet machine learning 9:10–10:00 SESSION 25, Room V/W: Nonlinear waves, singularities, vortices, and turbulence in hydrodynamics, physical, and biological systems – Part VI/VII Chairs: Alexander O. Korotkevich and Pavel Lushnikov 9:10–9:35 Avadh Saxena and Avinash Khare: Family of potentials with power-law kink tails 9:35–10:00 Taras Lakoba and Jeffrey Jewell: Higher-order Runge--Kutta-type schemes based on the method of characteristics for hyperbolic equations with crossing characteristics

10:00 - 10:30 COFFEE BREAK 10:30–12:10 SESSION 7, Masters Hall: Stability and traveling waves – Part VIII/IX Chairs: Bernard Deconinck, Anna Ghazaryan, Mat Johnson, Stephane Lafortune, Yuri Latushkin, Jeremy Upsal, Samuel Walsh 10:30–10:55 Miles Wheeler: Coriolis forces and particle trajectories for waves with stratification and vorticity 10:55–11:20 Kristoffer Varholm: On the stability of solitary water waves with a point vortex 11:20–11:45 David Ambrose: Periodic traveling hydroelastic waves 11:45–12:10 Robin Ming Chen: Asymptotic stability of the Novikov peakons

10:30–12:10 SESSION 13, Room F/G: Recent advances in numerical methods of pdes and applications in life science, material science – Part II/II Chairs: Qi Wang and Xueping Zhao 10:30–10:55 Yi Sun and Qi Wang: A hybrid model for simulating sprouting angiogenesis in biofabrication. 10:55–11:20 Paula Vasquez and Erik Palmer: A parallel approach to kinetic viscoelastic modelling 11:20–11:45 Xiaofeng Yang: Efficient schemes with unconditionally energy stabilities for anisotropic phase field models: S-IEQ and S-SAV 11:45–12:10 Qi Wang and Xueping Zhao: A second order fully-discrete linear energy stable scheme for a binary compressible viscous fluid model

10:30–12:10 SESSION 8, Room Y/Z: Dispersive wave equations and their soliton interactions: Theory and applications – Part IV/V Chairs: Avraham Soffer, Gang Zhao, S. Gustafson 10:30–10:55 Leonid Chaichenets: Dirk Hundertmark, Peer Kunstmann and Nikolaos Pattakos: Knocking out teeth in one-dimensional periodic NLS: Local and Global wellposedness results 10:55–11:20 Nikolai Leopold and Soeren Petrat: Derivation of the Schroedinger-Klein-Gordon equations 11:20–11:45 Yifei Wu: Global well-posedness for mass-subcritical NLS in critical Sobolev space

10:30–12:10 SESSION 4, Room K: Fractional diferential equations – Part III/III Chairs: Dumitru Baleanu 10:30–10:55 Harihar Khanal: Variable Order Differential Equations, Solutions and Applications 10:55–11:20 Thiab Taha: IST Numerical Schemes for Solving Nonlinear Evolution Equations and their possible applications for solving time Fractional Differential Equations

10:30–12:10 SESSION 25, Room V/W: Nonlinear waves, singularities, vortices, and turbulence in hydrodynamics, physical, and biological systems – Part VII/VII Chairs: Alexander O. Korotkevich and Pavel Lushnikov 10:30–10:55 Stephen Gustafson: Chiral magnetic skyrmions for 2D Landau-Lifshitz equations 10:55–11:20 Benno Rumpf: Clebsch variables for stratified compressible fluids

12:10–1:40 LUNCH (attendees on their own)

1:40–3:20 SESSION 7, Masters Hall: Stability and traveling waves – Part IX/IX Chairs: Bernard Deconinck, Anna Ghazaryan, Mat Johnson, Stephane Lafortune, Yuri Latushkin, Jeremy Upsal, Samuel Walsh 1:40–2:05 Wesley Perkins: Modulational instability of viscous fluid conduit periodic waves 2:05–2:30 Hung Le: On the existence and instability of solitary water waves with a finite dipole 2:30–2:55 Peter Howard: Renormalized oscillation theory for linear Hamiltonian systems via the Maslov index 2:55–3:20 Jiayin Jin: Invariant manifolds of traveling waves of the 3D Gross-Pitaevskii equation in the energy space

1:40–3:20 SESSION 23, Room F/G: Nonlinear waves in optics, fluids and plasma – Part II/II Chairs: Sergey Dyachenko, Katelyn Leisman, Denis Silantyev 1:40–2:05 Mimi Dai: Non-uniqueness of Leray-Hopf weak solutions for 3D Hall-MHD system 2:05–2:30 Ezio Iacocca: A hydrodynamic formulation for solid-state ferromagnetism 2:30–2:55 Alexander Korotkevich: Inverse cascade of gravity waves in the presence of condensate: numerical results and analytical explanation 2:55–3:20 Alexey Cheskidov and Xiaoyutao Luo: Weak solutions for the 3D Navier-Stokes equations with discontinuous energy

1:40–3:20 SESSION 8, Room Y/Z: Dispersive Wave Equations and their Soliton Interactions: Theory and Applications – Part V/V Chairs: Avraham Soffer, Gang Zhao, . SGustafson 1:40–2:05 Scott Strong and Lincoln Carr: Nonlinear waves on vortex filaments in quantum liquids: A geometric perspective 2:05–2:30 Svetlana Roudenko, Anudeep Kumar Arora and Kai Yang: Stable blow-up dynamics in the generalized L2-critical Hartree equation 2:30–2:55 M. Burak Erdogan, William R. Green and Ebru Toprak: The effect of threshold energy obstructions on the 1L to L-infinity dispersive estimates for some Schrodinger type equations 2:55–3:20 Yanqiu Guo and Edriss Titi: Backward behavior of a dissipative KdV equation

1:40–3:20 SESSION 14, Room K: Nonlinear kinetic self-organized plasma dynamics driven by coherent, intense electromagnetic fields – Part II/II Chairs: Bedros Afeyan,Shadwick Brad,Wilkening Jon 1:40–2:05 B. A. Shadwick, Alexander Stamm and Bedros Afeyan: Nonlinear instabilities due to drifting species and magnetic fields in high energy density plasmas 2:05–2:30 Richard Sydora, Bedros Afeyan and Bradley A. Shadwick: Impact of cyclotron harmonic wave instabilities on stability of self-organized nonlinear kinetic plasma structures 2:30–2:55 Frank Lee, Michael Allshouse, Harry Swinney and Philip Morrison: Internal wave energy flux from density perturbations

3:20–3:50 COFFEE BREAK KEYNOTE ABSTRACTS America), J. Holmes (Ohio State), H. Kalisch and S. Selberg (Uni- versity of Bergen, Norway), C. Kenig (University of Chicago), G. Vortex sheets, Boussinesq equations, and other Misiołek and F. Yan (University of Notre Dame), D. Mantzavinos problems in the Wiener algebra (University of Kansas), G. Ponce (University of California, Santa David M. Ambrose Barbara), R. Thompson (University of North Georgia), F. Tiglay (Ohio State). Drexel University, Department of Mathematics Philadelphia, PA 19104 USA [email protected] Nonlinear PDEs describing real experiments: There are several approaches to proving the ill-posedness of vor- recurrences, solitons, and shock waves tex sheets; we will explore the version due to Duchon and Robert. Stefano Trillo Interestingly, the Duchon and Robert result is really about global Department of Engineering, University of Ferrara, Via Saragat 1, 44122 existence of small solutions. The functional setting is a space-time Ferrara, Italy version of the Wiener algebra with exponential weights, and func- [email protected] tions in this space, at any time after the initial time, are spatially analytic. This existence theorem becomes an ill-posedness result The Fermi-Pasta-Ulam-Tsingou (FPUT) paradox discovered in the upon reversing time, finding small analytic solutions which lose 50s, i.e., the fact that a nonlinear system with many or even in- analyticity arbitrarily quickly. Both aspects of this proof – exis- finite degrees of freedom might exhibit near or exact recurrences tence of solutions and ill-posedness – are of interest for other prob- to the initial condition instead of a transition to equipartition of lems, and we will describe several applications. These applications energy between the modes, is still the driving force of many re- may include nonlinear ill-posedness of linearly ill-posed Boussi- search avenues in nonlinear physics. Historically, integrability of nesq equations, some small global solutions of the 2D Kuramoto- the underlying models, their soliton solutions, and the generating Sivashinsky equation, small global solutions for a problem in epi- mechanisms such as shock formation have commonly believed to taxial growth, and existence of solutions for non-separable mean play a key role. field games. This includes joint work with Jerry Bona, Anna Maz- In this paper, we will review the recent achievements obtained in zucato, and Timur Milgrom. understanding FPUT recurrence phenomena with special emphasis on the theoretical results that explain real experimental observations. Two main mechanisms will be discussed. The first entails Initial and boundary value problems for evolution the fission of solitons from periodic initial data [1] akin to the fa- equations mous numerical experiment performed by Zabusky and Kruskal Alex Himonas (1965) for the KdV equation. The case of the KdV and the defocusing NLS equations will be contrasted to illustrate similarities Department of Mathematics, University of Notre Dame and differences. Notre Dame, IN 46556 [email protected] A second scenario involves modulational instability in the focusing NLS equation where recurrences are mediated by the inter- In the first part of the talk we shall discuss questions of existence, action with a background according to a complicated homoclinic uniqueness, dependence on initial data, and regularity of solutions structure where breather play a key role [2]. Latest developments to the initial value problem of Camassa-Holm and related equa- of such scenario and related open problems will be discussed also tions in a variety of function spaces. Some of these equations can in connection to parametric resonance governed by non-integrable be thought as “toy” models for the Euler equations governing the models [3]. motion of an incompressible fluid, and the analytic techniques developed for these equations have been in some cases transferable 1. S. Trillo, G. Deng, G. Biondini, M. Klein, G. Clauss, A. Chabchoub, to the Euler equations. In the second part of the talk we shall fo- and M. Onorato, Experimental observation and theoretical description cus on the advancement of the Unified Transform Method of Fokas of multisoliton fission in shallow water, Phys. Rev. Lett., 117 (2016), for solving the initial-boundary value problem (ibvp) of nonlinear 144102. evolution equations in one and two space dimensions. Although 2. A. Mussot, C. Naveau, M. Conforti, A. Kudlinski, F. Copie, P. Szrift- introduced as the ibvp analogue of the renowned Inverse Scatter- giser, and S. Trillo, Fibre multiwave-mixing combs reveal the broken ing Transform method for integrable nonlinear evolution equations, symmetry of Fermi-Pasta-Ulam recurrence, Nat. Photonics, 10 (2018), Fokas’ approach can also be used to produce novel solution formu- 303–308. las for the linear versions of such equations. Replacing in Fokas’ 3. A. Mussot, M. Conforti, S. Trillo, F. Copie, and A. Kudlinski, Modu- solution formulas the forcing with the nonlinearity provides a new lation instability in dispersion oscillating fibers, Adv. Opt. Photon., 10 framework for the analysis of nonlinear equations with a variety of (2018), 1–42. boundary conditions in appropriate solution spaces. The talk is based on work in collaboration with R. Barostichi, R.O. Figueira and G. Petronilho (Federal University of Sao˜ Car- los, Brazil) A. Fokas (University of Cambridge, UK), J. Gorsky (University od San Diego), C. Holliman (Catholic University of

1 SESSION ABSTRACTS many current research topics in this area are multi-dimensional systems, boundary value problems, discretization issues, connec- SESSION 1 tions with algebraic and differential geometry, number theory and Nonlinear waves different areas of mathematics, etc. This session aims at bringing together leading researchers in the Jerry Bona ﬁelds of nonlinear wave equations and integrable systems, and at Department of Mathematics, Statistics, and Computer Science offering a broad overview of some of the current research activities University of Illinois at Chicago, Chicago, IL 60607 at the frontier of pure and applied mathematics. [email protected] SESSION 3 Min Chen Department of Mathematics, Purdue University, West Lafayette, IN 47907 Recent developments in mathematical studies of wa- [email protected] ter waves Shu-Ming Sun John D. Carter Department of Mathematics, Virginia Tech, Blacksburg, VA 24061 Mathematics Department, Seattle University [email protected] [email protected] Bing-yu Zhang This minisymposium brings together mathematicians, engineers, Department of Mathematical Sciences, University of Cincinnati and oceanographers. With a focus on nonlinear water waves, the Cincinnati, OH 45221 speakers in this session will present experimental, analytical, and [email protected] numerical results from mathematical models of waves on shallow and/or deep water. This session will focus on the propagation of waves in water and other media where nonlinearity, dispersion and sometimes dissipa- SESSION 4 tion and capillarity are all acting. Featured in the session will be theoretical work, such as existence of solitary wave solutions and Fractional differential equations existence of two dimensional standing waves, well-posedness of Andrei Ludu dispersion-managed nonlinear systems, higher order model equa- Embry-Riddle Aeronautical University tions and equations with dispersive terms, and numerical investi- Dept. Mathematics & Wave Lab, Daytona Beach, FL, USA gation on the solutions of various systems and equations. [email protected]

SESSION 2 Changpin Li Shanghai University, Department of Mathematics, Shanghai, China Novel challenges in nonlinear waves and integrable [email protected] systems Thiab Taha 1,2, 2 Barbara Prinari ∗ and Alyssa K. Ortiz University of Georgia, Computer Science Department, Athens, GA, USA 1 Department of Mathematics, University at Buffalo [email protected] 2 Department of Mathematics, University of Colorado Colorado Springs [email protected], [email protected] Fractional Calculus is a field of pure and applied mathematics that deals with derivatives and integrals of arbitrary orders and their ap- The study of physical phenomena by means of mathematical mod- plications in science, engineering, mathematics, and other fields. els leads in many cases to nonlinear wave equations. A special In recent years considerable interest in fractional calculus has been class of nonlinear wave equations is represented by the so-called stimulated by the applications that this calculus finds in numerical soliton equations, which are infinite-dimensional completely inte- analysis and different areas of physics and engineering, possibly grable Hamiltonian systems that admit an infinite number of con- including fractal phenomena. In this Special Session the talks will served quantities, and whose initial-value problem can be linearized cover areas from pure mathematical fractional calculus and theo- via a method called the inverse scattering transform. At the same rems for existence, uniqueness and stability of fractional differen- time, in realistic settings most nonlinear wave equations of phys- tial equations, to time-dependent fractional order differential equa- ical interest are non-integrable, and integrability can only offer a tions, and to several various fields of science including data sci- partial picture. Understanding integrable as well as non-integrable ence, viscoelasticity, rheology, electrical engineering, biophysics, nonlinear wave equations and their solutions and investigating their signal and image processing, quantum physics, and control theory. remarkably rich mathematical structure requires a variety of techniques from different branches of mathematics, such as exact meth- SESSION 5 ods, approximations, asymptotics and perturbation techniques, symmetry analysis, numerical simulations, etc. Evolution equations and integrable systems Over the past fifty years, a large body of knowledge has been ac- Alex A. Himonas cumulated on nonlinear waves and integrable systems, which con- Department of Mathematics, University of Notre Dame, Notre Dame, IN tinue to be extensively studied worldwide and to offer interesting 46556 research problems and new venues for applications. Among the [email protected]

2 SESSION 7 Curtis Holliman Stability and traveling waves Department of Mathematics, The Catholic University of America Aquinas Hall 116, Washington, DC 20064 Bernard Deconinck [email protected] Department of Applied Mathematics, University of Washington, Seattle, Dionyssis Mantzavinos WA 98195 [email protected] Department of Mathematics, University of Kansas Lawrence, KS 66045 Anna Ghazaryan [email protected] Department of Mathematics, Miami University, Oxford, OH 45056 [email protected] Linear and nonlinear evolution equations have been at the fore- front of advances in partial differential equations for a long time. Mat Johnson They are involved in beautiful, yet extremely challenging prob- Department of Mathematics, University of Kansas, Lawrence, KS 66045 lems, with a strong physical background, for which progress is [email protected] achieved through a mixture of techniques lying at the interface be- Stephane´ Lafortune tween analysis and integrable systems. Topics studied for these equations include, among others, traveling waves, initial-boundary Department of Mathematics, College of Charleston, Charleston, SC 29401 value problems, local and global well-posedness, inverse scatter- [email protected] ing, stability, and integrability. Yuri Latushkin Mathematics Department, University of Missouri, Columbia, MO 65211 SESSION 6 [email protected] Random matrices, Painleve´ equations, and Jeremy Upsal integrable systems Department of Applied Mathematics, University of Washington, Seattle, Vladimir Dragovic´ WA 98195 Department of Mathematical Sciences, [email protected] The University of Texas at Dallas, Richardson, TX 75080, USA Samuel Walsh [email protected] Mathematics Department, University of Missouri, Columbia, MO 65211 Anton Dzhamay and Virgil U. Pierce [email protected] School of Mathematical Sciences, This session will bring together researchers who study fronts, pulses, University of Northern Colorado, Greeley, CO 80639, USA wave trains and patterns of more complex structure which are re- [email protected], [email protected] alized as special solutions of nonlinear partial differential equa- The theory of Integrable Systems is well-known for employing tions. Existence, stability, dynamic properties, and bifurcations of tools from many different branches of mathematics and mathemat- those solutions will be discussed, from both analytical and numerical physics to perform qualitative and quantitative analysis of a ical point of views. wide range of important natural phenomena. In this special session we plan to primarily focus on the theory of discrete and con- SESSION 8 tinuous Painleve´ equations, and the interaction between Random Dispersive wave equations and their soliton interac- Matrices and Integrable Systems. Painleve´ equations are a special tions: Theory and applications class of nonlinear ordinary differential equations whose solutions satisfy the Painleve´ property that their only movable singularities Avraham Soffer, Gang Zhao, S. Gustafson are poles. Thus, solutions of Painleve´ equations have good an- In this session we focus on modern results and approaches to the alytic properties and form a class of genuinely nonlinear special large time complex dynamics of dispersive equations. In particular, functions known as the Painleve´ transcendents. The importance large time global existence and scattering problems will be consid- of these special functions has been steadily growing and we now ered, for rough initial data, nonlinear dynamics in the presence of know that many interesting models can be described in terms of noise, dispersive estimates in the presence of threshold singular- such Painleve´ transcendents. In recent years there had been many ities, soliton-potential collision dynamics and more. The mathe- interesting developments in the theory of discrete Painleve´ equa- matical tools presented come from analytic, computational and nu- tions, whose theory is founded on deep ideas from the algebraic merical approaches. On the more technical level, the emphasis will geometry. A large class of such examples occurs in the theory of be on dynamics which at large times is not asymptotically stable: Random Matrices as well as discrete dynamical systems of Ran- the solutions wander far away from the initial coherent structures, dom Matrix type, such as determinantal point processes. Among be it solitons or other objects. powerful tools for studying such problems, as well as for studying the asymptotics of Painleve´ transcendents, is the Riemann-Hilbert Problem approach. Talks in our session will give a broad overview of this research area, highlight important recent developments, and outline possible new research directions.

3 SESSION 9 practical point of view. On theoretical side, the study of DSW can involve a variety of techniques such as Whitham modulation the- Nonlinear evolutionary equations: Theory, numer- ory, the inverse scattering transform, oscillatory Riemann-Hilbert ics and experiments problems, the algebro-geometric approach, and the Deift-Zhou non- Efstathios G. Charalampidis and Foteini Tsitoura linear steepest descent method. On the practical side, DSWs have Department of Mathematics and Statistics, University of Massachusetts been experimentally observed in many physical contexts such wa- Amherst, MA 01003-9305 ter waves, nonlinear optics, plasmas and Bose-Einstein conden- [email protected] and [email protected] sates among others. This session aims at presenting several recent results on the subject. This session will focus on the study of nonlinear waves in a broad array of fields. It will bring together caliber experts studying the- SESSION 13 oretically, numerically as well as experimentally nonlinear waves in novel physical settings. Furthermore, this session will highlight Recent advances in numerical methods of PDEs and some of the newest findings in these settings as well as introduce applications in life science, material science novel theoretical and computational techniques associated with the Qi Wang underlying equations. Among the topics to be discussed in the ses- 1523 Greene Street, Office 313C, sion include, bifurcation analysis, stability problems motivated by Columbia, South Carolina, 29208 biological settings, discontinuous shock wave solutions, flows in [email protected] Boussinesq models, and experimentally observed patterns on deep water, among others. Xueping Zhao 1523 Greene Street, Office 313A, SESSION 10 Columbia, South Carolina, 29208 [email protected] Recent advances in PDEs from fluid dynamics and other dynamical models This session will focus on the recent advances in numerical approaches of PDEs and their applications in life science and material Robin Ming Chen science. Department of Mathematics Partial differential equation is a powerful tool to investigate the University of Pittsburgh, underlying mechanisms of various phenomena in nature. Due to USA the lack of analytical solutions, accurate and efficient numerical [email protected] methods of PDEs and their applications are required to promote Runzhang Xu the development of various fields. This session includes multiple College of Science numerical methods developed very recently and their applications, Harbin Engineering University, such as group behavior, dendritic solidification. The following is a Harbin, P R China list of titles of the talks and information of the authors. [email protected] SESSION 14 This session will mainly focus on a number of recent developments in the very active areas of fluid mechanics, integrable systems, clas- Nonlinear kinetic self-organized plasma dynamics sical physical models and other dynamical systems. The topics in- driven by coherent, intense electromagnetic fields clude Hamiltonian structures; derivation of physical model equa- Bedros Afeyan and Jon Wilkening tions; well-posedness; formation of singularities; stability analy- Polymath Research Inc. sis and geometric aspects, etc.. Directions of work related to new University of California, Berkeley methods and their applications to nonlinear PDEs will be empha- [email protected] and [email protected] sized, with the aim of bringing together a number of researchers at all career stages working in on these topics. Brad Shadwick∗ University of Nebraska, Lincoln SESSION 12 [email protected] Dispersive shocks, semiclassical limits and applica- Session 14 is dedicated to the study of nonlinear kinetic plasma tions wave structures in phase space and analogues in fluids. The presentations focus strogly on numerical methods that either extend Gino Biondini current state of the art towards spectral accuracy, or adaptive refine- State University of New York at Buffalo, Department of Mathematics ment tied to machine learning, or variational techniques or tradi- [email protected] tional particle-in-cell codes but in new regimes. Emphasis is given to strong, coherent (laser) field interactions with high energy den- The combination of nonlinearity and weak dispersion can often re- sity plasmas. Here, both one dimensional and higher dimensional sult in the formation of dispersive shock waves (DSW). The study models are treated in both the electrostatic limit and for fully elec- of the formation and interaction of dispersive shocks has received tromagnetic cases. Self-consistent and externally imposed mag- renewed interest in recent years, both from a theoretical and from a netic fields play an important role in most of the presentations.

4 Two fluid analogues are also included so as to show the breadth equation and the short pulse equation. In this session, different as- and potential impact of these works on nonlinear wave science in pects regarding the existence and stability of such peaked or cusped general. solutions will be discussed using analytical as well as numerical methods. SESSION 15 SESSION 17 Waves in topological materials Nonlinear dynamics of mathematical models in Hailong Guo neuroscience School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia Pamela B. Pyzza [email protected] Ohio Wesleyan University 61 S. Sandusky Street Xu Yang Delaware, OH 43015 Department of Mathematics, University of California, Santa Barbara, CA, [email protected] 93106, USA [email protected] Nonlinear dynamics appear often in neuroscience and thus mathematical modeling lends itself as an effective approach to investi- Yi Zhu gating neuronal phenomena. This session will feature recent con- Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Bei- tributions of mathematics to neuroscience and neuronal networks, jing, 100084, China including innovations in modeling and in the analysis of models. [email protected] The speakers in this session will present work applying compu- In recent several years, there have been intense efforts toward ex- tational, analytical, and experimental tools to address a variety of ploiting the topological protected wave propagation–immune from problems in mathematical neuroscience. scattering by defects and disorder. These novel and subtle wave patterns are investigated in different physical systems, which in- SESSION 18 clude, but not limited to, matter waves in quantum systems, acous- Negative flows, peakons, integrable systems and their tic waves in nano-systems, electromagnetic waves in photonic sys- applications tems. A vast of new experiments and theories come out to describe the wave phenomena in topological materials. The goal of this spe- Stephen Anco cial session is to bring together theoretical and applied researchers Brock University in these areas to discuss some recent advances in the mathematical St. Catharines, ON, L2S 3A1, Canada theories and physical applications. Topics include, but not limited [email protected] to, the analysis of the underlying governing equations, numerical Zhijun (George) Qiao methods on computing the edge states, experimental realizations. University of Texas Rio Grande Valley SESSION 16 Edinburg, TX 78539 [email protected] Existence and stability of peaked waves in nonlinear ´ evolution equations Stephane Lafortune College of Charleston Anna Geyer Charleston, South Carolina 29424 Address: Delft Institute of Applied Mathematics, [email protected] Delft University of Technology, Nonlinear dispersive wave equations appear in many fields, includ- Mekelweg 4, 2628 CD Delft, The Netherlands ing fluid mechanics, plasma physics, optics, and differential geom- [email protected] etry. There has been much recent work on the study of these equa- Dmitry E. Pelinovsky tions, especially ones that describe negative flows and peakons, yet Department of Mathematics and Statistics, many interesting questions and problems remain to be solved. This McMaster University, session will bring together researchers at all career stages to share Hamilton, Ontario, Canada, L8S 4K1 their recent results on various topics related to integrable systems, [email protected] negative flows, peakons, and nonlinear soliton models. Specific topics will focus on (but not be restricted to) peakons and other This session will focus on peaked waves in nonlinear evolution soliton solutions, negative flows and their integrability structure, equations. reciprocal/Liouville transformations, Hamiltonian structures, con- Several well-known nonlinear dispersive equations arising for in- servation laws relating negative flows and peakon equations, as stance from models for water waves allow for solutions which ex- well as other developments connected with these types of equa- hibit singularities in the derivatives. Examples of such equations tions and their solutions. are the Camassa-Holm equation, the Whitham equation and other members of the fractional KdV equation, the reduced Ostrovsky

5 SESSION 19 system of magnetization vectors, random arrangements of gran- Network dynamics ular material and stochastic gradient methods for training neural networks or Bayesian inferences with big data. This session will Thomas Carty address common challenges in stochastic systems across broadly Department of Mathematics, Bradley University, Peoria, IL, 61625 different applications, for example, high dimensionality, multiple [email protected] time-scales, and nonlinear multiplicative noise. This special section will concentrate on network dynamics. Recent SESSION 22 analytical advances have led to an explosion of the use of dynamical systems on graphs in the modeling of natural phenomena. In Modern methods for dispersive wave equations the last ten years alone, we have seen network models applied to neurochemisty in the study of brain neurons, to social science in Robert J. Buckingham models of group decision making and group participation, to eco- Department of Mathematical Sciences, University of Cincinnati nomics as a model of electrical power distribution on an energy P.O. Box 210025, Cincinnati, OH 45221-0025 grid, and more. One focus of this special session is on Kuramoto- [email protected] type models for finite networks. The Kuramoto variants have been Peter D. Miller indispensable in modeling dynamics where synchrony of oscilla- Departmentof Mathematics, University of Michigan tory behavior arise. Interesting and difficult problems arise as the 530 Church St., Ann Arbor, MI 48109 complexity of the oscillatory behavior of the individual actors in- [email protected] creases. This special session will bring together researchers developing the SESSION 20 latest analytic, asymptotic, and numerical techniques for under- Dynamical systems and integrability standing dispersive wave equations. Nalini Joshi SESSION 23 School of Mathematics and Statistics, The University of Sydney, New South Wales 2006, Australia. Nonlinear waves in optics, fluids and plasma [email protected] Sergey A. Dyachenko and Katelyn Leisman Nobutaka Nakazono and Giorgio Gubbiotti Department of Mathematics, 273 Altgeld Hall, 1409 W. Green Street (MC-382), Department of Physics and Mathematics, Aoyama Gakuin University Urbana, IL 61801 USA Sagamihara, Kanagawa 252-5258, Japan. [email protected], [email protected] School of Mathematics and Statistics, The University of Sydney New South Wales 2006, Australia. Denis Silantyev [email protected] and [email protected] Department of Mathematics, Milena Radnovic and Yang Shi Courant Institute of Mathematical Sciences, 251 Mercer Street, School of Mathematics and Statistics, The University of Sydney New York, NY 10012 USA New South Wales 2006, Australia. [email protected] College of Science and Engineering, Flinders University Adelaide 5042, Australia. We present some of the recent advancements in numerical meth- [email protected] and yang.shi@flinders.edu.au ods, and theoretical results in the field of nonlinear science. We The study of integrability and integrable systems addresses impor- focus mainly on plasma, fluids, and nonlinear optics. We demon- tant questions from mathematics and physics. Many of these ques- strate recent advances in turbulence theory of water waves, in par- tions arise from the study of models involving finite operations, ticular the corrections to Kolmogorov spectra due to interaction and require the analysis of discrete integrable systems in order to with condensate. We present recent advances on motion constants be answered. In this session we are bringing together researchers that give hint of 2D water waves may be integrable system after all. working with various aspects of integrable systems with purpose of intensifying the exchange of experience, methods and ideas. SESSION 24 Mathematical perspectives in quantum mechanics SESSION 21 and quantum chemistry Stochastic dynamics in nonlinear systems Jianfeng Lu Katie Newhall Mathematics Department, Duke University, Durham, NC, USA UNC Chapel Hill [email protected] [email protected] Israel Michael Sigal Stochastic dynamics arise in the modeling of biological and phys- Department of Mathematics, University of Toronto, Toronto, Canada ical systems but also from optimization algorithms. Examples in- [email protected] clude transport within a cell by molecular motors, dynamics of a

6 This session is aimed at a review of the current progress and a dis- SESSION 26 cussion of outstanding issues in quantum mechanics and quantum chemistry. The emphasis is on understanding behaviour of large Physical applied mathematics systems of quantum particles, such as atoms, molecules, solids, Ziad Musslimani and Matthew Russo etc. Department of Mathematics To give an account of complex quantum systems one uses effective Florida State University theories (in which large number of degrees of freedom are swapped Tallahassee, FL 32306, USA for the nonlinearity) and the main problems here are the derivation, [email protected], [email protected] mathematical analysis and application of such theories. Nonlinear waves pervade nature over a wide range of scales and One of the most prominent examples of above is the density func- across many disciplines. In many cases their properties, includ- tional theory (DFT). Despite the intensive use of this theory in ing their evolution, may be exactly or approximately determined physics, chemistry, materials science and biology, with thousands by one or more nonlinear evolution equations. This session will of papers published every month, and considerable progress include recent analytical and numerical work on nonlinear systems achieved, the time-dependent DFT is still in an initial stage of de- such as the classical and PT-symmetric NLS, KdV equation, and velopment. This gap is even more daunting since the theory is a generalized Heisenberg ferromagnet equation, with an emphasis being recently applied for understanding energy transfer and de- on their role in describing physical phenomena. Applications will signing energy storage. include hydrodynamics, nonlinear optics, aerodynamics, and other In this session we concentrate on mathematical underpinning of areas. the key effective theories, their justifications, applications and the computational techniques used. The topics to be discussed will SESSION 28 include the rigorous analysis of the density functional theory, the meso/macroscopic theories, described by the Ginzburg-Landau, Recent advances in analytical and computational Landau-Lifshitz equations, two-dimensional quantum systems and methods for nonlinear partial differential equations their geometrical and topological properties, measurement and de- coherence and the interaction of radiation and matter. Muhammad Usman Department of Mathematics, University of Dayton, Dayton OH 45469- SESSION 25 2316, USA [email protected] Nonlinear waves, singularities, vortices, and turbulence in hydrodynamics, physical, and biological sys- Chaudry Masood Khalique tems Department of Mathematical Sciences, North-West University, Mafikeng Campus, Alexander O. Korotkevich and Pavel M. Lushnikov Private Bag X 2046, Department of Mathematics and Statistics, Mmabatho 2735, South Africa MSC01 1115, 1 University of New Mexico, [email protected] Albuquerque, NM 87131-0001 USA [email protected], [email protected] Nonlinear differential/partial differential equations (NDEs/NPDEs) describe many physical phenomena arising in science and engi- Waves dynamics is one of the most interesting phenomena in ap- neering. Thus, finding their solutions play a vital role in provid- plied mathematics and physics. We encounter waves in all areas of ing information to understand and interpret the structure of such our everyday lives, from ripples on the surface of a cup of coffee physical phenomena. Researchers have developed many analytical and sound waves to the extremely powerful laser pulses propaga- and numerical methods to solve these equations. Recent numeri- tion in controlled fusion and plasma excitations in super novas. cal methods include finite difference methods, collocation meth- In wast majority of interesting cases the problem of wave propa- ods and finite element methods. While testing numerical tech- gation should be solved not only in the linear approximation but niques, when exact solutions of initial and boundary value problem also with nonlinear effects taken into account. Powerful tools of of NPDEs are not available, conservation laws play an important modern applied mathematics and theoretical physics together with role. Well-known analytical tools include Lie symmetry method, rapidly emerging computational power leads to new amazing ad- Backland¨ transformation method, and inverse scattering transfor- vances in the study of waves dynamics in different media. Com- mation method. mon approaches stimulate intensive cross fertilization of ideas in This special session is dedicated to showcase recent progress in the field which accelerates the development of the wave dynamics finding analytical and numerical solutions to nonlinear differen- even further. Our session is devoted to new advances in the theory tial/partial differential equations by various methods and to stimu- of waves and demonstrates vividly the similarity of approaches in late collaborative research activities. a broad spectrum of important applications.

7 PAPER ABSTRACTS Small Debye length limit for the Euler-Poisson system Chang-Yeol Jung and Bongsuk Kwon SESSION 1: “Nonlinear waves” ∗ Department of Mathematical Sciences, UNIST Three-dimensional steady water waves with Ulsan, 44919 Korea vorticity [email protected] and [email protected] Masahiro Suzuki Evgeniy Lokharu, Douglas Svensson Seth∗, Erik Wahlen´ Department of Computer Science and Engineering, Nagoya Institute of Matematikcentrum, Box 118, 221 00 Lund, Sweden Technology [email protected], douglas.svensson [email protected], Nagoya, 466-8555 Japan [email protected] [email protected] We will consider the nonlinear problem of steady gravity-driven We discuss existence, time-asymptotic behavior, and quasi-neutral waves on the free surface of a three-dimensional flow of an incom- limit for the Euler-Poisson equations. Specifically, under the Bohm pressible fluid. In the talk we will discuss a recent progress on criterion, we construct the global-in-time solution near the sta- three-dimensional waves with vorticity, which is a relatively new tionary solution of plasma sheath, and also investigate its time- subject. The rotational nature of the flow is modeled by the as- asymptotic behavior and small Debye length limit. If time permits, sumption on the velocity field, that it is proportional to its curl. some key features of the proof and related problems will be dis- Such vector fields are known in magnetohydrodynamics as Bel- cussed. This is joint work with C.-Y. Jung (UNIST) and M. Suzuki trami fields. We plan to provide a necessary background on the (Nagoya Tech.). topic and prove the existence of three-dimensional doubly periodic waves with vorticity. Wave equations with infinite memory Strong error order of time-discretization of the Filippo Dell’Oro, Vittorino Pata stochastic gBBM equation Politecnico di Milano - Dipartimento di Matematica Via Bonardi 9, 20133 Milano, Italy Guillaume Fenger , Olivier Goubet and Youcef Mammeri ∗ fi[email protected] and [email protected] LAMFA CNRS UMR 7352 Universite´ de Picardie Jules Verne, Olivier Goubet∗, Youcef Mammeri 33, rue Saint-Leu, 80039 Amiens, France. Laboratoire Amienois´ de Mathematique´ Fondamentale et Appliquee´ [email protected], [email protected] and CNRS UMR 7352, Universite´ de Picardie Jules Verne, 80039 Amiens, [email protected] France [email protected] and [email protected] We consider a Crank-Nicolson scheme to approximate analytical solutions to the generalized Benjamin-Bona-Mahony equation We introduce a new mathematical framework for the time discretiza- (gBBM) with white noise dispersion introduced in [2]. This equation of evolution equations with memory. As a model, we focus on tion reads, for p 1, x T the one-dimensional torus an abstract version of the equation ≥ ∈ ∞ p du duxx + ux dW + u uxdt = 0, ∂tu(t) g(s)∆u(t s) ds = 0 − ◦ − Z0 − where (Wt)t 0 is a standard real valued Brownian motion and is ≥ ◦ with Dirichlet boundary conditions, modeling hereditary heat con- the so-called Stratonovich product. duction with Gurtin-Pipkin thermal law. Well-posedness and ex- We choose a functional space in which the problem is well posed ponential stability of the discrete scheme are shown, as well as the and we study the strong error order of the time-discrete approxi- convergence to the solutions of the continuous problem when the mation. Due to the presence of a brownian motion we prove that time-step parameter vanishes. the strong error order of this Crank-Nicolson scheme is 1 instead We consider also the nonlinear integrodifferential Benjamin-Bona- of 2 for the determinist equation. Mahony equation

1. R. Belaouar, A. de Bouard and A. Debussche, Numerical analysis of ∞ ut utxx + ux g(s)uxx(t s)ds + uux = f the nonlinear Schrodinger¨ equation with white noise dispersion, Stoch. − − Z0 − Partial Differ. Equ. Anal. Comput., 3 no.1 (2015), 103-132. where the dissipation is entirely contributed by the memory term. 2. M. Chen, O. Goubet and Y. Mammeri, Generalized regularized long Under a suitable smallness assumption on the external force f , we waves equations with white noise dispersion, Stoch. Partial Differ. Equ. prove the convergence of trajectories to some global attractor. Anal. Comput., 5 no. 3 (2017), 319-342. 3. G. Fenger, O. Goubet, Y. Mammeri, Numerical Analysis of the Crank- 1. F. Dell’Oro, O. Goubet, Y. Mammeri, V. Pata, Global attractor for the Nicolson scheme for the Generalized Benjamin-Bona-Mahony Equa- Benjamin-Bona-Mahony equations with memory, Accepted to Indiana tion with White Noise Dispersion, (upcoming in 2019). University J. of Math. 2. F. Dell’Oro, O. Goubet, Y. Mammeri, V. Pata, A semidiscrete scheme for evolution equations with memory, to appear

8 Mathematical analysis of Bump to Bucket problem 4. S. Li, M. Chen, and B. Zhang. A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane. Dis- Min Chen∗ crete and Continuous Dynamical Systems - Series A, 38(5):2505–2525, Department of Mathematics, Purdue University, 2018. West Lafayette, IN 47907, USA 5. S. Li, M. Chen, and B. Zhang. Wellposedness of the sixth order Boussi- [email protected] nesq equation with non-homogeneous boundary value on a bounded Olivier Goubet domain. Accepted by Physica D: Nonlinear Phenomena LAMFA UMR 7352 CNRS, Universitede Picardie Jules Verne, 80039 Amiens CEDEX 1, France Well-posedness for a higher-order, nonlinear, disper- [email protected] sive equation: new approach Shenghao Li Jerry Bona and Hongqiu Chen Department of Mathematics, Purdue University, West Lafayette, IN 47907, University of Illinois at Chicago and University of Memphis USA [email protected] and [email protected] [email protected] Colette Guillope´∗ In numerical simulations of surface water waves, when there is a University Paris-Est Creteil´ deformation on the bottom, it is a common practice to transform [email protected] form the boundary deformation data to the free surface. In this talk, we investigate this procedure, by comparing the waves generated A class of higher-order models for unidirectional water wave of the by the moving bottom (Bump) and by the initial surface variation form (Bucket), using linear and nonlinear Boussinesq-type models. η + η γ βη + γ βη + δ β2η + δ β2η t x − 1 xxt 2 xxx 1 xxxxt 2 xxxxx Lower regularity solutions of non-homogeneous 3 2 2 7 2 1 2 3 + α(η )x + αβ γ(η )xx ηx α (η )x = 0 (1) boundary value problems of the sixth order 4 − 48 x − 8 Boussinesq equation in a quarter plane was derived by Bona, Carvajal, Panthee and Scialom [1]. With ap- Shenghao Li, and Min Chen propriate choices of the parameters γ1, γ2, δ1, δ2 and γ, this equa- Department of Mathematics, Purdue University, West Lafayette, IN 47907 tion serves as a model for the propagation of small-amplitude, long- [email protected] and [email protected] crested surface waves moving to the direction of increasing values of the spatial variable x. Here α is a typical ratio of wave am- Bingyu Zhang plitude to depth, β is a representative value of the square of the Department of Mathematics, University of Cincinnati, Cincinnati, OH 45221 depth to wavelength and t is proportional to elapsed time. The [email protected] dependent variable η = η(x, t) is a real-valued function of x ( ∞, ∞), t 0 representing the deviation of the free surface∈ We study an initial-boundary-value problem of the sixth-order − ≥ Boussinesq equation on a half line with nonhomogeneous bound- from its undisturbed position at the point corresponding to x at ary conditions: time t. This model subsists on the assumptions that α and β are comparably-sized small quantities while η and its ﬁrst few partial 2 utt uxx + βuxxxx uxxxxxx +(u )xx = 0, x > 0, t > 0, derivatives are of order one. Moreover, γ1 and γ2 are restricted by − − 1  γ1 + γ2 = . u(x, 0)= ϕ(x), ut(x, 0)= ψ′′(x), 6  7  The new result is that when γ = , δ2 > δ1 > 0 and the initial u(0, t)= h (t), uxx(0, t)= h2(t), uxxxx(0, t)= h3(t), 48 1 data  where β = 1. It is shown that the problem is locally well-posed η(x, 0)= η0(x, 0) (2) s R+ ± 1 < 1 in H ( ) for 2 s 0 with initial condition (ϕ, ψ) lies in H and not too big, then the initial-value problem of (1)-(2) s R+ s 1−R+ ≤ ∈ 1 H ( ) H − ( ) and boundary condition (h1, h2, h3) in the is globally well posed and the H -norm of the solution is uniformly × s+1 + s 1 + s 3 + product space H 3 (R ) H −3 (R ) H −3 (R ). bounded for t 0. × × ≥ 1. J. L. Bona, S. M. Sun, and B.-Y. Zhang. A non-homogeneous 1. J. L. Bona, X. Carvajar, M. Panthee and M. Scialom, Higher-order boundary-value problem for the Korteweg-de Vries equation in a Hamiltonian model for unidirectional water waves, Journal of Nonlin- quarter plane. Transactions of the American Mathematical Society, ear Science, 28 (2018), no. 2, 543-577. 354(2):427–490, 2002. 2. E. Compaan and N. Tzirakis. Well-posedness and nonlinear smoothing Solitary-wave solutions for some BBM-type of equa- for the good Boussinesq equation on the half-line. Journal of Differen- tions with inhomogeneous nonlinearity tial Equations, 262(12):5824–5859, 2017. Shu-Ming Sun 3. A. Esfahani and L. G. Farah. Local well-posedness for the sixth-order Boussinesq equation. Journal of Mathematical Analysis and Applica- Department of Mathematics tions, 385(1):230–242, 2012. Virginia Tech Blacksburg, VA 24061 email: [email protected]

9 The talk considers the existence of solitary-wave solutions of some 2. B. Prinari, F. Demontis, S. Li, and T. Horikis, Inverse scattering trans- higher-order Benjamin-Bona-Mahony (BBM) equations, whose lin- form and soliton solutions for square matrix nonlinear Schrodinger¨ ear parts are pseudo-differential operators and nonlinear parts in- equations with non-zero boundary conditions, Physica D, 368 (2017), volve inhomogeneous polynomials of solutions and their deriva- 22-49. tives, which have not been studied before. One of such equations can be derived from water-wave problems as the second-order ap- Real Lax spectrum implies stability proximate equation from fully nonlinear governing equations. Un- der some conditions on the symbols of pseudo-differential oper- Bernard Deconinck and Jeremy Upsal∗ ators and the nonlinear terms, it is shown that the equation has Department of Applied Mathematics, University of Washington solitary-wave solutions. Numerical study of the solitary-wave so- Seattle, WA 98195-2420, USA lutions for some special fifth-order BBM equations will also be [email protected] and [email protected] discussed. (This is a joint work with J. Bona, H. Chen, and J.-M. We consider the dynamical stability of elliptic solutions to inte- Yuan). grable equations that belong to the AKNS hierarchy. The spectrum of the differential operator obtained through linearization is important for determining the stability of solutions. When the spectrum is on the imaginary axis, the solutions are spectrally stable. To de- SESSION 2: “Novel challenges in nonlinear waves and integrable termine this stability spectrum, we use the integrability properties systems” of the underlying equation. The spatial component of the Lax pair for members of the AKNS We consider a slowly decaying oscillatory potential such that the hierarchy naturally defines an eigenvalue problem for the Lax pa- corresponding 1D Schrodinger¨ operator has a positive eigenvalue rameter, ζ. The collection of these eigenvalues is called the Lax embedded into the absolutely continuous spectrum. This potential spectrum, σL. When the eigenvalue problem is self adjoint, σL does not fall into a known class of initial data for which the Cauchy R. If it is not self adjoint, significantly more work is required⊂ to problem for the Korteweg-de Vries (KdV) equation can be solved resolve the Lax spectrum. We define a function whose solutions by the inverse scattering transform. We nevertheless show that the determine the Lax spectrum as well as an explicit construction of KdV equation with our potential does admit a closed form classical the eigenfunctions. Using the eigenfunctions, we present a method solution in terms of Hankel operators. Comparing with rapidly de- for constructing the Floquet discriminant, a commonly used tool caying initial data our solution gains a new term responsible for the for computing the Lax spectrum. positive eigenvalue. To some extend this term resembles a positon For stationary solutions of equations in the AKNS hierarchy, we (singular) solution but remains bounded. Our approach is based use knowledge of the Lax spectrum to determine spectral stability. upon certain limiting arguments and techniques of Hankel opera- In particular, we find that (1) R σL when the problem is not self tors. adjoint, and (2) for self-adjoint⊂ or non self-adjoint problems, real Lax spectra gives rise to imaginary, and hence stable, eigenvalues. Soliton solutions of certain reductions of the matrix nonlinear Schrodinger¨ equation with non-zero Studying DNA transcription pulses with refinements boundary conditions of a [discrete] Sine-Gordon approximation 1, 1,2 Alyssa K. Ortiz ∗ and Barbara Prinari Brenton LeMesurier* and Alex Kasman 1 Department of Mathematics, University of Colorado Colorado Springs College of Charleston, Charleston SC 29424 2 Department of Mathematics, University at Buffalo [email protected]/[email protected] [email protected] and [email protected] Transcription from DNA to RNA involves a traveling opening the We will present soliton solutions for two novel reductions of the double helix, and with a great many approximations and assump- matrix nonlinear Schrodinger¨ equation (MNLS), introduced in [1], tions, Englander et al [1] proposed modeling this by the equations which are integrable and are the analog of the modified Manakov of the pendulum chain model, and thence via continuum limit by system with mixed signs of the nonlinear coefficients, i.e. a nonlin- the Sine-Gordon equation. Indeed, the kink solutions of the latter earity in the norm which is of Minkowski type instead of Euclidean are a fair qualitative approximation of the phenomenon, and these type. kinks are “topological”, so they might be expected to be robust In this presentation we will develop the Inverse Scattering Trans- under deviations from the exactly integrable and continuum form. form (IST) for these novel reductions of MNLS in the case of non- In this work in progress, starting with the work of Kasman [2], zero boundary conditions, using similar methods as those shown we consider the effects of more accurate modeling, developing on in [2]. We will also discuss the resulting one-soliton solutions of ideas of Salerno, Yakushevich, et al [3, 4]. In particular we account such equations under various conditions on the norming constant for (a) the non-uniformity in the masses and sizes of the four nu- matrices. cleobases and in the base pair bond strengths (A-T pairs have two hydrogen bonds; C-T pairs have three), and (b) asymmetry in the 1. B. Prinari and A. Ortiz, Inverse Scattering Transform and Solitons motions of the two nucleobases within each pair. for Square Matrix Nonlinear Schrodinger¨ Equations, Studies in Appl. Questions include whether sustained propagation persists and Math., 141 (2018), 308-352. whether some of the many possible encodings of a given protein

10 are evolutionarily preferred due to easier propagation of this open- [email protected] ing “kink”; some preliminary observations will be made. A more Barbara Prinari basic question is whether the symmetry assumed in the basic model so as to get a single DOF per base pair is stable. In fact it is not, Dept. of Mathematics, University of Buffalo, USA but discrete Sine-Gordon style approximations are seen to return in [email protected] another more stable form! After a quick review of the direct and inverse scattering theory of the focusing Zakharov-Shabat system with symmetric nonvanish- 1. S.W. Englander, N.R. Kallenbach, A.J. Heeger, J.A. Krumhansl, and ing boundary conditions, we derive the exact expressions for its re- S. Litwin. Nature of the open state in long polynucleotide double he- flectionless solutions using Marchenko theory. Since the Marchenko lices: possibility of soliton excitations. Proceedings of the National Academy of Sciences, 77(12):7222–7226, 1980. integral kernel has separated variables, the matrix triplet method consisting of representing the Marchenko integral kernel in the 2. A. Kasman. DNA solitons and codon bias. In Mathematics of DNA form Structure, Function and Interactions. IMA conference, 2007. (x+y)A tH F(x + y, t)= Ce− e B 3. M. Salerno. Discrete model for DNA-promoter dynamics. Phys. Rev. A, 44(8):5292–5297, 1991. is applied to express the multisoliton solutions of the focusing non- 4. L. V. Yakushevich. Nonlinear Physics of DNA. Wiley, 2004. linear Schrodinger¨ equation with symmetric nonvanishing boundary conditions in terms of the matrix (A, B, C). Since these exact Integrable evolutions of twisted polygons in centro- expressions contain matrix exponentials and matrix inverses, com- affine space puter algebra can be used to “unpack” and graph them. Here A has only eigenvalues with positive real part, H is a suitable func- Annalisa Calini* and Gloria Mari-Beffa tion of A, and B and C are size compatible rectangular matrices. College of Charleston/University of Wisconsin-Madison The 2p 2q matrices involved are p q matrices with its entries × × [email protected]/[email protected] belonging to a division ring of 2 2 matrices that is isomorphic with Hamilton’s quaternion algebra,× thus supplying an application Many classical objects in differential geometry are described by of quaternion linear algebra [1] integrable systems: nonlinear partial differential equations (PDE) with infinitely many conserved quantities that are (in some sense) 1. L. Rodman, Topics in Quaternion Linear Algebra (Princeton Univer- solvable. Beginning in the 1980s, studies of curve evolutions that sity Press, 2014). are invariant under the action of a geometric group of transformations have unveiled more connections between geometric curve flows and well-known integrable PDE (among them are the KdV, Extreme superposition: Rogue waves of infinite or- mKdV, sine-Gordon, and NLS equations). More recently, efforts der and the Painleve-III´ hierarchy have been directed towards understanding geometric discretiza- Deniz Bilman and Peter D. Miller tions of surfaces and curves and associated evolutions. Department of Mathematics, University of Michigan This talk focuses on a natural geometric flow for polygons in centro- 530 Church St. Ann Arbor, MI, USA affine geometry derived from discretizations of the Adler-Gel’fand- [email protected] and [email protected] Dikii flows for curves in projective space. Such discretizations, together with a pair of Hamiltonian structures, were introduced in Liming Ling Mar227-Beffa´ and Wang [2]. We prove the compatibility of the South China University of Technology two Hamiltonian structures in arbitrary dimension by lifting them [email protected] to a pair of pre-symplectic forms on the moduli space of centro- affine arc length parametrized polygons. We also describe their We study the fundamental rogue wave solutions of the focusing kernels and discuss implications on the integrability of the polygo- nonlinear Schrodinger¨ equation in the limit of large order. Using nal flows. a recently-proposed Riemann-Hilbert representation of the rogue wave solution of arbitrary order k, we establish the existence of 1. G. Mari Beffa and A. Calini, Integrable evolutions of twisted polygons a limiting profile of the rogue wave in the large-k limit when the in centro-affine Rm, Preprint. solution is viewed in appropriate rescaled variables capturing the 2. G. Mari Beffa and J.P. Wang, Hamiltonian structures and integrable near-field region where the solution has the largest amplitude. The evolutions of twisted gons in cn, Nonlinearity 26 (2013) 2515-2551. limiting profile is a new particular solution of the focusing nonlinear Schrodinger¨ equation in the rescaled variables — the rogue Exact solutions of the focusing nonlinear Schrodinger¨ wave of infinite order — which also satisfies ordinary differential equation with symmetric nonvanishing boundary equations with respect to space and time. The spatial differential conditions equations are identified with certain members of the Painleve-III´ hierarchy. We compute the far-field asymptotic behavior of the Francesco Demontis and Cornelis van der Mee∗ near-field limit solution and compare the asymptotic formulæwith Dip. Matematica e Informatica, Universita` di Cagliari, Italy the exact solution with the help of numerical methods for solving [email protected] and [email protected] Riemann-Hilbert problems. In a certain transitional region for the asymptotics the near field limit function is described by a specific Giovanni Ortenzi globally-defined tritronquee´ solution of the Painleve-II´ equation. Dip. Matematica e Applicazioni, Universita` di Milano Bicocca, Italy

11 These properties lead us to regard the rogue wave of infinite order In this talk, we reconstruct the velocity potential and surface dis- as a new special function. placement from NLS coordinates in order to compute particle trajectories in physical coordinates. We use these particle trajectories 1. D. Bilman, L. Ling, and P. D. Miller, Extreme Superposition: Rogue to compute the mean transport properties of modulated wave trains. Waves of Infinite Order and the Painleve-III´ Hierarchy, Preprint., Additionally, we present particle trajectories and mean transport Arxiv-id: 1806.00545, 2018 properties for the Dysthe equation and two dissipative generalizations of NLS. Spectral properties of matrix-valued AKNS systems with steplike potentials 1. C.W. Curtis, J.D. Carter, and H. Kalisch, Deep water particle paths in the presence of currents, Journal of Fluid Mechanics, 855 (2018), 322- Martin Klaus 350. Department of Mathematics, Virginia Tech 2. J.D. Carter, C.W. Curtis, and H. Kalisch, Particle trajectories in nonlin- Blacksburg, Virginia, USA ear Schrodinger¨ models, arXiv:1809.08494 [physics.flu-dyn]. [email protected] We consider AKNS systems of the form Asymptotics and numerics for modulational instabilities of traveling waves iξIn Q v′ = − v, x R (1) R iξIm ∈ Benjamin F. Akers Air Force Institute of Technology where Q and R are n m and m n complex valued matrix func- × × benjamin.akers@afit.edu tions and ξ is a complex-valued eigenvalue parameter (In, Im are n n, m m identity matrices). We are particularly interested in The spectral stability problem for periodic traveling waves for wa- the× case where× Q and R tend to nonzero and possibly different lim- ter wave models is considered. The structure of the spectrum is dis- its as x ∞. Our focus will be on the location and existence of cussed from the perspective of resonant interaction theory. Modu- eigenvalues→± and spectral singularities. The motivation for studying lational asymptotic expansions are used to predict the location of spectral singularities comes from the fact that they cause techni- instabilities in frequency-amplitude space. These predictions ex- cal difficulties in the application of the inverse scattering transform plain numerical results in [1]. Asymptotics results are presented in (IST) to the matrix nonlinear Schrodinger¨ equation associated with the potential flow equations [2] as well as weakly nonlinear mod- (1). We hope to provide some insights that will lead to a better un- els [3]. The asymptotic predictions are compared to the results of derstanding of the conditions under which the IST can be applied a direct numerical simulation of the modulational spectrum. to AKNS systems. 1. Nicholls, David P., Spectral data for travelling water waves: singularities and stability Journal of Fluid Mechanics, 624 (2009), 339-360. 2. Akers, Benjamin F., Modulational instabilities of periodic traveling SESSION 3: “Recent developments in mathematical studies of wa- waves in deep water, Physica D: Nonlinear Phenomena, 300 (2015), ter waves” 26-33. 3. Akers, Benjamin F. and Milewski, Paul A., A Model Equation for Particle paths and transport properties of NLS and Wavepacket Solitary Waves Arising from Capillary-Gravity Flows, its generalizations Studies in Applied Mathematics, 122 (2009), 249-274.

John D. Carter∗ Mathematics Department Fully dispersive model equations for hydroelastic Seattle University waves [email protected] Evgueni Dinvay and Henrik Kalisch∗ Chris W. Curtis Dept. of Mathematics, University of Bergen, Norway [email protected] Department of Mathematics & Statistics San Diego State University Emilian Parau [email protected] School of Mathematics, University of East Anglia, UK Henrik Kalisch [email protected] Department of Mathematics In 1967, G. Whitham put forward a simple nonlinear nonlocal model University of Bergen equation for the study of gravity waves at the free surface of an [email protected] inviscid ﬂuid [9]. The advantage of this equation was that it de- The nonlinear Schrodinger¨ equation (NLS) is well known as a uni- scribed the propagation of small amplitude waves nearly perfectly, versal equation in the study of wave motion. In the context of wave and in addition was able to feature some nonlinear effects. motion at the free surface of an incompressible ﬂuid, NLS accu- In this lecture we review Whitham’s idea and present recent de- rately predicts the evolution of modulated wave trains with low to velopments on formal asymptotics. We then present a model of moderate wave steepness. Whitham type for hydro-elastic waves [4, 7], which is similar to

12 the systems given in [1, 3, 5]. The model is tested in the case higher-order Nonlinear Schrodinger (NLS) equation [2]. During of wave-sea-ice interactions and the response of an ice sheet to a the squall the wave action increases, the spectrum broadens, the moving load [2, 6, 8]. spectral mean shifts up and the Benjamin-Feir index (BFI) and kurtosis increase. Conversely, after the squall, due to viscous dissipa- 1. P. Aceves-Sanchez,´ A.A. Minzoni and P. Panayotaros, Numerical study tion, the opposite effect for each quantity occurs. of a nonlocal model for water-waves with variable depth. Wave Motion Kurtosis is considered the main parameter indicating if rogue waves 50 (2013), 80-93. are likely to occur in a sea state. In turn, the BFI is often mentioned 2. J.W. Davys, R.J. Hosking and A.D. Sneyd, Waves due to a steadily moving source on a floating ice plate, J. Fluid Mech. 158 (1985), 269- as a means to predict the kurtosis. We confirm that there is indeed 287. a quadratic relation between these these two quantities. However, 3. P. Guyenne and E.I. Par˘ au,˘ Finite-depth effects on solitary waves in a this relation depends on the intensity of wind forcing and damping, floating ice sheet, J. Fluids and Structures 49, (2014), 242-262. and is therefore not general. Instead, we find a simple and robust 4. A.K. Liu and E. Mollo-Christensen, Wave propagation in a solid ice exponential relation between the spectral mean and kurtosis, and pack, J. Phys. Oceanogr. 18 (1988), 1702-1712. between the spectral width and kurtosis, which are independent of 5. D. Moldabayev, H. Kalisch and D. Dutykh, The Whitham equation as any other quantity. Because of this simple relation, a single spec- a model for surface water waves, Physica D, 309 (2015), 99–107. trum allows to assess the risk of rogue wave occurrence. 6. E. Par˘ au˘ and F. Dias, Nonlinear effects in the response of a floating ice plate to a moving load, J. Fluid Mech. 460 (2002), 281-305. 7. V.A. Squire, R.J. Hosking, A.D. Kerr and P.J. Langhorne, Moving 1. S. Y. Annenkov, V. I. Shrira, On the predictability of evolution of sur- Loads on Ice Plates (Kluwer, Dordrecht, 1996). face401gravity and gravity-capillary waves, Physica D: Nonlinear Phe- 8. T. Takizawa, Field studies on response of a floating sea ice sheet to nomena 152-153 (2001) 665 - 675 a steadily moving load, Contrib. Inst. Low Temp. Sci. A 36 (1987), 2. D. Eeltink, A. Lemoine, H. Branger, O. Kimmoun, C. Kharif, J. D. 31-76. Carter, A. Chabchoub, M. Brunetti, J. Kasparian, Spectral up- and 9. G.B. Whitham, Variational methods and applications to water waves, downshifting of Akhmediev breathers under wind forcing, Physics of Proc. Roy. Soc. London A 299 (1967), 6-25. Fluids 29 (10) (2017) 107103

The stability of stationary solutions of the focusing Faraday waves with bathymetry NLS equation Diane Henderson & Austin Red Wing Bernard Deconinck∗ and Jeremy Upsal William G. Pritchard Fluid Mechanics Laboratory Department of Applied Mathematics, University of Washington Department of Mathematics Seattle, WA 98195-2420, USA Penn State University University Park, PA 16803 [email protected] and [email protected] [email protected] and [email protected]

We examine the stability of the elliptic solutions of the focusing Azar Eslam Panah∗ nonlinear Schrdinger equation (NLS) with respect to subharmonic Department of Mechanical Engineering perturbations. Using integrability properties of NLS, we discuss Penn State Berks the spectral stability of the solutions. We show that the spectrally Reading, PA 19610 stable solutions are orbitally stable by constructing a Lyapunov [email protected] functional using higher-order conserved quantities of NLS. This follows earlier work on the stability of elliptic solutions of We study, experimentally, parametrically excited surface waves in integrable equations, but in all these previous works, the Lax pair of water of finite depth with bathymetry. The bathymetry varies with the integrable equation was self adjoint, significantly simplifying respect to the long dimension of the tank (1 ft) and is uniform with the study. respect to the width of the tank (1in). It is either symmetric or anti- symmetric with respect to the centerline of the tank. Measurements Effect of viscosity and sharp wind increase of neutral stability, wave amplitude evolution, and fluid particle on ocean wave statistics velocities are presented. A theoretical framework for the normal modes of standing waves with such bathymetry is given by [1] . D. Eeltink*, A. Armaroli, Y.M. Ducimetiere,` J. Kasparian and M. Brunetti 1. J. Yu and L. N. Howard, Exact Floquet theory for waves over arbitrary GAP-Nonlinearity and Climate, University of Geneva, periodic topographies. J. Fluid Mech. 712, (2012), 451–470. Bd Carl-Vogt 66, CH1205 Geneva, Switzerland [email protected] Water waves under ice The evolution of gravity waves is very sensitive to initial condi- Olga Trichtchenko tions. That is, after a certain time, no information is left on the ∗ The Department of Physics and Astronomy initial conditions [1], even in the absence of irreversible processes The University of Western Ontario such as wave breaking. Therefore, a statistical approach is needed. [email protected] We study the statistical properties of narrow-banded waves propagating in one direction, during and after a squall (a sudden episode Emilian I. Par˘ au˘ of wind). The model is initialized with a Gaussian shaped spec- School of Mathematics trum with random phases, and propagated using a forced-damped University of East Anglia

13 [email protected] point-vortex approximations and the use of fast-multipole methods for updating point-vortex velocities. We then present results which Jean-Marc Vanden-Broeck show the impact of varying types of vortex patches on nonlinear- Department of Mathematics shallow-water wave propagation. A key result we find is that the University College London more nonlinear a surface wave, the more robust it is with respects [email protected] to the influence of submerged eddies. In contrast, nearly linear Paul Milewski waves can be strongly deformed, possibly to the point of breaking Department of Mathematical Sciences by underwater vorticity patches. University of Bath [email protected] In this talk, we present solutions for models of three-dimensional nonlinear flexuralgravity waves, propagating at the interface be- SESSION 4: “Fractional differential equations” tween a fluid and an ice sheet. The fluid is assumed to be inviscid and incompressible, and the flow irrotational resulting in Euler Time dependent order differential equations equations. We present the details of the numerical method based Andrei Ludu on boundary integral equations used for computing both forced and Embry-Riddle Aeronautical University, Dept. Mathematics & Wave Lab solitary wave solutions, show results in different regimes, and com- 600 S. Clyde Morris Blvd. Daytona Beach, FL 32114 USA pare different models for the ice sheet [1]. [email protected]

1. O. Trichtchenko, E. I. Par˘ au,˘ J.-M. Vanden-Broeck, and P. Milewski, Differential equations with space/time-dependent order of differ- Solitary flexural–gravity waves in three dimensions, Philosophical entiation recently emerge as valid predictive models for physical or Transactions of the Royal Society A, 376 (2018), 20170345. social systems with fast changing dynamics like population growth, anomalous phase transitions, laws of evolution of technology, emer- Tsunami gency of novelty and the adjacent possible [1, 2]. Such new differential tool can provide answers to deeper mathematical-physical Harvey Segur∗ questions like Hamiltonian flows on pseudo-manifolds, dimension Department of Applied Mathematics spectrum, fractional dimensions in homological algebra, or the re- University of Colorado at Boulder lation between fractional cohomology and fractal boundaries. The [email protected] natural frame for variable order equations is provided by fractional differential equations. We present as an application the reduction Diego Arcas of such a variable order equation (from 1st order ODE to 3rd order NOAA Center for Tsunami Research ODE) to a Volterra integral equation of second kind with singular Seattle, WA integrable kernel, and we solve the initial condition and the ex- [email protected] istence and uniqueness of solutions for such equation, or similar Tsunami have received great deal of public interest in the last 20 types. years, because of two very destructive tsunami – one in the Indian Ocean in December 2004 and the other off the eastern coast of 1. A. Ludu, Differential Equations of Time Dependent Order, Technical Japan in March 2011. Tsunami are often generated by undersea and Natural Sciences-AMiTaNS15, AIP Publishing, 1684 (2015). earthquakes that occur at the common boundary of adjacent tec- 2. A. Ludu, and H. Khanal, Differential Equations of Dynamical Order, tonic plates. The moment magnitude is a measure of the energy Electronic J. Diff. Eqs., 24 (2017) 47-61. released by an earthquake, and that same measure is also used to characterize the resulting tsunami, when the earthquake generates a On fractional calculus and nonlinear wave phenom- tsunami. The objective of this talk is to show two significant short- ena comings of this procedure, both of which were demonstrated by the tsunami of 2004. We are now trying to construct other measures of Dumitru Baleanu tsunami, which we hope will provide more useful information. Department of Mathematics, Cankaya University, Anakara, Turkey Institute of Space Sciences, Magurele-Bucharest, Romania Nonlinear waves over patches of vorticity [email protected]

Christopher W. Curtis ∗ Fractional calculus deals with the study of fractional order integral San Diego State University and derivative operators over real or complex domains [1, 2]. It is [email protected] an emerging ﬁeld with important real world applications in various Henrik Kalisch areas of science and engineering [3, 4, 5]. To accurately describe the non-local, frequency and history dependent properties of power University of Bergen law phenomena, some modeling tools have to be introduced such [email protected] as fractional calculus. In this talk, we present a method for numerically simulating freely In this paper, we show a new fractional extension of regularized evolving surface waves over patches of vorticity. This is done via long-wave equation. Besides, the existence and uniqueness of the

14 solution of the regularized long-wave equation within fractional and scale-free distributions. The poster child for fractional deriva- derivative having Mittag-Leffler type kernel is analyzed. The re- tives is anomalous diffusion, where in the superdiffusive regime lated numerical results are also given. . particles are allowed to jump farther than in a Gaussian-distributed random walk [1]. 1. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and One of the most useful fractional derivatives in modeling physical derivatives: Theory and applications, Gordon and Breach, Yverdon, systems is the Caputo fractional derivative, which ensures the con- (1993). vergence of the fractional derivative at the origin. In this paper we 2. A. A. Kilbas, M. H. Srivastava and J. J. Trujillo, Theory and applica- formulate the Caputo fractional derivative in terms of the general- tion of fractional differential equations, North Holland Mathematics ized Euler’s integral transform, which allows us to take fractional Studies 204, (2006). derivatives of a wide class of functions expressible as generalized 3. I. Podlubny, Fractional differential equations, Academic Press: San hypergeometric functions with a power law argument [2]. These Diego CA, (1999). functions can take on the form of many common functions, such as 4. D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional cal- trigonometric, hyperbolic, and a family of Gaussian and Lorentzian culus models and numerical methods, Series on Complexity, Nonlin- functions. earity and Chaos, World Scientific, (2012). The conventional Euler’s integral transform (developed in 1778) 5. A. Atagana and D. Baleanu, New fractional derivative with non-local integrates a power law with a linear argument hypergeometric func- and non-singular kernel, Thermal Sci. 20 (2006), 763-769. tion, which is commensurate with taking a Caputo fractional derivative of the hypergeometric function. Here we develop a method to Variable order differential equations, solutions and take the Caputo fractional derivative of a power law argument hy- applications pergeometric function by expanding the hypergeometric function into its constituent hypergeometric series and utlizing the proper- Harihar Khanal ties of the Pochhammer symbol. This allows us to extend the scope Embry-Riddle Aeronautical University, Department of Mathematics of the transform. 600 S. Clyde Morris Blvd., Daytona Beach, FL 32114 USA Furthermore, the generalized Euler’s integral transform can be used [email protected] to solve linear fractional differential equations by assuming a gen- Recently we introduced a special type of ordinary differential equa- eralized hypergeometric test function with a power law argument. tions whose order of differentiation is a continuous function of the For example, we can use the framework of the generalized Eu- independent variable [1, 2]. We show that such dynamical order ler’s integral transform to solve the fractional Schrodinger¨ equa- of differentiation equations can be approached by using the for- tion, which effectively models quantum transport in multiscale po- malism for Volterra integral equations of second kind with singu- tentials. lar integrable kernel. We present the numeric approach and solutions for particular cases when order of differentiation changes 1. Y. Sagi, M. Brook, I. Almog, and N. Davidson, Observation of anoma- smoothly from 1 to 2 and backwards, and we discuss the asymp- lous diffusion and fractional self-similarity in one dimension, Phys. totic approach of the solutions towards the limiting classical ODE. Rev. Lett. 108, 093002 (2012). We study the numerical solutions by collocation method based on 2. G. Shchedrin, N. C. Smith, A. Gladkina, and L. D. Carr, Exact results a Taylor expansion of the solution and identification of the series for a fractional derivative of elementary functions, SciPost Phys. 4, 029 coefficients [3]. In this way the critical open problem of the initial (2018). conditions is solved in an efficient way. The model is applied to models for social systems with fast changing dynamics like popu- One dimensional sub-hyperbolic equation via sequen- lation growth, emergency of novelty and world computer networks. tial Caputo fractional derivative 1. A. Ludu, Differential Equations of Time Dependent Order, Technical Aghalaya Vatsala and Natural Sciences-AMiTaNS15, AIP Publishing, 1684 (2015). The representation form for sub hyperbolic one dimensional equa- 2. A. Ludu, and H. Khanal, Differential Equations of Dynamical Order, tion for Caputo derivative can be easily obtained by the usual stan- Electronic J. Diff. Eqs., 24 (2017) 47-61. dard procedure of eigen function expansion method. See reference 3. A. Zacharias, H. Khanal, and A. Ludu, Variable Order Differential [1] below. However, this result does not yield the integer result as a Equations and Applications, in print (2018). special case. In order to obtain the integer result as a special case, in this work we assume that the Caputo derivative involved is se- Generalized Euler’s integral transform quential of order q, where 0.5 q 1. Here, we will assume that the Caputo fractional partial derivative≤ ≤ of order 2q, with respect to G. Shchedrin, N. C. Smith, A. Gladkina∗, and L. D. Carr t is sequential. This means that the Caputo derivative of order 2q Physics Department at Colorado School of Mines can be taken as the Caputo derivative of a function of order q fol- 1400 Illinois St, Golden, Colorado 80401, USA lowed by the Caputo derivative of order q. The reason this helps [email protected], [email protected], [email protected], us to obtain the integer result as a special case is that the integer [email protected] derivative is sequential. In addition, the initial conditions need to Fractional calculus is recognized as a ubiquitous tool to character- be modified also accordingly. In general, the initial and bound- ize the dynamics of complex nonlocal systems, described by spa- ary conditions involving Caputo derivative has the same initial and tial heterogeneity, non-Gaussian statistics, non-Fickian transport, boundary conditions as that of the integer derivative. In order to

15 obtain, the integer derivative results as a special case, our initial [email protected] and [email protected] conditions should be given at u(x, 0) and at the Caputo fractional We extend the factorization method to the case of fractional-differ- q derivative of u(x, t) at t = 0 should be known as functions of x. ential Hamiltonians [1, 2]. Taking the quantum harmonic oscillator We will obtain a representation form for the sub hyperbolic equa- as a primary example for this fractional-factorization framework, tion in one dimensional space, using the sequential Caputo partial we present two such factorizations, one with a single Levy´ index derivative with respect to t. The representation form is obtained by [3] and the other with two Levy´ indices. Proceeding like in super- eigen function expansion method and followed by Laplace trans- symmetric quantum mechanics, we also revert the fractional factor- form method for Sequential Caputo derivatives. See reference [2]for ization brackets in order to introduce the fractional supersymmetric Laplace transform method for ordinary sequential fractional differ- partner problem. Nonlinear oscillators of the type xm, m N, are ential equations. If q = 1, our result yields the classical linear also discussed in the same context. ∈ hyperbolic equation as a special case. If q = .5, then we get the linear parabolic result as a special case. The Green’s function will 1. K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, involve the fractional trigonometric functions of Sin and cosine New York, 1974. functions. These trigonometric functions arise from some combination of Mittag-Lefﬂer functions instead of the usual exponential 2. N. Laskin, Fractional quantum mechanics, Phys. Rev. E , 62 (2000), 3135; Fractional Schrodinger¨ equation, Phys. Rev. E, 66 (2002), function. The main reason we want to obtain the integer result as a 056108. special case is to establish that the fractional differential equations represent as a better mathematical model and yield better results 3. F. Olivar-Romero, O. Rosas-Ortiz, Factorization of the quantum frac- compared with the integer derivative models. tional oscillator J. Phys: Conf. Series, 698 (2016), 012025.

Closed-form projection method for regularizing a 1. Donna. S. Stutson, Aghalaya S. Vatsala, Sub Hyperbolic linear Partial function defined by a discrete set of noisy data and Fractional Differential Equation in One Dimensional Space with Nu- for estimating its derivative and fractional derivative merical Results, NONLINEAR STUDIES., V 20, No 4, (2013),483- 492. Timothy J. Burns and Bert W. Rust 2. Aghalaya S. Vatsala, Bhuvaneswari Sambandham, Laplace Trans- Applied and Computational Mathematics Division form Method for Sequential Caputo Fractional Dofferential Equations, National Institute of Standards and Technology Mathematics, in Engineering, Science and Aerospace V 7, No 2, 100 Bureau Drive, Stop 8910 (2016), 339-347. Gaithersburg, MD 20899-8910 [email protected] The gravity of light travel: Riding the fractional wave We present a finite-dimensional projection method for regulariz- of a visible universe from h to c-squared ing a smooth function that has been defined by a discrete set of Christina Nevshehir measurement data, which have been contaminated by random, zero 820 N. Center Ave mean errors. Our approach extends a statistical time series tech- Gaylord, Michigan 49735, USA nique for separating signal from noise in the data, that was orig- [email protected] inally developed by Rust [1] for the study of Fredholm integral equations of the first kind, with a smooth kernel. We then show As light travels fastest of all, all else must be relative to it. How how to obtain closed-form estimates of the derivative and fractional all else relates to light is explored first by establishing a mathe- derivative of the data function, by finding approximate solutions of matical foundation on terra firma. Infinite regress objections of set the Volterra integral equations of the first kind which correspond theory with Russell’s and Zeno’s paradoxes can be set aside. Re- to integration and fractional integration, respectively. These esti- lying on the concept of Ulams spiral and Perelman’s solution of mates are finite linear combinations of trigonometric or Legendre the Poincare conjecture, this approach paves the way computation- polynomials of low degree. ally for fractional differential equations to solve the Riemann hypothesis and other unsolved Millennium problems. This approach 1. B.W. Rust, Truncating the singular value decomposition for ill-posed proposes a Theory of Everything which unifies Einstein’s relativity problems, NISTIR 6131, National Institute of Standards and Technol- quanta with the overarching cosmos. This leads deterministically ogy, Gaithersburg, MD, July, 1998. rather than with uncertainty to the most optimal solutions for any and all future systems and their applications like AI. IST numerical schemes for solving nonlinear evolution equations and their possible applications for The factorization method for fractional quantum os- solving time fractional differential equations cillators Thiab Taha Haret C. Rosu and Stefan C. Mancas ∗ Computer Science Department IPICYT, Instituto Potosino de Investigacion Cientifica y Tecnologica, Univeristy of Georgia, Athens, GA Camino a la presa San Jose´ 2055, Col. Lomas 4a Seccion,´ 78216 San Luis [email protected] Potos038,´ S.L.P., Mexico Department of Mathematics, Embry-Riddle Aeronautical University, Day- In this talk a survey and a method of derivation of certain class tona Beach, FL 32114-3900, USA of numerical schemes and an implementation of these schemes

16 will be presented. These schemes are constructed by methods re- Enhanced existence time of solutions to the fractional lated to the Inverse Scattering Transform (IST) and can be used as Korteweg–de Vries equation numerical schemes for their associated nonlinear evolution equa- Mats Ehrnstrom¨ and Yuexun Wang tions. They maintain many of the important properties of their original partial di?erential equations such as in?nite numbers of Department of Mathematical Sciences, Norwegian University of conservation laws and solvability by IST. Numerical experiments Science and Technology, 7491 Trondheim, Norway have shown that these schemes compare very favorably with other [email protected] and [email protected] known numerical methods.In addition,I will talk about their possi- We consider the fractional Korteweg–de Vries equation ut + uux α − ble applications for solving time Fractional Differential Equations. D ux = 0 in the range of 1 < α < 1, α = 0. Using basic Fourier| | techniques in combination− with the normal6 form transformation and modiﬁed energy method we extend the existence time of classical solutions in Sobolev space with initial data of size ε 2 SESSION 5: “Evolution equations and integrable systems” from 1/ε to a time scale of 1/ε .

Integrable systems in 4+2 dimensions 1. M. Ehrnstrom,¨ and Y. Wang, Enhanced existence time of solutions and their reduction to 3+1 dimensions to the fractional Korteweg-de Vries equation, ARXIV:1804.06297, 2018. M.C. van der Weele∗ and A.S. Fokas Department of Applied Mathematics and Theoretical Physics, Low regularity stablity for the KdV equation University of Cambridge, Cambridge CB3 0WA, United Kingdom [email protected] and [email protected] Brian Pigott Telephone: +447452866946 Wofford College [email protected] One of the main current topics in the field of integrable systems concerns the existence of nonlinear integrable evolution equations Sarah Raynor∗ in more than two spatial dimensions. The fact that such equations Wake Forest University exist has been proven by one of the authors [1], who derived equa- [email protected] tions of this type in four spatial dimensions, which however had In this talk, we consider the stability of solitons for the KdV equa- the disadvantage of containing two time dimensions. The associ- tion below the energy space, using spatially-exponentially-weighted ated initial value problem for such equations, where the dependent norms. We discuss known results including our own recent work variables are specified for all space variables at t1 = t2 = 0, can demonstrating arbitrarily long time stability in this setting, as well be solved by means of a nonlocal d-bar problem. as new progress towards full asymptotic stability. The next step in this program is to formulate and solve nonlinear integrable systems in 3+1 dimensions (i.e., with three space vari- Revivals and fractalisation in the linear free space ables and a single time variable) in agreement with physical reality. Schrodinger¨ equation The method we employ is to first construct a system in 4+2 dimen- Peter J. Olver and Natalie E. Sheils sions, which we then aim to reduce to 3+1 dimensions. ∗ In this paper we focus on the Davey-Stewartson system [2] and School of Mathematics, University of Minnesota the 3-wave interaction equations. Both these integrable systems have their origins in fluid dynamics where they describe the evolu- [email protected] and [email protected] tion and interaction, respectively, of wave packets on e.g. a water David A. Smith surface. We start from these equations in their usual form in 2+1 Division of Science, dimensions (two space variables x, y and one time variable t) and Yale-NUS College, we bring them to 4+2 dimensions by complexifying each of these [email protected] variables. We solve the initial value problem of these equations in 4+2 dimensions. Subsequently, in the linear limit we reduce this We consider the one-dimensional linear free space Schrodinger¨ analysis to 3+1 dimensions to comply with the natural world. Fi- equation on a bounded interval subject to homogeneous linear nally, we discuss the construction of the 3+1 reduction of the full boundary conditions. We prove that, in the case of pseudoperi- nonlinear problem, which is currently under investigation. odic boundary conditions, the solution of the initial-boundary value problem exhibits the phenomenon of revival at specific (“rational”) 1. A.S. Fokas, Integrable Nonlinear Evolution Partial Differential Equa- times, meaning that it is a linear combination of a certain number tions in 4 + 2 and 3 + 1 Dimensions, Phys. Rev. Lett. 96 (2006), of copies of the initial datum. Equivalently, the fundamental solu- 190201. tion at these times is a finite linear combination of delta functions. 2. A.S. Fokas and M.C. van der Weele, Complexification and integrability At other (“irrational”) times, for suitably rough initial data, e.g., in multidimensions, J. Math. Phys. 59 (2018), 091413. a step or more general piecewise constant function, the solution exhibits a continuous but fractal-like profile. Further, we express the solution for general homogenous linear boundary conditions in terms of numerically computable eigenfunctions. Alternative solution formulas are derived using the Uniform Transform Method

17 (UTM), that can prove useful in more general situations. We then [email protected] investigate the effects of general linear boundary conditions, in- Dionyssios Mantzavinos cluding Robin, and find novel “dissipative” revivals in the case of ∗ energy decreasing conditions. Department of Mathematics University of Kansas Well-posedness of initial-boundary value problems [email protected] for dispersive equations via the Fokas method Fangchi Yan A. Alexandrou Himonas Department of Mathematics University of Notre Dame Department of Mathematics [email protected] University of Notre Dame [email protected] The initial value problem for nonlinear evolution equations has been studied extensively and from many points of view over the Dionyssios Mantzavinos last fifty years. On the other hand, the analysis of initial-boundary Department of Mathematics value problems for these equations is rather limited, despite the University of Kansas fact that such problems arise naturally in applications. This talk [email protected] will be devoted to a new approach for the well-posedness of non- Fangchi Yan∗ linear initial-boundary value problems, which combines the linear Department of Mathematics solution formulae produced via the unified transform method of University of Notre Dame Fokas with suitably adapted harmonic analysis techniques. Con- [email protected] crete examples to be discussed include the nonlinear Schrodinger¨ and Korteweg-de Vries equations, as well as a reaction-diffusion We shall discuss the initial-boundary value problem for dispersive equation with power nonlinearity. equations. First, by applying the unified transform method (UTM), which is also known as the Fokas method [F3], we shall solve the Inverse scattering transform for the defocusing initial-boundary value problem with forcing to obtain a formula Ablowitz-Ladik equation with arbitrary nonzero for the solution. Then, replacing the forcing with the nonlinearity background we will define the iteration map for the nonlinear equation. Fi- 1,2, 2 nally, following the methodology developed for the cubic NLS in Barbara Prinari ∗ and Alyssa K. Ortiz [FHM2] or the KdV in [FHM1] (see also [HMY]), we shall prove 1 Department of Mathematics, University at Buffalo well-posedness in Sobolev spaces. 2 Department of Mathematics, University of Colorado Colorado Springs [email protected], [email protected] F3. A.S. Fokas, A unified approach to boundary value problems, SIAM, In this talk we discuss the inverse scattering transform (IST) for the 2008. defocusing Ablowitz-Ladik equation with arbitrarily large nonzero FHM1. A.S. Fokas, A. Himonas and D. Mantzavinos, The Korteweg-de boundary conditions at infinity. The IST was developed in the past Vries equation on the half-line. Nonlinearity 29 (2016), 489-527. [1, 2] under the assumption that the amplitude of the background FHM2. A.S. Fokas, A. Himonas and D. Mantzavinos, The nonlinear intensity Qo satisfies a “small norm” condition 0 < Qo < 1. As Schrodinger¨ equation on the half-line. Trans. Amer. Math. Soc. 369 recently shown by Ohta and Yang [3], the defocusing AL system, (2017), 681-709. which is modulationally stable for 0 Q < 1, becomes un- ≤ o HMY. A. A. Himonas, D. Mantzavinos, F. Yan The nonlinear stable if Qo > 1. And, in analogy with the focusing case, when Schrodinger¨ equation on the half-line with Neumann boundary cond- Qo > 1 the defocusing AL equation admits discrete rogue wave tions. Appl. Numer. Math (2018). solutions, some of which are regular for all times. Therefore, it KPV1. C.E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial is clearly of importance to develop the IST for the defocusing AL value problem for the Korteweg-de Vries equation. J. AMS 4 (1991), with Qo > 1, analyze the spectrum and characterize the soliton and 323-347. rational solutions from a spectral point of view. Both the direct and the inverse problems are formulated in terms of a suitable uniform variable; the inverse problem is posed as a Riemann-Hilbert prob- Analysis of nonlinear evolution equations in domains lem in the complex plane, and solved by properly accounting for with a boundary the asymptotic dependence of eigenfunctions and scattering data Athanassios S. Fokas on the Ablowitz-Ladik potential. Department of Applied Mathematics and Theoretical Physics University of Cambridge 1. V.E. Vekslerchik and V.V. Konotop, Discrete nonlinear Schrodinger¨ equation under non-vanishing boundary conditions, Inv. Probl., 8, [email protected] (1992) 889. A. Alexandrou Himonas 2. M.J. Ablowitz, G. Biondini and B. Prinari, Inverse scattering transform Department of Mathematics for the integrable discrete nonlinear Schrodinger¨ equation with non- University of Notre Dame vanishing boundary conditions, Inv. Probl., 23, (2007) 1711. 3. Y. Ohta and J. Yang, General rogues waves in the focusing and defocusing Ablowitz-Ladik equations, J. Phys. A, 47, (2014) 255201.

18 4. B. Prinari and F. Vitale, Inverse scattering transform for the focusing Dahlonega, GA Ablowitz-Ladik system with nonzero boundary conditions, Stud. App. [email protected] Math., 137, (2016) 28. Dark matter is deﬁned as nonluminous matter not yet directly de- 5. B. Prinari, Discrete solitons of the Ablowitz-Ladik equation with tected by astronomers that is hypothesized to exist to account for nonzero boundary conditions via inverse scattering, J. Math. Phys., 57, (2016) 083510. various observed gravitational effects. In this talk, we will provide a brief history of the observations made by renowned scien- Non-uniqueness and norm-inﬂation for tists since the late nineteenth century and the subsequent data col- Camassa-Holm-type equations lected that led to the proposed concept of dark matter in the Uni- verse. We now know that this invisible nondissipative dark matter Curtis Holliman plays a decisive role in the formation of large scale structures in the Department of Mathematics Universe such as galaxies, clusters of galaxies, and superclusters. The Catholic University of America Since the corresponding nonlinear dynamics may be modeled by Washington, DC 20064 hydrodynamic-like equations, this is where we shall focus the rest [email protected] of our attention and discuss results regarding these systems.

Alex Himonas 1. A. V. Gurevich, K.P. Zybin, Nondissipative gravitational turbulence, Department of Mathematics Soviet Phys. JETP 67 No. 1 (1988), 1-12. The University of Notre Dame 2. A. V. Gurevich, K.P. Zybin, Large-scale structure of the Universe: An- Notre Dame, IN 46556 alytic Theory, Soviet Phys. Usp. 38 No. 7 (1995), 687-722. [email protected] 3. J. H. Jeans, Astronomy and Cosmology, Cambridge University Press, London and New York, 1969. We consider a number of equations related to the Camassa-Holm equation and will examine how well-posedness fails when the ini- 4. Ya. B. Zeldovich, I.D. Novikov, Structure and Evolution of the Uni- verse, Moscow, “Nauka” (1975), 736 pp. tial data are taken in Sobolev spaces with exponents less than 3/2. Depending on the structure of the equation, the ill-posedness is either norm-inflation or non-uniqueness and typically depends on the Isometric immersions and self-similar buckling in Sobolev exponent. non-Euclidean elastic sheets John A. Gemmer and Maximilian Rezek 1. A. Himonas and C. Holliman Non-Uniqueness for the Fokas-Olver- Wake Forest University, Department of Mathematics Rosenau-Qiao equation. Journal of Mathematical Analysis and Appli- cations 470 (1), 647-658. 127 Manchester Hall, Winston Salem, NC 27109 [email protected] and [email protected] 2. A. Himonas, C. Holliman and C. Kenig Construction of 2-peakon solutions and ill-posedness for the Novikov Equation. Siam J. Math. Anal. The edges of torn elastic sheets and growing leaves often display Vol. 50, No. 3, pp. 2968–3006. hierarchical self-similar like buckling patterns. Within non- 3. A. Himonas, C. Holliman and K. Grayshan, Norm inflation and ill- Euclidean plate theory this complex morphology can be under- posedness for the Degasperis-Procesi equation. Comm. Partial Differ- stood as low bending energy isometric immersions of hyperbolic ential Equations 39, 2198–2215, 2014. Riemannian metrics. With this motivation we study the isometric 4. A. Himonas, K. Grayshan and C. Holliman, Ill-posedness for the b- immersion problem in a strip with an asymptotically decaying met- family of equations. J. Nonlinear Sci. 26 (2016), 1175–1190. ric. By finding explicit piecewise smooth solutions of hyperbolic Monge-Ampere equations on, we show there exist periodic isomet- The Cauchy problem for evolution equations with ric immersions of hyperbolic surfaces in the small slope regime. analytic data We extend these solutions to exact isometric immersions through Alex A. Himonas resummation of a formal asymptotic expansion. Using this construction, we identify the key role of branch-point (or monkey- Department of Mathematics, University of Notre Dame saddle) singularities, in complex wrinkling patterns within the class Notre Dame, IN 46556 of finite bending energy isometric immersions. Using these de- [email protected] fects we give an explicit construction of strain-free embeddings of In this talk we will discuss analyticity properties in the spatial and hyperbolic surfaces that are fractal like and have lower elastic en- time variables for solutions to the Cauchy problem of evolution ergy than their smooth counterparts.For hyperbolic non-Euclidean equations with analytic initial data. In particular, lower bound esti- sheets, complex wrinkling patterns are thus possible within the mates for the uniform radius of spatial analyticity will be presented class of finite bending energy isometric immersions. Further, our for Camassa-Holm and Korteweg-de Vries type equations. The results identify the key role of the degree regularity of the isometric talk is based on works with Professors G. Petronilho, R. Baros- immersion in determining the global structure of a non-Euclidean tichi, S. Selberg, H. Kalisch. elastic sheet in 3-space. On the evolution of dark matter Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity Ryan C. Thompson Department of Mathematics Fredrik Hildrum University of North Georgia Department of Mathematical Sciences,

19 Norwegian University of Science and Technology, 1. J.L. Bona and R. Sachs, Global existence of smooth solutions and sta- 7491 Trondheim, Norway bility theory of solitary waves for a generalized Boussinesq equation, [email protected] Comm. Math. Phys., 118 (1988), 15-29. 2. L.G. Farah, Local solutions in Sobolev spaces with negative indices for We show existence of small-amplitude solitary and periodic traveling- the “good” Boussinesq equation, CPDE, 34 (2009), 52-73. wave solutions in fractional Sobolev spaces Hs to a class of nonlinear, dispersive integro-differential equations of the form 3. F. Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Diff. Equations, 106 (1993), 257-293.

ut + (Lu + n(u))x = 0, Gevrey regularity in time variable for solutions to where L is a Fourier multiplier operator of any negative order whose the “good” Boussinesq equation symbol is of KdV type at the origin and has integrable inverse Fourier transform—so that L becomes convolution with integrable Renata Figueira∗ kernel—and n is an inhomogeneous power-type nonlinearity of or- Department of Mathematics der strictly greater than 1 at the origin. Notably, this class includes Federal University of Sao Carlos Whitham’s model equation for surface gravity water-waves featur- Sao Carlos, SP- Brazil ing the exact linear dispersion relation, in which we obtain periodic [email protected] waves for s > 0 and solitary waves for s > 1 . Our tools involve 6 Alex Himonas constrained variational methods, Lions’ concentration-compactness principle, a fractional chain rule and a cut-off argument for n, Department of Mathematics > 1 The University of Notre Dame which enables us to go below the typical s 2 regime. More- over, we prove that most of the nonlocal estimates follow directly Notre Dame, IN 46556 from integrability of the kernel. [email protected] Rafael Barostichi 1. M. Ehrnstrm, M. D. Groves and E. Wahln, “On the existence and sta- Department of Mathematics bility of solitary-wave solutions to a class of evolution equations of The University of Notre Dame Whitham type”, Nonlinearity, 25(10), 2903–2936, 2012. Notre Dame, IN 46556 2. P. L. Lions, “The concentration-compactness principle in the calculus [email protected] of variations. The locally compact case. I”, Ann. Inst. H. Poincare´ Anal. Non Lineaire´ , 1(2), 109–145, 1984. We shall consider the Cauchy problem for the “good” Boussinesq 3. T. Runst and W. Sickel, “Sobolev Spaces of Fractional Order, Ne- equation and enunciate a result about its well-posedness in a class mytskij Operators, and Nonlinear Partial Differential Equations”, of analytic Gevrey spaces, which guarantees the Gevrey regularity De Gruyter Series in Nonlinear Analysis and Applications, Wal- of the solutions in space variable. The main discussion of this talk ter de Gruyter, 1996. concerns about regularity in time of these solutions. This work is in collaboration with Rafael Barostichi and Alex Himonas.

The Cauchy problem for the “good” Boussinesq equa- 1. L.G. Farah, Local solutions in Sobolev spaces with negative indices for tion with analytic and Gevrey initial data the “good” Boussinesq equation. CPDE 34 (2009), 52–73.

Rafael Barostichi∗ 2. L.G. Farah and M. Scialom, On the periodic “good” Boussinesq equa- University of Notre Dame tion. Proceedings of the American Math. Soc. 138(3) (2010), 953–964. [email protected] 3. J. Gorsky, A. Himonas, C. Holliman and G. Petronilho, The Cauchy problem of a periodic higher order KdV equation in analytic Gevrey Alex Himonas spaces. J. Math. Anal. Appl. 405 (2013), 349–361. University of Notre Dame 4. H. Hannah, A. Himonas and G. Petronilho, Gevrey regularity of the [email protected] periodic gKdV equation. J. Diff. Equations 250 (2011), 2581–2600. Renata Figueira Federal University of Sao Carlos - Brazil Existence of solutions for conservation laws [email protected] John Holmes We shall consider the initial value problem for the “good” Boussi- 231 West 18th Avenue nesq equation with initial data belonging in a class of Gevrey func- Columbus OH, 43210-1174 tions on both the line and the circle, which includes a class of ana- [email protected] lytic functions that can be extended holomorphically in a symmetric strip of the complex plane around the real axis. Systems of conservation laws in one spacial variable are locally We shall talk about the history of this equation and present some well-posed in the space of functions with bounded total variation details of the proof of the local well-posedness in theses analytic- (BV). It is well known that classical solutions break down in ﬁnite Gevrey spaces. This is work in collaboration with Alex Himonas time; in particular, smoothness is lost and shocks form. However, and Renata Figueira. if the initial data is sufﬁciently small in BV, weak solutions exist (and when entropy conditions are imposed) are unique. There

20 have been several proofs of this result including the random choice 2. Himonas, A., Misiolek, G. Non-Uniform Dependence on Initial Data method and the vanishing viscosity method. These methods have of Solutions to the Euler Equations of Hydrodynamics. Commun. Math. been extended to systems with forcing. Global in time results are Phys. 296 (2010), 285-301. also found for systems with forcing under suitable constraints. We 3. Kato, T., Ponce, G., On non-stationary flows of Viscous and Ideal Flu- will discuss some new results concerning the existence of solutions ids. Duke Mathematical Journal, Vol. 55, No.3, 487-499 (1987) to these systems, and the relationship between our results and these 4. Kenig, C., Ponce, G., Vega, L. On the (generalized) KdV Equation. previous methodologies. Duke Mathematical Journal, Vol. 59, No.3, 585-610 (1989) 5. Misiolek, G., Yoneda, T. Continuity of the solution map of the Euler Unified transform method with moving interfaces Equations in Holder¨ spaces and weak norm inflation in Besov spaces.; Trans. Amer. Math. Soc. 370 (2018), no. 7, 4709-4730. Dave Smith∗ Yale-NUS College, Singapore [email protected] Non-uniform dependence of the data-to-solution map for the Hunter–Saxton equation in Besov spaces Tom Trogdon University of California, Irvine CA F. Tiglay, J. Holmes [email protected] Department of Mathematics, The Ohio State University, Columbus, OH 43210 Vishal Vasan [email protected], [email protected] International Centre for Theoretical Sciences, Bengaluru, India [email protected] The Cauchy problem for the Hunter-Saxton equation is known to s be locally well posed in Besov spaces B2,r on the circle. We prove The unified transform method was extended to interface problems that the data-to-solution map is not uniformly continuous from any in the past 5 years, particularly by Sheils. Earlier work by Pelloni s s bounded subset of B2,r to C([0, T]; B2,r). We also show that the so- and Fokas implemented the unified transform method on domains lution map is Holder¨ continuous with respect to a weaker topology. with moving boundaries. We present a synthesis and extension of these approaches, and an application to a new linearization of Well-posedness and analyticity of solutions to a wa- the Korteweg-de Vries equation with step-like initial datum that ter wave problem with viscosity produces linear dispersive shocks. Marieme Ngom and David P. Nicholls∗ Non-uniform continuous dependence for Euler equa- Department of Mathematics, Statistics, and Computer Science tions in Besov spaces University of Illinois at Chicago Chicago, IL 60607 Jose´ Pastrana [email protected] and [email protected] Department of Mathematics University of Notre Dame The water wave problem models the free–surface evolution of an [email protected] ideal fluid under the influence of gravity and surface tension. The governing equations are a central model in the study of open ocean The Cauchy problem governing the motion of an incompressible wave propagation, but they possess a surprisingly difficult and sub- d and ideal fluid, in a domain Ω R , is given by the system of tle well–posedness theory. In this talk we discuss the existence partial differential equations: ∂ u⊆+(u )u + p = 0. Where t ·∇ ∇ and uniqueness of spatially periodic solutions to the water wave incompressibility translates to div u = 0 and u0(x) := u(x, 0) is equations augmented with physically inspired viscosity suggested the initial configuration. in the recent work of Dias et al. (2008). As we show, this viscosity For local well-posedness theory see Bahouri, Chemin and Danchin (which can be arbitrarily weak) not only delivers an enormously [1]. Ever since the papers of Kato and Ponce [3] there has been a lot simplified well–posedness theory for the governing equations, but of interest in the regularity properties of the data to solution map, also justifies a greatly stabilized numerical scheme for use in study- u0 u. We make use of the approximate solutions technique ing solutions of the water wave problem. which→ traces back to Kenig, Ponce and Vega [4] (when working on KdV type equations) and a construction due to Himonas and Energy decay for the linear damped Klein-Gordon Misiolek [2]; to show that continuity of such map is the best you equation on unbounded domain s can expect for the Besov spaces, Bp,q. This is done for all relevant scales in the periodic case and partially in Euclidean space; we Satbir Malhi and Milena Stanislavova restrict to dimension d = 2. As a consequence we obtain the result University of Kansas for the little Holder¨ class, c1,σ(T2) ( C1,σ(T2), σ (0, 1) where [email protected] ∈ Misiolek and Yoneda [5] proved local well posedeness in the sense In this talk, we consider energy decay for the damped Klein-Gordon of Hadamard. equation.

1. Bahouri, H., Chemin, J., Danchin, R., Fourier Analysis and Nonlinear utt + γ(x)ut uxx + u = 0. (x, t) R R (3) Partial Differential Equations, Springer, New York 2011. − ∈ × Where γ(x)ut represents a damping force proportional to the velocity ut.

21 We give an explicit necessary and sufficient condition on the con- [email protected] tinuous damping functions λ 0 for which the energy E(t) = ∞ 2 2 2 ≥ Sevak Mkrtchyan∗ ux + u + ut dx decays exponentially, whenever ∞ | | | | | | Department of Mathematics, University of Rochester, Rochester, NY,USA (Ru−(0), u (0)) H2(R) H1(R). The approach we use in this pa- t ∈ × [email protected] per is based on the asymptotic theory of C0 semigroups, in particular, the results by Gearhart-Pruss, and later Borichev and Tomilov Maria Shcherbina in which one can relate the decay rate of energy and the resolvent Institute for Low Temperature Physics Ukr. Ac. Sci., Kharkov, Ukraine growth of the semigroup generator. A key ingredient of our proof [email protected] is a projection method, in which we project the frequency domain Alexander Soshnikov on appropriate regions and estimate the resolvent norms through Department of Mathematics, University of California at Davis, Davis, Fourier transformation. At the end of the talk, I will also show USA some result on Fractional type Klein Gordon equation. [email protected] Riemann problems, solitons and dispersive shocks in We will study the asymptotic properties of the density functions modulationally unstable media of beta ensembles that arise in random matrix theory. We will Gino Biondini show that the ensembles have the asymptotic equipartition property (AEP), and discuss the analogy with the Shannon-McMillan- State University of New York at Buffalo Breiman theorem and entropy. In addition to the AEP, the density [email protected] of the eigenvalues of these ensembles satisfy a Central Limit The- The study of Riemann problems — i.e., the evolution of a jump orem. We will discuss the results in detail in the case of several discontinuity between two uniform values of the initial datum — classical ensembles, and give a sketch for the case of beta ensem- is a well-established part of fluid dynamics, since understanding bles with generic real analytic potential. the response of a system to such inputs is a step in characterizing its behavior. When nonlinearity and dissipation are the dom- Rational solutions of Painleve´ equations inant physical effects, these problems can give rise to classical Peter D. Miller shocks. Conversely, when dissipation is negligible compared to ∗ Dept. of Mathematics, University of Michigan dispersion, Riemann problems can give rise to dispersive shock 530 Church St., Ann Arbor, MI 48109 waves (DSWs). This talk will discuss Riemann problems and DSW [email protected] formation in self-focusing media, using the cubic one-dimensional nonlinear Schrodinger¨ equation as a prototypical example. I will All of the six Painleve´ equations except the first have rational solu- show how a broad class of problems can bs effectively studied tions for certain parameter values. We survey some recent results using Whitham modulation theory. At the same time, however, obtained in collaboration with T. Bothner, R. Buckingham, and Y. the full power of the inverse scattering method and the Deift-Zhou Sheng on the asymptotic behavior of rational solutions of Painleve´ nonlinear steepest descent method must be used in order to obtain II, III, and IV when the parameters are large. These results are ob- rigorous results. tained by first computing the correct isomonodromy data for the Jimbo-Miwa Lax pair associated with the family of rational solu- 1. “Universal nature of the nonlinear stage of modulational instability”, tions with the help of classical special functions, their connection G. Biondini and D. Mantzavinos, Phys. Rev. Lett. 116, 043902 (2016) formulæ, and Schlesinger transformations. Then it becomes possi- 2. “Universal behavior of modulationally unstable media”, G. Biondini, ble to apply the Deift-Zhou steepest descent method to an appro- S. Li, D. Mantzavinos and S. Trillo, SIAM Review 60, 888–908 (2018) priate Riemann-Hilbert problem characterizing the rational solu- 3. G. Biondini, S. Li and D. Mantzavinos, “Soliton transmission, trapping tions at hand. This allows the transitions between pole-free regions and wake in modulationally unstable media”, Phys. Rev. E 98, 042211 and regions containing regular lattices of poles to be characterized (2018) in terms of bifurcations of a suitable g-function, and provides ac- 4. G. Biondini, “Riemann problems and dispersive shocks in self- curate asymptotic formulæ for the rational solutions valid in both focusing media”, Phys. Rev. E 98, 052220 (2018) types of regions. 5. G. Biondini and J. Lottes, “Nonlinear interactions between solitons and dispersive shocks in focusing media”, submitted (2019) A representation of joint moments of CUE characteristic polynomials in terms of a Painleve-V solution

Robert Buckingham∗ Department of Mathematical Sciences University of Cincinnati SESSION 6: “Random matrices, Painleve equations, and integrable [email protected] systems” We establish a representation of the joint moments of the char- Entropy of beta random matrix ensembles acteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the σ-Painleve´ V equation. The deriva- Alexander Bufetov tion involves the analysis of a formula for the joint moments in Laboratoire d’Analyse, Topologie, Probabilites,´ CNRS, Marseille

22 terms of a determinant of generalised Laguerre polynomials us- type of quad-equation, but also the combinatorial structure of the ing the Riemann-Hilbert method. We use this connection with the lattice before reduction. Then, we reconstruct the lattice from a σ-Painleve´ V equation to derive explicit formulae for the joint mo- CAC cubic lattice via reduction. ments and to show that in the large-matrix limit the joint moments For the lower types of discrete Painleve´ equations in the Sakai’s are related to a solution of the σ-Painleve´ III equation. This is joint classiﬁcation, this approach works well. However, in a study of a work with Estelle Basor, Pavel Bleher, Tamara Grava, Alexander higher type of discrete Painleve´ equation, a different lattice struc- Its, Elizabeth Its, and Jonathan Keating. ture from CAC cubic lattice appeared. The lattice can be obtained from a reduction of the lattice. The new lattice locally has a cuboc- Skew-orthogonal polynomials and continuum limits tahedron structure (CACO property) instead of the CAC cubic struc- of the Pfaff lattice ture, but such a structure has not been investigated until now. Virgil U. Pierce In this talk, we give a more detailed description of the CACO prop- University of Northern Colorado, School of Mathematical Sciences erty and show a classiﬁcation of quad-equations on a cuboctahe- [email protected] dron using the CACO property.

The partition function of the Gaussian Orthogonal and Gaussian SLk character varieties and quantum cluster alge- Symplectic Ensembles (GOE and GSE) can be expressed in terms bras of the skew-orthogonal polynomials with respect to a perturbed Leonid O. Chekhov and Michael Z. Shapiro Gaussian measure. As in the case of the Gaussian Unitary Ensem- ∗ ble that has been studied extensively, this provides a connection Michigan State University, East Lansing, MI, 48824 between the random matrix ensemble and a family of integrable lat- [email protected] and [email protected] tice hierarchies. In the case of GOE and GSE those hierarchies are We describe quantum algebras of monodromies of SLk Fuchsian the so-called Pfaff lattices. In this presentation we will review re- systems using the Fock-Goncharov construction [3] of higher Te- sults about the skew-orthogonal polynomials and their asymptotic ichmuller spaces. We prove that the monodromy matrices in the expansions. The goal is a description of the continuum limits of the disc with three marked points on the boundary, which corresponds Pfaff lattice hierarchies as they pass from a differential-difference to configurations of three flags in Rn, satisfy the Lie-Poisson semi- system to a differential system by passing the discrete variable to a classical and quantum commutation relations, whereas a particular continuous one. Ideally this computation is based upon a rigorous A T combination of these matrices = M1 M2 enjoys the quantum foundation of the existence of such a limit, and will result in ex- reflection equation. It is known that this equation naturally ap- pressions for the generating functions enumerating Mobius¨ maps pears as a Poisson structure on the set of matrices of upper triangu- (ribbon graphs embedded on unoritented surfaces). lar groupoid studied by A. Bondal [1] that is compatible with the braid-group action and with the dynamics governed by transforma- Classification of quad-equations on a cuboctahedron tions of bilinear forms A BABT studied by Chekhov and Maz- zocco [2]. In the mathematical7→ physics literature particular Poisson Nalini Joshi and Nobutaka Nakazono∗ School of Mathematics and Statistics, The University of Sydney, New leaves of these algebras were identified by J. Nelson and T. Regge South Wales, Australia. [4] with algebras of geodesic functions on Riemann surfaces with holes. Our approach enables us to find canonical (Darboux) coor- ∗Department of Physics and Mathematics, Aoyama Gakuin University, Kanagawa, Japan. dinate representation for general Poisson leaves of these algebras, classify their central elements both in the upper-triangluar and in [email protected] and ∗[email protected] the general case, and construct the cluster algebra representations In the theory of discrete integrable systems, the classification of for the corresponding braid-group action. integrable partial difference equations (PDEs) by Adelr-Bobenko- Suris (2003, 2009) and Boll (2011) are well known. They classi- 1. A. Bondal, A symplectic groupoid of triangular bilinear forms and the fied quad-equations1 on a cube using the CAC property. The CAC braid groups, preprint IHES/M/00/02 (Jan. 2000). property means a local property of Backlund¨ transformations of 2. L.O. Chekhov, M. Mazzocco, Poisson algebras of block-upper- some integrable PDEs, including discrete Schwarzian KdV equa- triangular bilinear forms and braid group action, Commun. Math. Phys. tion, lattice modified KdV equation, lattice potential KdV equation 332 (2013) 49–71. and so on. Thus, repeated translation of a cube which has the CAC 3. V. V. Fock and A. B. Goncharov, Moduli spaces of local systems and property (CAC cube) leads to a space-filling cubic lattice (CAC cu- higher Teichmuller¨ theory, Publ. Math. Inst. Hautes Etudes´ Sci. 103 bic lattice), on which integrable PDEs are iterated. Such PDEs are (2006), 1-211. collectively called ABS equations. 4. Nelson J.E., Regge T., Homotopy groups and (2+1)-dimensional In our recent works, the mathematical connection between two quantum gravity, Nucl. Phys. B 328 (1989), 190–199. longstanding classifications of ABS equations and discrete Painleve´ equations by Sakai (2001) have been investigated by using their lattice structures. Our approach is as follows. First, we derive a The space of initial conditions for some 4D Painleve´ lattice, where quad-equations are observed, from the theory of dis- systems crete Painleve´ equation. The derived lattice provides not only the Tomoyuki Takenawa

1An equation Q(x, y, z, w)= 0, where Q is an irreducible multi-afﬁne polyno- Faculty of Marine Technology, Tokyo University of Marine Science and mial, is called a quad-equation. Technology,

23 2-1-6 Etchu-jima, Koto-ku, Tokyo, 135-8533, Japan 5. K. Takasaki, Painleve-Calogero´ correspondence revisited, J. Math. [email protected] Phys., 42, (2001), 1443. In recent years, research on 4D Painleve´ systems have progressed mainly from the viewpoint of isomonodromy deformation of lin- Algebro-geometric solutions to Schlesinger systems ear equations. In this talk we study the geometric aspects of 4D Vladimir Dragovic´ Painleve´ systems by investigating the space of initial conditions in Department of Mathematical Sciences, University of Texas at Dallas, 800 Okamoto-Sakai’s sense, which was a powerful tool in the original West Campbell Road, Richardson TX 75080, USA. 2D case. Speciﬁcally, starting from known discrete symmetries, we Mathematical Institute SANU, Kneza Mihaila 36, 11000 Belgrade, Serbia. construct the space of initial conditions for some 4D Painleve´ sys- [email protected] tems, and using the Neron-Severi bi-lattice, clarify the whole group of their discrete symmetries. The examples include the directly Renat Gontsov (1) M.S. Pinsker Laboratory no.1, Institute for Information Transmission Prob- coupled 2D Painleve´ equations, Noumi-Yamada’s A5 system and the 4D Garnier system. The spaces of initial conditions for the ﬁrst lems of the Russian Academy of Sciences, Bolshoy Karetny per. 19, two equations are obtained by 16 blow-ups from (P1)4, while for build.1, Moscow 127051 Russia. the last equation, it is obtained by 21 blow-ups from (P2)2. [email protected]

Vasilisa Shramchenko∗ Asymptotic of solution of three-component Painleve-´ Department of mathematics, University of Sherbrooke, 2500, boul. de II equation. l’Universite,´ J1K 2R1 Sherbrooke, Quebec, Canada. Alexander Its and Andrei Prokhorov* [email protected] Indiana University-Purdue University Indianapolis We construct various algebro-geometric solutions to the Schlesinger 402 N Blackford St., Indianapolis, IN, 46202, USA system. First, we discuss a rank two four point Schlesinger system Saint Petersburg State University which we solve using a special meromorphic differential on an el- Universitetskaya emb. 7/9, 199034, St. Petersburg, Russia liptic curve presented as a ramified double covering of the Riemann [email protected] and [email protected] sphere. This differential has a remarkable property: the common We consider the three-component Painleve´ equation. It was ob- projection of its two zeros on the base of the covering, regarded tained in [5] as degeneration of higher rank Inozemtsev rational as a function of the only moving branch point of the covering, is extension of Calogero system. It can be interpreted as the equation a solution of a Painleve´ VI equation. This differential provides an of motion of 3 interacting particles in the external potential. invariant formulation of one particular Okamoto transformation for We are interested in its application in random matrix theory. More the Painleve´ VI equations. precisely Tracy-Widom beta distribution with even β = 2r was de- Next, we study the Schlesinger system of partial differential equations in the case when the unknown matrices of arbitrary size (p scribed in [4] using the particular solution of r-component Painleve-´ × II equation. Tracy-Widom beta law is the limiting distribution p) are triangular and the eigenvalues of each matrix, called the of the largest eigenvalue of Hermite and Laguerre β-ensembles exponents of the system, form an arithmetic progression with a ra- of random matrices when the size of the matrix tends to infinity. tional difference q, the same for all matrices. We show that such This distribution is well studied for β = 1, 2, 4 and is described a system possesses a family of solutions expressed via periods of in these cases using Hastings-McLeod solution of one-component meromorphic differentials on the Riemann surfaces of superellip- Painleve-II´ equation. For arbitrary β > 0 the leading term in the tic curves. We determine the values of the difference q, for which tail asymptotics was obtained rigorously in [3]. The full asymp- our solutions lead to explicit polynomial or rational solutions of the Schlesinger system. As an application of the (2 2)-case, we totic expansion for left and right tail asymptotics was conjectured × in [2]. obtain explicit sequences of rational solutions and one-parametric We study the solution of three-component Painleve-II´ equation men- families of rational solutions of Painleve´ VI equations. tioned above. We use the Riemann-Hilbert problem for multi- component Painleve´ equations found recently in [1]. We perform Two discrete dynamical systems are discussed and analyzed whose nonlinear steepest descent analysis to get asymptotic results. trajectories encode significant explicit information about a number of problems in combinatorial probability. In this talk we will fo- 1. M. Bertola, M. Cafasso, V. Roubtsov, Noncommutative Painleve´ equa- cus on applications to random walks in random environments. The tions and systems of Calogero type, Commun. Math. Phys. , 363:2, two models are integrable and our analysis uncovers the geometric (2018), 503–530. sources of this integrability and uses that to conceptually explain 2. G. Borot, C. Nadal, Right tail asymptotic expansion of Tracy-Widom the rigorous existence and structure of elegant closed form expres- beta laws,Random Matrices: Theory and Applications, 01:03, 1250006 sions for the probability distributions for physically meaningful (2012). random variables of these walks. The work here brings together 3. J. Ramirez, B. Rider, and B. Virag, Beta ensembles, stochastic Airy ideas from a variety of fields including dynamical systems theory, spectrum, and a diffusion, J. Amer. Math. Soc., 2011, (2011), 919–944. probability theory, classical analogues of quantum spin systems,

4. I. Rumanov, Painleve´ Representation of Tracy-Widomβ distribution addition laws on elliptic curves, and links between randomness and for β = 6 , Commun. Math. Phys. , 342:3, (2016), 843–868. symmetry.

24 Discrete Painleve´ equations in tiling problems I will present a survey of the work done by Paolo Lorenzoni and myself in the last few years developing the theory of bi-flat F- Anton Dzhamay ∗ manifolds and exploring their relationships with integrable hierar- School of Mathematical Sciences, University of Northern Colorado, Gree- chies (dispersionless and dispersive), with Painleve´ transcendents, ley, CO 80639, USA and with complex reflection groups. If there is enough time, I [email protected] will address also very recent results about the existence of inte- Alisa Knizel grable dispersive deformations in the non-Hamiltonian setting us- Department of Mathematics, Columbia University, New York, NY, USA ing tools from the so called cohomological field theory (these latter [email protected] results are being developed together also with Alexander Buryak and Paolo Rossi). The notion of a gap probability is one of the main characteristics of a probabilistic model. In [3] Borodin showed that for some discrete probabilistic models of Random Matrix Type discrete gap probabilities can be expressed through solutions of discrete Painleve´ equations, which provides an effective way to compute them [1]. SESSION 7: “Stability and traveling waves” We discuss this correspondence for a particular class of models of lozenge tilings of a hexagon. For uniform probability distribu- On the existence and instability of solitary tion, this is one of the most studied models of random surfaces. water waves with a finite dipole Borodin, Gorin, and Rains [2] showed that it is possible to assign Hung Le a very general elliptic weight to the distribution and degenerations Department of Mathematics, University of Missouri, Columbia, MO 65211 of this weight correspond to the degeneration cascade of discrete [email protected] polynomial ensembles, such as Racah and Hahn ensembles and their q-analogues. This also correspond to the degeneration scheme his paper considers the existence and stability properties of two- of discrete Painleve´ equations, due to the work of Sakai. Con- dimensional solitary waves traversing an infinitely deep body of tinuing the approach of Knizel [4], we consider the q-Hahn and water. We assume that above the water is vacuum, and that the q-Racah ensembles and corresponding discrete Painleve´ equations waves are acted upon by gravity with surface tension effects on the (1) (1) air–water interface. In particular, we study the case where there is of types q P(A ) and q P(A ) [5]. We show how to use 2 1 a finite dipole in the bulk of the fluid, that is, the vorticity is a sum the algebro-geometric− techniques− of Sakai’s theory to pass from the of two weighted -functions. Using an implicit function theorem isomonodromic coordinates of the model to the discrete Painleve´ δ argument, we construct a family of solitary waves solutions for this coordinates that is compatible with the degeneration. system that is exhaustive in a neighborhood of 0. Our main result is that this family is conditionally orbitally unstable. This is proved 1. Alexei Borodin and Dmitriy Boyarchenko, Distribution of the first particle in discrete orthogonal polynomial ensembles, Comm. Math. Phys. using a modification of the Grillakis–Shatah–Strauss method re- 234 (2003), no. 2, 287–338. cently introduced by Varholm, Wahlen,´ and Walsh. 2. Alexei Borodin, Vadim Gorin, and Eric M. Rains, q-distributions on boxed plane partitions, Selecta Math. (N.S.) 16 (2010), no. 4, 731– Double-periodic waves of the focusing NLS equation 789. and rogue waves on the periodic background 3. Alexei Borodin, Discrete gap probabilities and discrete Painleve´ equa- Jinbing Chen tions, Duke Math. J. 117 (2003), no. 3, 489–542. School of Mathematics, Southeast University, Nanjing, Jiangsu 210096, 4. Alisa Knizel, Moduli spaces of q-connections and gap probabilities, In- P.R. China ternational Mathematics Research Notices (2016), no. 22, 1073–7928. [email protected] 5. Kenji Kajiwara, Masatoshi Noumi, and Yasuhiko Yamada, Geometric Dmitry E. Pelinovsky∗ aspects of Painleve´ equations, J. Phys. A 50 (2017), no. 7, 073001, Department of Mathematics, McMaster University, Hamilton, Ontario, 164. Canada, L8S 4K1 [email protected] A survey of Bi-flat F-manifolds We address Lax–Novikov equations derived from the cubic NLS Alessandro Arsie equation. Lax-Novikov equations of the lowest orders admit ex- Department of Mathematics and Statistics, plicit periodic and double-periodic solutions expressed as rational The University of Toledo, 43606, Toledo, OH, USA functions of Jacobian elliptic functions. By applying an algebraic [email protected] method which relates the periodic potentials and the squared periodic eigenfunctions of the Lax operators, we characterize explicitly Paolo Lorenzoni the location of eigenvalues in the periodic spectral problem away Dipartimento di Matematica e Applicazioni, from the imaginary axis. We show that Darboux transformations University of Milano-Bicocca, 20126 Milano, Italy with the periodic eigenfunctions remain in the class of the same [email protected] periodic waves of the NLS equation. On the other hand, Darboux transformations with the non-periodic solutions to the Lax equations produce rogue waves on the periodic background which are

25 formed in a finite region of the time-space plane. The results are the spectrum is stabilized by using an exponential weight. A-priori based on the recent papers [1, 2, 3]. estimates for the nonlinear terms of the equation governing the evolution of the perturbations of the front are obtained when perturba- 1. J. Chen and D.E. Pelinovsky, “Rogue periodic waves in the modified tions belong to the intersection of the exponentially weighted space Korteweg-de Vries equation”, Nonlinearity 31 (2018), 1955–1980. with the original space without a weight. These estimates are then 2. J. Chen and D.E. Pelinovsky, “Rogue periodic waves in the focusing used to show that in the original norm, initially small perturbations nonlinear Schrodinger¨ equation”, Proceeding A of Roy. Soc. Lond. 474 to the front remain bounded, while in the exponentially weighted (2018), 20170814 (18 pages). norm, they algebraically decay in time. 3. J. Chen and D.E. Pelinovsky, “Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background”, (2018), Asymptotic stability for spectrally stable arXiv:1807.11361 (40 pages). Lugiato-Lefever solutions in periodic waveguides Milena Stanislavova and Atanas Stefanov Formation of extreme events in NLS systems Department of Mathematics, University of Kansas [email protected], [email protected] Efstathios G. Charalampidis Department of Mathematics and Statistics, University of Massachusetts We consider the Lugiato-Lefever model of optical fibers in the pe- Amherst riodic context. Spectrally stable periodic steady states were con- Amherst, MA 01003-4515, USA structed recently in [2] and [3], also by S. Hakkaev, M. Stanislavova [email protected] and A. Stefanov, [5]. The spectrum of the linearization around such solitons consists of simple eigenvalues 0, 2α < 0, while the rest his talk will focus on the formation and spatio-temporal evolution of it is a subset of the vertical line µ : −µ = α . Assuming of extreme events, called rogue waves in nonlinear Schrodinger¨ such property abstractly, we show that{ theℜ linearized− } operator gen- (NLS) equations and discrete variants thereof. Motivated by the erates a C0 semigroup and more importantly, the semigroup obeys physics of ultracold atoms, i.e., atomic Bose-Einstein condensates (optimal) exponential decay estimates. Our approach is based on (BECs), we will attempt to address the question about what type the Gearhart-Pruss¨ theorem, where the required resolvent estimates of experimental initial conditions should be utilized for producing may be of independent interest. These results are applied to the waveforms which are strongly reminiscent of the Peregrine soli- proof of asymptotic stability with phase of the steady states. ton. To do so, we will consider the initial boundary value problem (IBVP) with Gaussian wavepacket initial data for the scalar (NLS) 1. Y.K. Chembo, C.R. Menyuk, Spatiotemporal Lugiato-Lefever formal- and novel features will be presented. In particular, it will be shown ism for Kerr-comb generation in whispering-gallery-mode resonators, Phys. Rev. A 87, (2010), 053852. that as the width of the relevant Gaussian is varied, large amplitude excitations strongly reminiscent of Peregrine, Kuznetsov-Ma 2. L. Delcey, M. Haragus, Periodic waves of the Lugiato-Lefever equation or regular solitons appear to form. This analysis will be comple- at the onset of Turing instability, Phil. Trans. R. Soc. A 376, (2018), 20170188. mented by considering the Salerno model interpolating between the discrete NLS (DNLS) and Ablowitz-Ladik (AL) models where 3. L. Delcey, M. Haragus, Instabilities of periodic waves for the Lugiato- Lefever equation, to appear in Rev. Roumaine Maths. Pures Appl. similar phenomenology is observed. Finally, and if time permits, recent results on the stability of discrete Kuznetsov-Ma solitons 4. F. Gesztesy, C. K. R. T. Jones, Y. Latushkin, M. Stanislavova, (via the use of Floquet theory) will be discussed as well. The A spectral mapping theorem and invariant manifolds for nonlinear Schrodinger¨ equations, Indiana Univ. Math. J. 49, (2000), no. 1, p. findings presented in this talk might be of particular importance 221–243. towards realizing experimentally extreme events in BECs. 5. S. Hakkaev, M. Stanislavova, A. Stefanov, On the generation of stable Kerr frequency combs in the Lugiato-Lefever model of periodic optical Stability of planar fronts in a class of reaction- waveguides, submitted, available at arXiv:1806.04821. diffusion systems 6. L. Lugiato, R. Lefever, Spatial dissipative structures in passive optical Anna Ghazaryan systems. Phys. Rev. Lett. 58, (1987), p. 2209–2211. Department of Mathematics, Miami University, Oxford, OH 45056, USA 7. R. Mandel, W. Reichel, A priori bounds and global bifurcation results [email protected] for frequency combs modeled by the Lugiato-Lefever equation. SIAM J. Appl. Math. 77 (2017), no. 1, p. 315–345. Yuri Latushkin Mathematics Department, University of Missouri, Columbia, MO 65211, Fisher-KPP dynamics in diffusive USA Rosenzweig-MacArthur and Holling-Tanner models [email protected] Hong Cai Xinyao Yang Department of Physics, Brown University, Xi’an Jiaotong-Liverpool University, Suzhou, Jiangsu, P. R. China 182 Hope Street, Providence, RI 02912, USA, [email protected] Hong [email protected] For a class of reaction-diffusion equations we study the planar Anna Ghazaryan fronts with the essential spectrum of the linearization in the direc- Department of Mathematics, Miami University, tion of the front touching the imaginary axis. At the linear level, 301 S. Patterson Ave, Oxford, OH 45056, USA

26 [email protected] symmetric operators related to abstract boundary triples. We will also discuss Hadamard’s type formulas expressing the derivative Vahagn Manukian of eigenvalues with respect to a parameter in terms of the respec- Department of Mathematical and Physical Sciences, Miami University, tive Maslov crossing forms. Applications are given to multidimen- 1601 University Blvd, Hamilton, OH 45011, USA sional Schrodinger¨ operators on periodic and star-shaped domains. [email protected] The Maslov index is a geometric characteristic defined as the signed We prove existence of traveling fronts in two known population dy- number of intersections of a path in the space of Lagrangian planes namics models, Rosenzweig-MacArthur and Holling-Tanner, and with the train of a given plane. The problem of relating this quan- investigated the relation of these fronts with fronts in scalar Fisher- tity to the spectral count is rooted in Sturm’s Theory and has a long KPP equation. More precisely, we prove existence of traveling history going back to the classical work by Arnold, Bott and Smale, fronts in a modified diffusive Rosenzweig-MacArthur predator- and has attracted recent attention of several groups of mathemati- prey model in the two situations. One situation is when the prey cians. diffuses at the rate much smaller than that of the predator. In the second situation both the predator and the prey diffuse very slowly. On some select Klein-Gordon problems: Internal Both situations can be captured as singular perturbations of the as- modes, fat tails, wave collisions and beyond sociated limiting systems. In the first situation we demonstrate a P.G. Kevrekidis clear relation of the fronts with the fronts in a scalar Fisher-KPP Department of Mathematics and Statistics equation. We show that a similar relation also holds for fronts in a University of Massachusetts, Amherst, MA 01003, USA diffusive Holling-Tanner population model. The analysis suggests [email protected] that the scalar Fisher-KPP equation may serve as a normal form for a variety of reaction-diffusion systems that rise in population In this work we will revisit the seemingly well-established story dynamics. of the φ4 kink collisions and discuss a (seemingly) fatal sign error. This will already expose some intriguing open questions for Bulk versus surface diffusion in highly amphiphilic what was previously thought to be well-known. This will serve polymer networks as a teaser for the development of further mathematical theory on the subject. However, the emphasis of the work will be on a num- Yuan Chen and Keith Promislow∗ ber of vignettes in cases that are even less well understood than Department of Mathematics, 4, namely 6, 8, 10 and 12 models. The first of these models Michigan State University φ φ φ φ φ [email protected] can have kinks with either 0 or (controllably) many internal modes. Some of the associated spectral and collisional phenomenology of Shibin Dai the relevant exponentially decaying kinks will be presented. Then, Department of Mathematics, University of Alabama we will venture into the remaining three models and unearth even [email protected] more complex features of the latter. For one thing, it is now possible to have power-law decaying kinks for which linearization Amphiphilic materials self assemble into complex networks, a fun- yields no information. In this case, many of the things we know damental example is the endoplasmic reticulum that serves as the and trust go out the window: sum ansatze¨ do not work to con- basis for intracellular transport and protein synthesis. A key prop- struct proper initial conditions for interactions; if used, they yield erty of the ER network is it grows by transport of the network ma- misleading results. Methods for evaluating interactions (including terial along the the network itself, by surface diffusion. This is variational ones etc.) do not properly work. Again, special care primarily due to the strongly hydrophobic nature of the lipids that needs to be used to unveil the power law interaction between the makes the energy of a single lipid in solvent prohibitively high. We kinks. We will thus attempt to provide a glimpse of the current state discuss several approaches to model this phenomena which include of understanding and to offer a number of intriguing directions for wells with limited smoothness that induce compactly supported bi- future study. layers. Regularizations that include asymptotically strong convexity support small densities of background lipids, but with enhanced Steady concentrated vorticity and its stability of the mobility that induces significant bulk flux. We show that balancing 2-dim Euler equation on bounded domains strong convexity with degenerate mobility arrives at a model with limited background density and weak bulk flux. Chongchun Zeng Georgia Tech Recent results on application of the Maslov index in [email protected] spectral theory of differential operators On a smooth bounded domain Ω R2, we consider steady solu- Yuri Latushkin and Selim Sukhtaiev tions of the incompressible Euler⊂ equation with concentrated vor- Department of Mathematics, University of Missouri, Columbia, MO 65211, ticity. More precisely, with prescribed integer n > 0, vortical do- USA main sizes r1,..., rn > 0, and vorticity strengths µ1,..., µn = 0, Department of Mathematics, Rice University, Houston, TX 77005, USA we seek steady vorticity distributions in the form of 6 [email protected] and [email protected] n ω = ∑j=1 ωj(x) where 1.) the vortical domains satisfy Ω = supp(ω ) B(p , 2r ǫ), We describe relations between the Maslov index and the count- j j ⊂ j j ing function for the spectrum of selfadjoint extensions of abstract Ω = πr2ǫ2, with 0 < ǫ << 1 and distinct p ,..., p Ω; | j| j | | 1 n ∈

27 and 2.) µj = ωjdx. Hamiltonian systems via the Maslov index Since the dynamicsR of localized vorticity is approximated by the Peter Howard∗ point vortex dynamics, we take p1,..., pn close to a non-degenerate steady configuration of the{ point vortex} system in Ω with pa- Department of Mathematics, Texas A&M University, College Station, TX 77843, USA rameters µ ,..., µn. Through a perturbation method applied to 1 [email protected] Ωj parametrized by conformal mappings, we obtained two types 2 of steady solutions with smooth ∂Ωj being O(ǫ ) perturbations Alim Sukhtayev to ∂B(pj, rjǫ): a.) infinitely many piecewise smooth solutions Department of Mathematics, Miami University, Oxford, OH 45056, USA ω C0,1(Ω); and b.) a unique steady vortex patch with piecewise [email protected] ∈ µj Ω constant vorticity, i.e. ωj = 2 2 χ( j). Moreover, the spectral Working with a general class of linear Hamiltonian systems on πrj ǫ and evolutionary properties (stability, exponential trichotomy, etc.) bounded intervals, we show that renormalized oscillation results of the linearized vortex patch dynamics at the latter is determined can be obtained in a natural way through consideration of the Maslov by those of the linearized point vortex dynamics at the steady con- index associated with appropriately chosen paths of Lagrangian C2n figuration p1,..., pn . This is a joint work with Yiming Long subspaces of . and Yuchen{ Wang at Nankai} University. Stability of traveling waves in a model for a thin A bifurcation analysis of standing pulses and the liquid film flow Maslov index Stephane´ Lafortune Paul Cornwell Department of Mathematics Johns Hopkins Applied Physics Laboratory College of Charleston [email protected] Charleston, SC 29424 [email protected] Claire Kiers∗ The University of North Carolina at Chapel Hill Anna Ghazaryan [email protected] Department of Mathematics Miami University, 301 S. Patterson Ave The Maslov index is a powerful and insightful tool that can be used Oxford, OH 45056, USA, Ph. 1-513-529-0582 to determine the stability of solutions for PDEs. We demonstrate [email protected] the robustness of a certain method of Maslov index calculation by applying it to standing pulse solutions of a three-component Vahagn Manukian reaction-diffusion system. The Maslov index shows exactly why Department of Mathematics the stability of a wave changes at a bifurcation due to the appear- Miami University, 301 S. Patterson Ave ance of a conjugate point. The calculation also indicates that the Oxford, OH 45056, USA Maslov index can see stable eigenvalues. [email protected] We consider a model for the flow of a thin liquid film down an Dynamics of frequency combs modeled by the inclined plane in the presence of a surfactant. The model is known Lugiato-Lefever equation to possess various families of traveling wave solutions. We use Mariana Haragus a combination of analytical and numerical methods to study the Institut FEMTO-ST, Univ. Bourgogne-Franche Comte,´ France stability of the traveling waves. We show that for at least some [email protected] of these waves the spectra of the linearization of the system about them are within the closed left-half complex plane. The Lugiato-Lefever equation is a nonlinear Schrodinger-type¨ equation with damping, detuning and driving, derived in nonlinear op- Rigorous verification of wave stability tics by Lugiato and Lefever (1987). While extensively studied in the physics literature, there are relatively few rigorous mathemat- Blake Barker∗ and Taylor Paskett ical studies of this equation. Of particular interest for the physi- Brigham Young University cal problem is the formation and the dynamical behavior of Kerr [email protected] and [email protected] frequency combs (optical signals consisting of a super-position of Kevin Zumbrun modes with equally spaced frequencies). The underlying mathe- Indiana University matical questions concern the existence and the stability of certain [email protected] particular steady solutions of the Lugiato-Lefever equation. In this talk, I’ll focus on periodic steady waves for which I’ll show how We discuss recent work regarding rigorous verification of stabil- tools from bifurcation theory can be used to study their existence ity properties of traveling waves. In particular, we describe our and stability. work developing computer assisted proof techniques to evaluate the Evans function in order to prove spectral stability of waves in Renormalized oscillation theory for linear the one-dimensional non-isentropic Navier-Stokes equations with an ideal, polytropic gas equation of state. For this system, spectral

28 stability implies nonlinear stability. Proving spectral stability is the [email protected] and [email protected] last piece of a program begun over 20 years ago for establishing the Kevin Zumbrun stability of traveling waves in this model. Indiana University Stability of travelling waves in a haptotaxis model [email protected] Kristen Harley, Peter van Heijster and Graeme Pettet By reduction to a generalized Sturm Liouville problem, we establish spectral stability of hydraulic shock profiles of the Saint- Queensland University of Technology Venant equations for inclined shallow-water flow, over the full pa- [email protected], [email protected] and rameter range of their existence, for both smooth-type profiles and [email protected] discontinuous-type profiles containing subshocks. Together with Robby Marangell∗, Tim Roberts and Martin Wechselberger work of Mascia-Zumbrun and Yang-Zumbrun, this yields linear University of Sydney and nonlinear H2 L1 H2 stability with sharp rates of decay in ∩ → [email protected], [email protected] and mar- Lp, p 2, the first complete stability results for large-amplitude ≥ [email protected] shock profiles of a hyperbolic relaxation system.

I will examine the spectral stability of travelling waves in a hap- Turning point principle for the stability of stellar totaxis model for tumor invasion [1]. In the process, I will show models how to apply Lienard coordinates to the linearised stability problem and show some further developments in a geometrically in- Zhiwu Lin spired method for numerically computing the point spectrum of a School of Mathematics linearised operator. Georgia Institute of Technology Atlanta, GA, 30332 1. K.E. Harley, P. v Heijster, R. Marangell, G. J. Pettet, and M. Wech- [email protected] selberger, Existence of traveling wave solutions for a model of tumor invasion. SIAM Journal on Applied Dynamical Systems. 13 1. (2014), I will discuss some recent results on the linear stability criterion of 366-396. spherically symmetric equilibria of several stellar models, including Euler-Poisson, Einstein-Euler and Einstein-Vlasov models. For Euler-Poisson and Einstein-Euler models, a turning point principle Solitary waves for weakly dispersive equations with for the sharp stability criterion will be given. For Vlasov-Einstein inhomogeneous nonlinearities model, the stability part of the turning point principle is obtained Ola Maehlen and the linear instability in the strong relativisitic limit will also be Department of Mathematical Sciences, discussed. For these models, a combination of first order and 2nd Norwegian University of Science and Technology, order Hamiltonian formulations is used to derive the stability crite- 7491 Trondheim, Norway rion and study the linearized equation for initial data in the energy [email protected] space. This is joint work with Chongchun Zeng (on Euler-Poisson) and with Hadzic and Rein (on Einstein-Euler and Einstein-Vlasov). We show existence of solitary-wave solutions to the equation Solitary wave solutions of a Whitham-Bousinessq u +(Lu n(u)) = 0 , t − x system for weak assumptions on the dispersion L and the nonlinearity n. Dag Nilsson∗ The symbol m of the Fourier multiplier L is allowed to be of low Norwegian University of Science and Technology positive order (s > 0), while n need only be locally Lipschitz and [email protected] asymptotically homogeneous at zero. We shall discover such solu- Evgueni Dinvay tions in Sobolev spaces contained in H1+s. University of Bergen 1. M. N. Arnesen, Existence of solitary-wave solutions to nonlocal equa- [email protected] tions, Discrete Contin. Dyn. Syst., 36 (2016), pp. 3483-3510. We consider a Whitham-Boussinesq type system that was recently 2. M. Ehrnstrom,¨ M. D. Groves, and E. Wahln, On the existence and sta- introduced in [2] as a fully dispersive model for bidirectional sur- bility of solitary-wave solutions to a class of evolution equations of face waves. Moreover, the system was shown to be locally well Whitham type, Nonlinearity, 25 (2012), pp. 2903-2936. posed in [1]. 3. M.I.Weinstein, Existence and dynamic stability of solitary wave solu- In this paper we prove existence of solitary wave solutions of this tions of equations arising in long wave propogation, Comm. Partial system, and in addition show that these solutions are approximated Differential Equations, 12 (1987), pp. 1133-1173. by scalings of KdV-type solitary waves. This is proved using a variational approach, where solitary waves are identified as critical Spectral stability of hydraulic shock profiles points of a certain functional, and proceed to show that there exist minimizers of this functional, using the concentration-compactness Alim Sukhtayev∗ and Zhao Yang theorem. Miami University and Indiana University

29 1. Dinvay, E., On well-posedness of a dispersive system of the Whitham– The Chen-Mckenna suspension bridge equation is a nonlinear PDE Boussinesq type, Applied Mathematics Letters, Volume 88, February which is 2nd order in time and is used to model traveling waves on 2019, Pages 13-20. a suspended beam. For certain parameter regimes, it admits multi- 2. Dinvay, E., Dutykh, D., Kalisch, H. A comparative study of bi- pulse traveling wave solutions, which are small perturbations of the directional Whitham systems. Applied Numerical Mathematics. stable, primary pulse solution. Linear stability of these multi-pulse solutions is determined by eigenvalues near the origin representing Viewing spectral problems through the lens of the the interaction between the individual pulses. Linearization about Krein matrix these multi-pulse solutions yields a quadratic eigenvalue problem. To study this problem, we use a reformulated version of the Krein Todd Kapitula∗ matrix, which was presented by Todd Kapitula in a previous talk. Department of Mathematics and Statistics Using an appropriate leading order expansion of the Krein matrix, Calvin College we are able to give analytical criteria for the stability of these multi- [email protected] pulse solutions. We also present numerical results to support our analysis. Ross Parker Division of Applied Mathematics Coriolis forces and particle trajectories for waves Brown University with stratification and vorticity ross [email protected] Miles H. Wheeler Bjorn¨ Sandstede ∗ University of Vienna Division of Applied Mathematics Faculty of Mathematics Brown University Oskar-Morgenstern-Platz 1 bjorn [email protected] 1080 Wien, Austria When considering the problem of finding point spectrum for the [email protected] linearization about a wave for a Hamiltonian system, it is of inter- In recent years there has been much mathematical interest in gen- est to not only find those eigenvalues with positive real part, but eralizations of the classical water wave problem which take into also those purely imaginary eigenvalues with negative Krein sig- account the Coriolis force due to the rotation of the Earth, and in nature. The Krein matrix is a meromorphic-valued function of the particular in a two-dimensional model for waves traveling along spectral parameter which has the property that it is singular. More- the equator. In the first part of this talk we will observe that, for over, it can be constructed so that the Krein signature of purely waves which travel at a constant speed, the Coriolis terms in this imaginary eigenvalues can be graphically determined via the sign two-dimensional model can in fact be completely removed by a of a derivative. Here we construct the Krein matrix for linear and change of variables. This fact does not seem to appear in the exist- quadratic eigenvalue problems, and show how it can be used: ing literature, and it allows for many proofs and calculations to be to locate possible Hamiltonian-Hopf bifurcations (collision dramatically simplified. • of purely imaginary eigenvalues with opposite Krein signa- If time permits we will also discuss ongoing work with Biswajit ture) Basu (University of Vienna) on the particle trajectories and related properties of solitary waves with vorticity and/or stratification. locate small eigenvalues which arise through some type of • bifurcation. On the stability of solitary water waves with a point More details associated with the applications will be presented by vortex Ross Parker in a subsequent talk. Kristoffer Varholm∗ Department of Mathematical Sciences, Norwegian University of Science Spectral stability of multi-pulses via the Krein and Technology, matrix 7491 Trondheim, Norway [email protected] Ross Parker∗ Division of Applied Mathematics Erik Wahlen´ Brown University Centre for Mathematical Sciences, Lund University, PO Box 118, 22100 ross [email protected] Lund, Sweden Todd Kapitula [email protected] Deparment of Mathematics and Statistics Samuel Walsh Calvin College Department of Mathematics, University of Missouri, Columbia, MO 65211, [email protected] USA Bjorn¨ Sandstede [email protected] Division of Applied Mathematics This paper investigates the stability of solutions of the steady water Brown University wave problem with a submerged point vortex. We prove that waves bjorn [email protected]

30 with sufficiently small amplitude and vortex strength are conditionally orbitally stable. In the process of obtaining this result, we de- In this talk, I will discuss Anderson localization for Bernoulli–type velop a quite general stability theory for bound state solutions of random models on metric and discrete radial graphs. Dynamical a large class of Hamiltonian systems in the presence of symmetry. localization is proved on compact intervals contained in the com- This is in the spirit of the seminal work of Grillakis, Shatah, and plement of a discrete set of exceptional energies. This is based on Strauss [2], but with hypotheses that are relaxed in a number of joint work with D. Damanik and J. Fillman. ways necessary for the point vortex system, and for other hydro- dynamical applications more broadly. In particular, we are able to allow the Poisson map to be state-dependent, and to have merely A Maslov index for non-Hamiltonian systems dense range. Graham Cox∗ As a second application of the general theory, we consider a fam- Department of Mathematics and Statistics ily of nonlinear dispersive equations that includes the generalized Memorial University of Newfoundland KdV and Benjamin–Ono equations. The stability (or instability) St. John’s, NL Canada of solitary waves for these systems has been studied extensively, [email protected] notably by Bona, Souganidis, and Strauss [1], who used a modification of the GSS method. We provide a new, more direct proof of The Maslov index is a powerful and well known tool in the study of these results that follows as a straightforward consequence of our Hamiltonian systems, providing a generalization of Sturm-Liouville abstract theory. theory to systems of equations. For non-Hamiltonian systems, one no longer has the symplectic structure needed to define the Maslov 1. J. L. BONA, P. E. SOUGANIDIS, AND W. A. STRAUSS, Stability and index. In this talk I will describe a recent construction of a “gener- instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. alized Maslov index” for a very broad class of differential equa- London Ser. A, 411 (1987), pp. 395–412. tions. The key observation is that the manifold of Lagrangian 2. M. GRILLAKIS,J.SHATAH, AND W. STRAUSS, Stability theory of planes can be enlarged considerably without altering its topolog- solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 ical structure, and in particular its fundamental group. This is (1987), pp. 160–197. joint work with Tom Baird, Paul Cornwell, Chris Jones and Robert Marangell.

Modulational instability of viscous fluid conduit Nonlinear stability of layers in precipitation models periodic waves Alin Pogan Mathew A. Johnson and Wesley R. Perkins∗ Miami University Department of Mathematics, University of Kansas, 1460 Jayhawk Boule- Department of Mathematics vard, Lawrence, KS 66045 301 S. Patterson Ave. [email protected] and [email protected] Oxford, OH 45056, USA [email protected] The Whitham modulation equations are widely used to describe the behavior of modulated periodic waves on large space and time Standing layers are known to exist in models arising in chemical scales; hence, they are expected to give insight into the stability conversion equations in closed reactors. We explore various con- of spatially periodic structures. However, the derivation of these cepts of stability such as spectral, linear and nonlinear stability. equations are based on formal asymptotic (WKB) methods, thus removing a layer of rigor that would otherwise support their pre- Periodic traveling hydroelastic waves dictions. In this study, we aim at rigorously verifying the predic- David M. Ambrose tions of the Whitham modulation equations in the context of the so- called conduit equation, a nonlinear dispersive PDE governing the Department of Mathematics, Drexel University evolution of the circular interface separating a light, viscous fluid Philadelphia, PA 19104 USA rising buoyantly through a heavy, more viscous, miscible fluid at [email protected] small Reynolds numbers. In particular, using rigorous spectral per- Recent work of the presenter, Benjamin Akers, and J. Douglas turbation theory, we connect the predictions of the Whitham mod- Wright developed a formulation for traveling waves in interfacial ulation equations to the rigorous spectral (in particular, modula- fluid dynamics which allows the free fluid surface to have multi- tional) stability of the underlying wave trains. This makes rigorous valued height. This formulation was shown to be amenable to recent formal results on the conduit equation obtained by Maiden efficient computation of bifurcation branches as well as develop- and Hoefer. ment of local and global bifurcation theory for interfacial capillary- gravity waves. All of this work has then been adapted to the hy- Localization for Anderson models on tree graphs droelastic case, allowing elastic effects at the fluid interface, such David Damanik, Selim Sukhtaiev* as those present in ice sheets, cellular membranes, or thin structures such as flags. With Akers and David Sulon, we have proved Department of Mathematics, Rice University, Houston, TX 77005, USA existence of families of traveling waves and computed the same. [email protected] The analysis in the hydroelastic case also proves existence in the Jake Fillman Wilton ripple case, in which the kernel of the relevant linearization Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA is two-dimensional.

31 Invariant Manifolds of Traveling Waves of the 3D Asymptotic stability of the Novikov peakons Gross-Pitaevskii Equation in the Energy Space Ming Chen∗ Jiayin Jin and Zhiwu Lin Department of Mathematics, University of Pittsburgh, PA 15260, USA School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332 [email protected] [email protected] and [email protected] Wei Lian Chongchun Zeng College of Science, Harbin Engineering University, Harbin 150001, P. R. School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332 China [email protected] lianwei [email protected] We study the local dynamics near general unstable traveling waves Dehua Wang of the 3D Gross-Pitaevskii equation in the energy space by con- Department of Mathematics, University of Pittsburgh, PA 15260, USA structing smooth local invariant center-stable, center-unstable and [email protected] center manifolds. We also prove that (i) the center-unstable mani- Runzhang Xu fold attracts nearby orbits exponentially before they go away from the traveling waves along the center or unstable directions and (ii) College of Science, Harbin Engineering University, Harbin 150001, P. R. if an initial data is not on the center-stable manifolds, then the for- China ward orbit leaves traveling waves exponentially fast. Furthermore, [email protected] under an additional non-degeneracy assumption, we show the or- We prove that the peakons of the Novikov equation are asymp- bital stability of the traveling waves on the center manifolds, which totically H1-stable in the class of functions with the momentum also implies the uniqueness of the local invariant manifolds. Our density m := u u belonging to the set of non-negative finite − xx method based on a geometric bundle coordinate Radon measure M+. The key ingredient in the argument is a Liou- ville property for the uniformly (up to translation) almost localized Generalized solitary wave solutions of the capillary- global solutions satisfying m M+, that is, we prove that such a gravity Whitham equation solution must be a peakon. ∈ J. Douglas Wright Drexel University [email protected] SESSION 8: “Dispersive wave equations and their soliton interac- “Whitham” equations have enjoyed a recent resurgence of popu- tions: Theory and applications” larity as models for free surface fluid flows. They are, roughly speaking, obtained by using the full linear part of the appropriate Backward behavior of a dissipative KdV equation Euler equation together with a simpler “KdV”- type nonlinearity. Generalized solitary waves are traveling wave solutions which are Yanqiu Guo and Edriss S. Titi the superposition of a classical solitary wave with a “small beyond Florida International University and Texas A&M University all orders” periodic wave. Such waves are known to exist for the yanguo@fiu.edu and [email protected] full capillary-gravity wave problem and in this talk we discuss re- In this talk, I will discuss the backward-in-time behavior of a KdV cent work on establishing their existence for the “Whithamized” equation influenced by dissipation and source terms. In particular, version. (This work is joint with A. Stefanov and M. Johnson.) we prove that every solution of a KdV-Burgers-Sivashinsky type equation blows up in the energy space, backward in time, if the so- Modulational dynamics of spectrally stable lution does not belong to the global attractor. In addition, we pro- Lugiato-Lefever periodic waves vide some physical interpretation of various backward behaviors Mathew A. Johnson∗ and Wesley R. Perkins of several perturbations of the KdV equation by studying explicit University of Kansas soliton-type solutions. This is a joint work with E. S. Titi. [email protected] and [email protected] On the energy cascade of acoustic wave turbulence: Mariana Haragus Beyond Kolmogorov-Zakharov solutions Univ. Bourgogne Franche?Comt?e [email protected] Avy Soffer and Minh-Binh Tran Mathematics Department, Rutgers University, New Brunswick, NJ 08903 e consider the dynamics of periodic steady waves of the Lugiato- USA Lefever equation, which is an equation derived in nonlinear optics Department of Mathematics, Southern Methodist University, Dallas, TX of NLS type with damping, detuning, and driving. Using Floquet- 75275, USA Bloch theory, we are able to describe at the linear level the modula- [email protected] and [email protected] tional dynamics of periodic steady waves that are spectrally stable to general bounded perturbations on the line. We will also discuss In weak turbulence theory, the Kolmogorov-Zakharov spectra is a important challenges towards describing the associated nonlinear class of time-independent solutions to the kinetic wave equations. dynamics.

32 In this paper, we construct a new class of time-dependent solu- where the asymptotic state of the system can be far from the initial tions to those kinetic equations. These solutions exhibit the inter- state in parameter space. Specifically, if we let a narrow soliton esting property that the energy is cascaded from small wavenum- state with initial velocity υ0 of order 1 to interact with an exter- bers to large wavenumbers. We can prove that starting with a reg- nal potential V(x), then the velocity υ+ of outgoing solitary wave ular initial condition whose energy at the infinity wave number in infinite time will in general be very different from υ0. In con- p = ∞ is 0, as time evolves, the energy is gradually accumulated trast to our present work, previous results proved that the soliton is at p = ∞ . Finally, all the energy of the system is concentrated asymptotically stable so that υ+ stays close to υ0 for all times. at { p = ∞} and the energy function becomes a Dirac function { } at infinity Eδ p=∞ , where E is the total energy. The existence of Stable blow-up dynamics in the generalized this class of solutions{ } is, in some sense, a rigorous mathematical L2-critical Hartree equation proof based on the kinetic description for the energy cascade phe- Svetlana Roudenko , Anudeep Kumar Arora and Kai Yang nomenon. We restrict our attention in this paper to the statistical ∗ description of acoustic waves. However, the technique is quite ro- Department of Mathematics and Statistics, DM430 bust and can be applied to other types of wave turbulence kinetic Florida International University, Miami, FL 33199 equations. sroudenko@fiu.edu and [email protected] and yangk@fiu.edu Keyword: weak turbulence theory, acoustic wave, Kolmogorov- We study stable blow-up dynamics in the nonlinear Schrodinger¨ Zakharov spectra, energy cascade (NLS) equation and generalized Hartree equation in the L2-critical regime. The NLS equation is with pure power nonlinearity iut + ∆u + u 2σu = 0, and the generalized Hartree equation is a Dynamics of a heavy quantum tracer particle in a Schrodinger-type¨ | | equation with a nonlocal, convolution-type non- (d 2) p p 2 Bose gas linearity in dimension d: iut + ∆u + x − − u u − u = | | ∗| | | | Thomas Chen∗ 0, p 2. ≥ 2 Department of Mathematics First, we consider the L -critical case of the NLS equation in di- University of Texas at Austin mensions 4 d 12 and give a numerically-assisted proof of the ≤ ≤ [email protected] spectral property, which completes the log-log blow-up theory of Merle-Raphael for the mass-critical NLS up to the dimension 12. Avy Soffer We next consider the generalized Hartree equation in the L2-critical Department of Mathematics regime and investigate spectral properties needed to understand the Rutgers University blow-up dynamics of the solutions. We then show that similar to [email protected] NLS, solutions with mass slightly above the corresponding ground We consider the dynamics of a heavy quantum tracer particle cou- state and negative energy, will blow-up with the “log-log” dynam- pled to a non-relativistic boson field in R3. The pair interactions of ics in the 3d generalized Hartree equation. the bosons are of mean-field type, with coupling strength proportional to 1/N where N is the expected particle number. Assuming Knocking out teeth in one-dimensional periodic NLS: that the mass of the tracer particle is proportional to N, we derive Local and global wellposedness results generalized Hartree equations in the limit where N tends to infinity. L. Chaichenets, D. Hundertmark Moreover, we prove the global well-posedness of the associated Karlsruhe Institute of Technology Cauchy problem for sufficiently weak interaction potentials. This [email protected] and [email protected] is joint work with Avy Soffer (Rutgers University). P. Kunstmann 1. T. Chen and A. Soffer, Mean field dynamics of a quantum tracer par- Karlsruhe Institute of Technology ticle interacting with a boson gas, J. Funct. Anal., 276 (3), 971-1006, [email protected] 2019. N. Pattakos Karlsruhe Institute of Technology Soliton Potential interaction of NLS in R3 [email protected] Qingquan Deng In this talk local and global wellposedness results of the 1-dimension- The School of Mathematics and Statistics, Central China Normal Univer- al nonlinear Schrodinger equation sity α 1 iut uxx u − u = 0 152 Luoyu Street, Wuhan, 430079, P. R. China − ±| | s s [email protected] will be discussed with initial data u0 H (R)+ H (T), where s 0, α (1, 5) and T is the one dimensional∈ torus. We consider the following equation ≥ ∈ In the case of the cubic nonlinearity, α = 3, local existence of weak solutions in the extended sense is shown through a differentiation i∂ ψ = 1/2∆ψ + Vψ F ( ψ 2)ψ. t − − ǫ | | by parts argument and in the case of the quadratic nonlinearity, In this work we mainly focus on the dynamics and scattering of a α = 2, global existence is established with the use of Strichartz narrow soliton of the above NLS equation with a potential in R3, type estimates and a conserved quantity argument.

33 1. L. CHAICHENETS,D.HUNDERTMARK, P. KUNSTMANNAND N. of the fermions is close to a Slater determinant with a certain semi- PATTAKOS, Knocking out teeth in one-dimensional periodic NLS. classical structure. We prove that the many-body state approxi- arXiv:1808.03055 (2018), submitted to Analysis and PDE. mately retains its Slater determinant and semiclassical structure at 2. L. CHAICHENETS,D.HUNDERTMARK, P. KUNSTMANNAND N. later times and that its time evolution can be approximated by the PATTAKOS, Global wellposedness of the quadratic NLS in one dimen- fermionic Schrodinger-Klein-Gordon¨ equations. These are a non- sion with initial data in L2(R)+ H1(T). preprint (2019). linear system of two equations: a nonlinear Schrodinger¨ equation and a wave equation with source term. We prove the convergence Conservation laws and asymptotics for the wave for reduced densities with explicit rates and for all semiclassical equation times. Stefanos Aretakis 1. N. Leopold and S. Petrat, Mean-field Dynamics for the Nelson Model University of Toronto, Toronto, Canada with Fermions, Preprint, [arXiv:1807.06781] (2018). [email protected] The effect of threshold energy obstructions on the We will present results regarding the precise late-time asymptotics L1 L∞ dispersive estimates for some Schrodinger¨ for solutions to the wave equation on black hole backgrounds. Our → method relies on purely physical space techniques and makes use type equations of conservation laws for the wave equation along null hypersur- M. Burak Erdogan˘ faces. We will present results for both extremal and sub-extremal University of Illinois at Urbana Champaign black hole backgrounds ([1, 2]). In the case of extremal black [email protected] holes, we will show that deriving precise asymptotics leads to some interesting conclusions such as the existence of observational sig- Willam. R. Green natures of extremal event horizons ([3]). Rose-Hulman Institute of Technology [email protected] 1. Yannis Angelopoulos and Stefanos Aretakis and Dejan Gajic, Late- Ebru Toprak ∗ time asymptotics for the wave equation on extremal Reissner- Rutgers University Nordstrom¨ backgrounds, arXiv:1807.03802 (2018) [email protected] 2. Yannis Angelopoulos and Stefanos Aretakis and Dejan Gajic, Late- time asymptotics for the wave equation on spherically symmetric, sta- In this talk, I will discuss the differential equation iut = Hu, tionary spacetimes, to appear in Advances in Mathematics, 323 (2018), H := H0 + V , where V is a decaying potential and H0 is a Lapla- 529-621 cian related operator. In particular, I will focus on when H0 is 3. Yannis Angelopoulos and Stefanos Aretakis and Dejan Gajic, Horizon Laplacian, Bilaplacian and Dirac operators. I will discuss how the hair of extremal black holes and measurements at null infinity , Phys. threshold energy obstructions, eigenvalues and resonances, effect 1 ∞ itH Rev. Lett., 121 (2018), 131102. the L L behavior of e Pac(H). The threshold obstructions are known→ as the distributional solutions of Hψ = 0 in certain dimension dependent spaces. Due to its unwanted effects on the Derivation of the Schrodinger-Klein-Gordon¨ dispersive estimates, its absence has been assumed in many works. equations I will mention our previous results on Dirac operator, [1, 2] and Nikolai Leopold recent results on Bilaplacian operator, [3] under different assump- IST Austria (Institute of Science and Technology Austria), Am Campus 1, tions on threshold energy obstructions. 3400 Klosterneuburg, Austria. 1. Erdogan,˘ M. B., and Green, W. R., Toprak, E. Dispersive estimates [email protected] for Dirac operators in dimension three with obstructions at threshold Soren¨ Petrat∗ energies , to appear American Journal of Mathematics, Jacobs University, Department of Mathematics, Campus Ring 1, 28759 2. Erdogan,˘ M. B., and Green, W. R., Toprak, E. Dispersive estimates Bremen, Germany. for massive Dirac operators in dimension two ,J. Differntial Equations [email protected] (2018), Volume 264, 5802–5837. 3. Green, W. R., Toprak, E. On the Fourth order Schrodinger equation This talk is about an example of how to derive non-linear Schro-¨ in four dimensions: dispersive estimates and zero energy resonances, dinger equations in a mathematically rigorous way, starting from arxiv.org/abs/1810.03678. the linear interacting many-body Schrodinger¨ equation. Here, I will present the recent result [1] where we start with the Quantization of energy of blow up for wave maps Nelson model with ultraviolet cutoff. This model is linear and describes a quantum system of non-relativistic particles coupled to Hao Jia a positive or zero mass quantized scalar field. We take the non- University of Minnesota relativistic particles to obey Fermi statistics and discuss the time [email protected] evolution in a mean-field limit of many fermions which is coupled The two dimensional wave map equation is an important geometric to a semiclassical limit. At time zero, we assume that the bosons wave equation. Soliton and dispersion are two fundamental fea- of the radiation field are close to a coherent state and that the state tures for wave maps. We will report a recent idea to show that

34 when a wave map blows up, all the concentrated energy is in the In this talk, I will discuss a two-dimensional nonlinear dispersive form of traveling waves. PDE arising in the study of water waves called the Davey-Stewartson system (DS), which is formally similar to the L2-critical cubic non- Local smoothing estimates for Schrodinger¨ equations linear Schrodinger¨ equation (NLS) but differs by an additional non- on hyperbolic space and applications local term. Specifically, I will discuss recent work on the global well-posedness and scattering for a particular case of DS with ini- Jonas Luhrmann¨ tial data in the critical L2 space, which is inspired by Benjamin Johns Hopkins University Dodson’s breakthrough work on the cubic NLS. Finally, I will dis- [email protected] cuss the question of the rigorous justification of DS as a multiple We establish frequency-localized local smoothing estimates for scales approximation for wave packet solutions to the water waves Schrodinger¨ equations on hyperbolic space. The proof is based on equation. the positive commutator method and a heat flow based Littlewood- Paley theory. Our results and techniques are motivated by appli- Semi-linear Schrodinger’s¨ equation with random cations to the problem of stability of solitary waves to nonlinear time-dependent potentials Schrodinger-type¨ equations on hyperbolic space. Marius Beceanu∗ This is joint work with Andrew Lawrie, Sung-Jin Oh, and Sohrab University at Albany SUNY Mathematics and Statistics Department Shahshahani. 1400 Washington Ave., Albany, NY 12222, USA [email protected] Nonlinear waves on vortex filaments in quantum Avy Soffer liquids: A geometric perspective Rutgers University Department of Mathematics Scott A. Strong and Lincoln D. Carr 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA [email protected] Department of Applied Mathematics and Statistics and Department of Physics Jurg¨ Frohlich¨ Colorado School of Mines Institute for Advanced Study School of Mathematics [email protected] and [email protected] 1 Einstein Drive, Prnceton, NJ 08540, USA [email protected] A vortex filament is modeled as a one-dimensional region of a quantum liquid about which the otherwise irrotational fluid circu- This talk will be a presentation of results, obtained together with lates. The vortex filament equation, i.e., the local induction ap- Jurg¨ Frohlich¨ and Avy Soffer, pertaining to the semi-linear Schrodinger¨ proximation, asserts that points on a vortex filament are transported equation with random time-dependent potential by the velocity field in the direction of the local binormal vector i∂ ψ ∆ψ + V ψ(x, t)ψ = N(ψ). and at a speed proportional to local curvature. Its simplest non- t − ω trivial prediction is that vortex rings with smaller curvature (larger In general, the interaction between a nonlinear term N(ψ) and the radius) travel slower than those with larger curvature. That said, bound states of a time-dependent linear potential can be compli- the result should be thought of as an arclength conserving flow cated to describe. There exist small standing-wave solutions and which evolves the curvature and torsion variables according to an growth in norm of the solutions is possible. However, for the case integrable Schrodinger equation. While this allows the vortex line of a random time-dependent short-range potential on Euclidean to support a wide variety of nonlinear waves, the integrability is space, driven by a Markov process, we show that, with proba- thought to restrict energy transfer between helical Kelvin modes. bility one, all solutions scatter (i.e. nonlinear wave operators are In this talk, we go beyond the local induction approximation by bounded) and disperse at the same rate as for the free equation. defining corrections which lead to a non-Hamiltonian evolution of the curvature and torsion variables. These corrections are asso- Long time dynamics for nonlinear dispersive equa- ciated with an emergent curvature gain/loss mechanism and en- tions hanced dispersion on the vortex medium. Altogether we find that Baoping regions of localized curvature seek to transport their bending into Peking University the vortex in the form of helical Kelvin waves, which provides a [email protected] necessary ingredient for modeling vortex dynamics in turbulent ultracold quantum fluids. Dispersive equations usually admit solutions with quite different asymptotic behaviors, such as scattering solutions and solitons. So Global well-posedness and scattering for the Davey- it is rather difficult to describe the long time dynamics for general Stewartson system at critical regularity solutions. In this talk, we will discuss few cases for which we are able to get a definite answer. Matthew Rosenzweig Department of Mathematics Global well-posedness for mass-subcritical NLS in University of Texas at Austin critical Sobolev space 2515 Speedway, Stop C1200 Austin, TX 78712 [email protected] Yifei Wu Center for Applied Mathematics

35 Tianjin University between two different values of the bifurcation parameter while [email protected] expanding the region of localization and hence ascending in norm. The mechanism that drives snaking in PDEs has been understood In this talk, we consider the mass-subcritical nonlinear Schrodinger by analyzing the evolution of the ordinary differential equation in equation. It was known that the solution is global if the initial the spatial variable governing steady-state solutions to the PDE. data is small in critical Sobolev space, or the solution is uniformly In this talk we extend this theory to lattice dynamical systems by bounded in whole lifespan in critical Sobolev space. In this talk, we showing that the associated steady-state equations in this context show that if any initial data in critical Sobolev space with compact can be written as a discrete dynamical system. We can then inter- suppoerted, then the corresponding solution is global. pret localized solutions to the lattice system as homoclinic orbits of the associated discrete dynamical system, and show that the bi- Higher order corrections to mean field dynamics of furcation structure is determined by bifurcations of nearby hetero- Bose cold gases clinic orbits. We supplement these results with examples from a Peter Pickl well-studied bistable lattice differential equation which has been Mathematical Institute LMU Munich// Theresienstr. 39//80333 Muenchen the focus of many works to date. [email protected] Growing stripes, with and without wrinkles It is well known that the the dynamics of ultra-cold Bose gases in the weak coupling regime is given by its respective mean-field Ryan Goh∗ limit, i.e. the Hartree equation. Recent developments in mathemat- Department of Mathematics and Statistics ical research made it possible to prove the validity of the next order Boston University correction, the Boguliubov dynamics, in many situations. [email protected] While convergence to the Hartree equation is typically proven to The interplay between growth processes and spatial patterns has be valid in trace norm, it has been shown by several authors that arisen as a topic of recent interest in many fields, such as directional the N-body solution is close to the solution of the Boguliubov quenching in alloy melts, growing interfaces in biological systems, 2 time evolution in L . In contrast to the mean-field description, the moving masks in ion milling, eutectic lamellar crystal growth, and Boguliubov time evolutions takes pair correlations into account. traveling reaction fronts, where such processes have been shown to In this talk I will prove the validity of higher order corrections in select spatially periodic patterns, and mediate the formation of de- the high density limit. We shall show that the rate of convergence fects. Mathematically, they can be encoded in a step-like parameter convergence gets better when higher order correlations are taken dependence that allows patterns in a subset of the spatial domain, into account. and suppresses them in the complement, while the boundary of the The estimates hold even in situations where the volume and the pattern-forming region propagates with fixed normal velocity. density of the gas go to infinity. It is a joint work with Lea Boß- In this talk, I will show how techniques from dynamical systems, mann, Natasa Pavlovic, and Avy Soffer based on [1] Our result is functional analysis, and numerical continuation, can be used to similar to but more explicit than recent findings by Paul and Pul- study the effect of these traveling heterogeneities on patterns left virenti [2]. in the wake; finding for example how the speed of the parameter interface affects orientation and deformation of stripes. I will 1. S. Petrat, P.Pickl, A. Soffer, Derivation of the Bogoliubov Time Evolu- also show how periodic wrinkles can form on top of pure stripes, tion for Gases with Finite Speed of Sound, arXiv:1711.01591 , (2017). with frequency behavior similar to that of a saddle-node on a limit 2. H. Paul, M. Pulvirenti, Asymptotic expansion of the mean-field ap- cycle. I will explain this approach in the context of the Swift- proximation, Disc. & Cont. Dyn. Sys. - A, 39 (4) (2019), 1891-1921. Hohenberg PDE, a prototypical model for many pattern forming systems, posed in one and two spatial dimensions. I will also discuss recent work which uses techniques from geometric desingu- larization and modulational theory to study the stability and dynamics of these structures. SESSION 9: “Nonlinear evolutionary equations: Theory, numerics and experiments” 1. Avery, M and Goh, R and Goodloe, O and Milewski, A and Scheel, A, Growing stripes, with and without wrinkles, arXiv preprint Snakes and lattices: Understanding the bifurcation arXiv:1810.08688, (2018). structure of localized solutions to lattice dynamical 2. R. Goh, A. Scheel. Pattern-forming fronts in a Swift-Hohenberg equa- systems tion with directional quenching - parallel and oblique stripes, J. London Math. Soc., 98 (2018), 104-128. Jason J. Bramburger∗ and Bjorn¨ Sanstede 170 Hope Street Providence, Rhode Island, 02906, USA Nonlinear eigenvalue problems in biologically moti- jason [email protected] and bjorn [email protected] vated PDEs

A wide variety of spatially localized steady-state solutions to par- Zoi Rapti∗ and Jared C. Bronski tial differential equations (PDEs) are known to exhibit a bifurca- Department of Mathematics tion phenomenon termed snaking. That is, these solutions bounce University of Illinois, Urbana-Champaign

36 [email protected] and [email protected] Traveling waves in the fifth order Korteweg-de Vries equation and discontinuous shock solutions of the Andrea K. Barreiro Whitham modulation equations Department of Mathematics Southern Methodist University Patrick Sprenger, Mark Hoefer [email protected] Department of Applied Mathematics, University of Colorado Boulder [email protected] This paper is focused on the spectral properties of certain classes of coupled nonlinear PDEs arising in biology. We will present results Whitham modulation theory is a powerful mathematical tool to de- that show the existence of only real spectrum in the corresponding scribe the slow evolution of a nonlinear, periodic wave. It yields non-selfadjoint eigenvalue problem. Our proof relies on the theory a system of hyperbolic partial differential equations for the evo- of operator pencils and Herglozt functions. Concrete applications lution of the wave’s parameters. The typical solution of interest will be demonstrated in models of rabies epidemics in fox popula- in applications to dispersive shock waves is a weak, self-similar tions, plant-herbivore interactions and morphogen diffusion. expansion wave solution to the hyperbolic Whitham system. This talk will focus on the fifth order Korteweg-de Vries (KdV5) equa- Grain boundaries of the Swift-Hohenberg equation: tion and its rich family of traveling wave solutions. It is shown that simulations and analysis discontinuous shock solutions of the Whitham modulation system which represent the zero dispersion limit of traveling wave solu- Joceline Lega tions of the KdV5 equation. These shock correspond to a rapid Department of Mathematics, University of Arizona, 617 N. Santa Rita transition joining two disparate periodic waves copropagating at a Avenue, Tucson, AZ 85721 fixed velocity. These traveling waves necessarily satisfy classical [email protected] jump conditions for the far-field wave parameters and shock ve- I will summarize the results of [1], which describes an analytical locity. These solutions have recently been observed numerically in and numerical investigation of the phase structure of some stable applications to water waves and nonlinear optics. grain boundary solutions of the Swift-Hohenberg equation. I will then introduce new analytical and numerical tools to explore Nonlinear instability of spectrally stable shifted states properties of the phase of the pattern in the vicinity of the disloca- on star graphs tions that form at the core of such grain boundaries in the strong Adilbek Kairzhan and Dmitry E. Pelinovsky bending limit. Department of Mathematics, McMaster University This work is joint with Nick Ercolani. Hamilton, Ontario L8S4K1, Canada [email protected] and [email protected] 1. Nicholas M. Ercolani, Nikola Kamburov, Joceline Lega, The phase structure of grain boundaries, Phil. Trans. R. Soc. A 376, 20170193 Roy Goodman (2018). Department of Mathematical Sciences New Jersey Institute of Technology Newark NJ, USA Bifurcations on an NLS dumbbell graph [email protected] Roy H. Goodman When coefficients of the cubic terms match coefficients in the Department of Mathematical Sciences, New Jersey Institute of Technol- boundary conditions at a vertex of a star graph and satisfy a cer- ogy, University Heights, Newark, NJ 07102 tain constraint, the nonlinear Schrodinger (NLS) equation on the [email protected] star graph can be transformed to the NLS equation on a real line. We consider the bifurcations of standing wave solutions to the Such balanced star graphs appeared in the context of reflectionless nonlinear Schrodinger¨ equation (NLS) posed on a quantum graph transmission of solitary waves. The steady states can be translated consisting of two loops connected by a single edge, the so-called along the edges of a balanced star graph with a translational pa- dumbbell, recently studied by Marzuola and Pelinovsky. The au- rameter and are referred to as the shifted states. When the star thors of that study found the ground state undergoes two bifurca- graph has exactly one incoming edge and several outgoing edges, tions, first a symmetry-breaking, and the second which they call a the steady states are spectrally stable if their monotonic tails are lo- symmetry-preserving bifurcation. We clarify the type of the cated at the outgoing edges. Nonlinear stability of these spectrally symmetry-preserving bifurcation, showing it to be transcritical. We stable states has been an open problem up to now. In this talk, then reduce the question, and show that the phenomena described we show that these spectrally stable states are nonlinearly unstable in that paper can be reproduced in a simple discrete self-trapping because of the irreversible drift along the incoming edge towards equation on a combinatorial graph of bowtie shape. This allows the vertex of the star graph. These spectrally stable states are de- for complete analysis both by geometric methods and by parame- generate minimizers of the action functional with the degeneracy terizing the full solution space. We then expand the question, and due to the symmetry of the NLS equation on a balanced star graph. describe the bifurcations of all the standing waves of this system, When the shifted states reach the vertex as a result of the drift, which can be classified into three families, and of which there ex- they become saddle points of the action functional, in which case ists a countably infinite set. the nonlinear instability leads to destruction of the shifted states. In addition to the rigorous mathematical results, we use numerical

37 simulations to illustrate the drift instability and destruction of the 1. V. I. Yudovich, ?Non-stationary flows of an ideal incompressible fluid?, shifted states on the balanced star graph. Zh. Vychisl. Mat. Mat. Fiz., 3:6 (1963), 1032-1066; U.S.S.R. Comput. Math. Math. Phys., 3:6, 1407-1456 (1963). Curve lengthening and shortening in strong FCH 2. T. Hou and G. Luo: Toward the finite-time blowup of the 3d axisymmetric Euler equations: A numerical investigation, Multiscale Model. Yuan Chen and Keith Promislow Simul., 12(4):1722–1776 (2014). Michigan State University [email protected] and [email protected] New PT-symmetric systems with solitons: nonlinear We show that nearly circular, codimension one interfaces evolving Dirac and Landau-Lifshitz equations under the L2-gradient flow of the strong scaling of the functional- Igor Barashenkov ized Cahn Hilliard gradient flow enjoy a sharp-interface limit corresponding to a curve shortening or regularized curve-lengthening Department of Mathematics, University of Cape Town, South Africa flow. Depending upon the distribution of mass, the interface by [email protected] absorbing or releasing mass from the far-field may expand against Although the spinor field in (1+1) dimensions has the right struc- interface that induces interfacial meandering or shrink. More pre- ture to model a dispersive bimodal system with gain and loss, the cisely, we show that the leading order interfacial evolution can be plain addition of gain to one component of the field and loss to described by an asymptotically large but finite dimension, Galerkin the other one results in an unstable dispersion relation. In this reduction of motion against curvature regularized by higher order talk, we advocate a different recipe for the PT-symmetric exten- Willmore terms. sion of spinor models — the recipe that does not produce instability of the Dirac equation. We consider the PT-symmetric ex- Observation of phase domain walls in deep water tensions of nonlinear spinor models and demonstrate a remarkable surface gravity waves sturdiness of spinor solitons in two dimensions. Another new class F. Tsitoura of PT-symmetric systems comprises the Heisenberg ferromagnet with spin torque transfer. In the vicinity of the exceptional point, Department of Mathematics and Statistics, University of Massachusetts the corresponding Landau-Lifshitz equation reduces to a nonlinear Amherst, Amherst, MA 01003-4515, USA Schroedinger equation with a quadratic nonlinearity. In the sim- [email protected] plest, isotropic, case the equation has the form iψ + ψ ψ + t xx − Experiments of nonlinear phase domain walls in weakly nonlin- ψ2 = 0. We show that this PT-symmetric Schrodinger¨ equation ear deep water surface gravity waves are presented. The domain has stable soliton solutions. walls presented are connecting homogeneous zones of weakly nonlinear plane Stokes waves of identical amplitude and wave vector 1. N V Alexeeva, I V Barashenkov and A Saxena, Spinor soli- but differences in phase. By exploiting symmetry transformations tons and their PT-symmetric offspring, Ann Phys (2018), within the framework of the nonlinear Schrodinger¨ equation we https://doi.org/10.1016/j.aop.2018.11.010. demonstrate the existence of exact analytical solutions represent- 2. I V Barashenkov and A Chernyavsky, A PT-symmetric Heisenberg fer- ing such domain walls in the weakly nonlinear limit. The walls are romagnet and a quadratic nonlinear Schrodinger¨ equation. Submitted in general oblique to the direction of the wave vector and stationary for publication. in moving reference frames. Experimental and numerical studies confirm and visualize the findings. Parity-time and other symmetries in optics and photonics 1. F. Tsitoura, U. Gietz, A. Chabchoub and N. Hoffmann, Phase Domain Demetrios Christodoulides Walls in Weakly Nonlinear Deep Water Surface Gravity Waves, Phys. Rev. Lett., 120 (2018), 224102. CREOL-The College of Optics & Photonics University of Central Florida Orlando, FL 32816, USA Models for 3D Euler’s equations [email protected] Hang Yang The prospect of judiciously utilizing both optical gain and loss has Euler’s equation is one of the most important mathematical prob- been recently suggested as a means to control the flow of light. lems in fluids. The global regularity of 2D Euler has been solved This proposition makes use of some newly developed concepts by Yudovich [1] in late 60’s. Yet in 3D, due to the competition of based on non-Hermiticity and parity-time (PT) symmetry-ideas first quadratic non-linear terms of different natures, the dynamics of Eu- conceived within quantum field theories. By harnessing such no- ler’s equations remains still unclear nowadays. In 2013, Hou-Luo tions, recent works indicate that novel synthetic structures and de- [2] investigated 3D Euler’s equations in the axisymmetric settings vices with counter-intuitive properties can be realized, potentially and observed numerical blow up. Their numerical simulation has enabling new possibilities in the field of optics and integrated pho- shed significant light on the study of a few important fluids prob- tonics. Non-Hermitian degeneracies, also known as exceptional lems centered around Euler’s equations. In this talk, we will in points (EPs), have also emerged as a new paradigm for engineer- particular discuss theoretical developments on Boussinesq equa- ing the response of optical systems. In this talk, we provide an tions and SQG equations that followed thereafter. overview of recent developments in this newly emerging field. The use of other type symmetries in photonics will be also discussed.

38 Generation, propagation and interaction of solitary write down the evolution equation for h for flows with h = 0, gv m 6 waves in integrable versus non-integrable nonlinear and show how hgv and hm are coupled via the shear tensor of the lattices background fluid flow. An application of the Godbillon-Vey helicity to the nonlinear force free fields is described. Guo Deng∗, Gino Biondini and Surajit Sen Department of Physics, University at Buffalo Dispersion and attenuation in a poroelastic model Department of Mathematics, University at Buffalo for gravity waves on an ice-covered ocean [email protected], [email protected], [email protected] Hua Chen , Robert P. Gilbert The study of lattice dynamics, i.e., the motion of a spatially discrete ∗ Department of Mathematical Sciences, University of Delaware system governed by a system of differential-difference equations, Institute of Mechanics and Materials, Ruhr-Universitat¨ Bochum is a classical subject. Of particular interest are lattices that support [email protected], [email protected] the propagation of solitary waves [1]. In this talk, we will compare the properties of two kinds of lattices, one integrable and one Philippe Guyenne non-integrable: the Toda lattice and the Hertzian chain. As is well Department of Mathematical Sciences, University of Delaware known, the Toda lattice is an integrable system and has exact soli- [email protected] ton solutions [2]. In contrast, the Hertzian chain, which has many physical and engineering applications, is a non-integrable system The recurrent interactions between ocean waves and sea ice are a and no exact solitary-wave solutions are known [3]. Here we will widespread feature of the polar regions, and their impact on sea- analyze the similarities and differences between the solitary waves ice dynamics and morphology has been increasingly recognized in these two systems, we will discuss how each of these systems as evidenced by the surge of research activity during the last two respond to a velocity perturbation, and we will compare the inter- decades. The rapid decline of summer ice extent that has occurred action dynamics of solitary wave. in the Arctic Ocean over recent years has contributed to the renewed interest in this subject. Continuum models have recently 1. G. Friesecke and J.A.D. Wattis, Existence theorem for solitary waves gained popularity to describe wave propagation in various types of on lattices, Commun. Math. Phys., 161, pp. 391–418 (1994) ice cover and across a wide range of length scales. In this talk, we propose a continuum wave-ice model where the floating ice cap is 2. M. Toda, Theory of nonlinear lattices, Springer-Verlag, (1981) described as a homogeneous poroelastic material and the underly- 3. V.F. Nesterenko, Dynamics of heterogeneous materials, Springer- ing ocean is viewed as a slightly compressible fluid. The linear dis- Verlag, (2001) persion relation for time-harmonic wave solutions of this coupled system is established and compared to predictions from existing theories.

SESSION 10: “Recent advances in PDEs from fluid dynamics and Wave model for Poiseuille flow of nematic liquid crys- other dynamical models” tals Geng Chen∗ Godbillon-Vey helicity in magnetohydrodynamics Department of Mathematics, University of Kansas, Lawrence, KS 66045, and fluid dynamics U.S.A. [email protected] G. M. Webb∗, Q. Hu and A. Prasad Center for Space Plasma and Aeronomic Research, Tao Huang The University of Alabama in Huntsville, Huntsville AL 35805 Department of Mathematics, Wayne State University, Detroit, MI, 48201, [email protected] U.S.A. S. C. Anco [email protected] Department of Mathematics, Brock University, St. Catharines Weishi Liu ON L2S3A1, Canada Department of Mathematics, University of Kansas, Lawrence, KS 66045, [email protected] U.S.A. [email protected] The Godbillon-Vey invariant occurs in the theory of foliations. The magnetic Godbillon-Vey invariant in magnetohydrodynamics In this talk, we will discuss the global existence of Holder¨ contin- (MHD) for the magnetic field B occurs if the magnetic helicity uous solution for the Poiseuille flow of full Ericksen-Leslie system density hm = A B = A A = 0. This implies that the Pfaf- modeling nematic liquid crystals. Different from many previous fian A dx = 0 admits· an integrating·∇× factor µ, where µA dx = dΦ results which omit the kinetic energy, the full system we consider and the· family of surfaces Φ(x, y, z) = const. is a foliation.· The includes a quasilinear wave equation, which may form cusp singu- Godbillon-Vey field η = A B/ A 2 lies in the surface and the larity in general. The strong coupling on the second order parabolic × | | Godbillon-Vey helicity density is defined as hgv = η η. We equation on the velocity of flow and the quasilinear wave equation obtain evolution equations for the Godbillon-Vey helicity·∇ × density on the direction field of mean orientation of the liquid crystal gives 3 hgv and the Godbillon-Vey invariant Hgv = V hgvd x for a vol- the main challenge for the global existence, which will be solved ume V moving with the fluid for the case whereR hm = 0. We also

39 by a new method. This is a joint work with Weishi Liu and Tao SESSION 11: Moved to Session 26 Huang.

Finite time blow up of compressible Navier-Stokes equations on half space or outside a fixed ball SESSION 12: “Dispersive shocks, semiclassical limits and appli- Dongfen Bian∗ cations” School of Mathematics and Statistics, Beijing Institute of Technology, Bei- jing 100081, China; Universal behavior of modulationally unstable me- Division of Applied Mathematics, Brown University, Providence, Rhode dia with non-zero boundary conditions Island 02912. Gino Biondini dongfen [email protected] and [email protected] State University of New York at Buffalo Jinkai Li [email protected] South China Research Center for Applied Mathematics and Interdisci- Sitai Li plinary Studies, South China Normal University, Zhong Shan Avenue West ∗ 55, Tianhe District, Guangzhou 510631, China University of Michigan [email protected] [email protected] Dionyssios Mantzavinos In this paper, we consider the initial-boundary value problem to the compressible Navier-Stokes equations for ideal gases without heat University of Kansas conduction in the half space or outside a fixed ball in RN, with [email protected] N 1. We prove that any classical solutions (ρ, u, θ), in the class Stefano Trillo 1≥ m Ω > N C ([0, T]; H ( )), m [ 2 ]+ 2, with bounded from below ini- University of Ferrara tial entropy and compactly supported initial density, which allows [email protected] to touch the physical boundary, must blow-up in finite time, as long as the initial mass is positive. This talk is divided into three parts. First, I will briefly describe the inverse scattering transform for the focusing nonlinear Schrodinger¨ Global well-posedness of coupled parabolic systems (NLS) equation with nonzero boundary conditions at infinity, and then I will present the long-time asymptotics of pure soliton solu- Wei Lian∗ tions on the nonzero background. Second, I will describe in detail College of Science, Harbin Engineering University, Harbin 150001, P. R. the properties of the asymptotic state of the modulationally unsta- China ble solutions of the NLS equation, including the number of oscilla- lianwei [email protected] tions and the local structure of the solution near each peak, showing Runzhang Xu in particular that in the long-time limit the solution tends to an en- College of Science, Harbin Engineering University, Harbin 150001, P. R. semble of classical (i.e., sech-shaped) solutions of the NLS equa- China tion. Third, I will show that a similar asymptotic state is shared [email protected] among a broad class of systems of NLS-type possessing modulational instability. Yi Niu School of Information Science and Engineering, Modulational instability of a plane wave in the pres- Shandong Normal University, Jinan 250001, P. R. China ence of localized perturbations: some experimental yanyee [email protected] results in nonlinear fiber optics

The initial boundary value problem of a class of reaction-diffusion Stphane Randoux∗, Adrien E. Kraych, Pierre Suret systems (coupled parabolic systems) with nonlinear coupled source Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes terms is considered in order to classify the initial data for the global et Molecules,´ F-59000 Lille, France existence, finite time blowup and long time decay of the solution. [email protected] and [email protected] and The whole study is conducted by considering three cases accord- [email protected] ing to initial energy: low initial energy case, critical initial energy Gennady El case and high initial energy case. For the low initial energy case and critical initial energy case the sufficient initial conditions of Department of Mathematics, Physics and Electrical Engineering, Northum- global existence, long time decay and finite time blowup are given bria University, Newcastle upon Tyne, NE1 8ST, United Kingdom to show a sharp-like condition. And for the high initial energy [email protected] case the possibility of both global existence and finite time blowup We report an optical fiber experiment in which we study nonlinear is proved first, and then some sufficient initial conditions of finite stage of modulational instability of a plane wave in the presence of time blowup and global existence are obtained respectively. a localized perturbation [1]. Using a recirculating fiber loop as experimental platform, we show that the initial perturbation evolves into expanding nonlinear oscillatory structure exhibiting some universal characteristics that agree with theoretical predictions based

40 on integrability properties of the focusing nonlinear Schrodinger¨ In this talk, I will describe a new type of the wave-mean flow in- equation [2]. Our experimental results demonstrate persistence of teraction whereby a short-scale wave projectile—a soliton or a lin- the universal evolution scenario, even in the presence of small dis- ear wave packet—is incident on the evolving large-scale nonlin- sipation and noise in an experimental system that is not rigorously ear dispersive hydrodynamic state: a rarefaction wave or a disper- of an integrable nature. sive shock wave (DSW). Modulation equations are derived for the coupling between the soliton (wavepacket) and the mean flow in 1. A. E. Kraych, P. Suret, G. El, S. Randoux Nonlinear evolution of the the nonlinear dispersive hydrodynamic state. These equations ad- locally induced modulational instability in fiber optics Accepted for mit particular classes of solutions that describe the transmission or publication in Phys. Rev. Lett. (2019) [arXiv:1805.05074] trapping of the wave projectile by an unsteady hydrodynamic state. 2. G. Biondini and D. Mantzavinos, Universal nature of the nonlinear Two adiabatic invariants of motion are identified in both cases that stage of modulational instability Phys. rev. Lett. 116, 043902 (2016) determine the transmission, trapping conditions and show that solitons (wavepackets) incident upon smooth expansion waves or com- Towards kinetic equation for soliton and breather pressive, rapidly oscillating DSWs exhibit so-called hydrodynamic gases for the focusing nonlinear Schroedinger equa- reciprocity. The latter is confirmed in a laboratory fluids experi- tion ment for soliton-hydrodynamic state interactions. The developed theory is general and can be applied to integrable Alexander Tovbis ∗ and non-integrable nonlinear dispersive wave equations in various Department of Mathematics, University of Central Florida, Orlando, FL, physical contexts including nonlinear optics and cold atom physics. USA As concrete examples we consider the Korteweg-de Vries and the [email protected] viscous fluid conduit equations. The talk is based on recent papers Gennady El [1, 2] Department of Mathematics, Physics and Electrical Engineering Northumbria University Newcastle, UK 1. M. D. Maiden, D. V. Anderson, N. A. Franco, G. A. El, & M. A. Hoe- [email protected] fer, Solitonic dispersive hydrodynamics: theory and observation. Phys. Rev. Lett., 120 (2018) 144101. inetic equation for a soliton gas for the Korteweg - de Vries equa- 2. T. Congy, G.A. El and M.A. Hoefer, Interaction of linear tion was first proposed by V. Zakharov and later derived by G. modulated waves with unsteady dispersive hydrodynamic states, El using the thermodynamic limit of the KdV-Whitham equations. arXiv:1812.06593. Later, G. El and A. Kamchatnov proposed kinetic equation for the soliton gas for the focusing Nonlinear Schroedinger (fNLS) equation using physical reasoning. The universality of the semi-classical sine-Gordon In this talk, we consider the large N limit of nonlinear N-phase equation at the gradient catastrophe wave solutions to the fNLS equation subject to a certain scaling of Bingying Lu∗ the corresponding bands and gaps. In this limit, we obtain integral Institute of Mathematics, Academia Sinica, Taipei equations for the scaled wavenumbers and frequences and, as a [email protected] consequence, derive the kinetic equation for soliton/breather gases, which takes into account soliton-soliton and soliton-background Peter Miller interactions. Our approach can be used to derive kinetic equation Department of Mathematics, University of Michigan in Ann Arbor for the soliton gas on the background of any finite gap solution. [email protected] This work is still in progress. We study the semi-classical sine-Gordon equation with pure im- 2 2 pulse initial data below the threshold of rotation: ǫ utt ǫ uxx + Wave-mean flow interactions in dispersive hydrody- − < namics sin(u) = 0, u(x, 0) 0, ǫut(x, 0) = G(x) 0, and G(0) 2. A dispersively-regularized≡ shock forms in≤ finite time.| Using| Gennady El and Thibault Congy Riemann–Hilbert analysis, we rigorously studied the asymptotics Department of Mathematics, Physics and Electrical Engineering near a certain gradient catastrophe. In accordance with a con- Northumbria University Newcastle, UK jecture made by Dubrovin et. al., the asymptotics in the this re- [email protected] and [email protected] gion is universally (insensitive to initial condition) described by the tritronquee´ solution to the Painleve-I´ equation. Furthermore, Mark Hoefer we are able to universally characterize the shapes of the spike-like Department of Applied Mathematics, University of Colorado Boulder, USA local structures (rogue wave on periodic background) on top of the [email protected] poles of the tritronquee´ solution. The interaction of waves with a mean flow is a fundamental and longstanding problem in fluid mechanics. The key to the study of Semiclassical Lax spectrum of Zakharov-Shabat sys- such an interaction is the scale separation, whereby the length and tems with periodic potentials time scales of the waves are much shorter than those of the mean Jeffrey Oregero and Gino Biondini flow. The wave-mean flow interaction has been extensively studied University at Buffalo, SUNY for the cases when the mean flow is prescribed externally—as a [email protected] and [email protected] stationary or time-dependent current (a “potential barrier”).

41 The semiclassical limit of the focusing nonlinear Schrodinger¨ equa- Nonlinear interactions between solitons and disper- tion with periodic initial conditions is studied analytically and nu- sive shocks in focusing media merically. First, through a comprehensive set of careful numerical Gino Biondini and Jonathan Lottes simulations, it is demonstrated that solutions arising from many different initial conditions share the same qualitative features, which University at Buffalo also coincide with those of solutions arising from localized initial [email protected] and [email protected] conditions. Rigorous bounds on the location of eigenvalues of the Nonlinear interactions in focusing media between traveling soli- associated scattering problem are derived and it is shown that the tons and the dispersive shocks produced by an initial discontinuity spectrum is a subset of the real and imaginary axes of the spectral are studied numerically and analytically using the one-dimensional variable in the semiclassical limit. Finally, by employing a suitable nonlinear Schrodinger¨ equation. Wentzel-Kramers-Brillouin expansion for the scattering eigenfunctions, asymptotic formulae are derived for the number and location Evolution of broad initial profiles—solitary wave fis- of the bands and gaps in the spectrum, as well as for the relative sion and solitary wave phase shift band and gap widths. Michelle Maiden and Mark A. Hoefer∗ Nonlinear Schrodinger¨ equations and the universal Department of Applied Mathematics, University of Colorado, Boulder, description of dispersive shock wave structure USA Thibault Congy and Gennady El [email protected] and [email protected] Department of Mathematics, Physics and Electrical Engineering Gennady El Northumbria University Newcastle, UK Department of Mathematics, Physics and Electrical Engineering, Northum- [email protected] and [email protected] bria University, Newcastle, UK Mark Hoefer [email protected] Department of Applied Mathematics The temporal evolution of a large, localized initial disturbance is University of Colorado, Boulder, USA considered in the context of scalar, dispersive nonlinear equations [email protected] in the dispersive hydrodynamic regime. Modulation theory for Michael Shearer solitary wave fission in long time evolution of the broad disturbance is developed and certain universal properties of the dynam- Department of Mathematics ics are identified. The theory, asymptotically valid for the gener- North Carolina State University, Raleigh, USA ation of a large number of solitary waves, yields predictions for [email protected] the number of solitary waves and their amplitude distribution. The A dispersive shock wave (DSW) is an expanding, modulated non- number of solitary waves universally depends linearly on the initial linear wavetrain that connects two disparate hydrodynamic states, profile’s width. The normalized cumulative amplitude distribution and can be viewed as a dispersive counterpart to the dissipative, function is independent of the initial profile’s width. These prop- classical shock. DSWs have raised a lot of interest in the recent erties are verified quantitatively in experiments involving the inter- years, due to the growing recognition of their fundamental na- facial dynamics of two miscible Stokes fluids with high viscosity ture and ubiquity in physical applications, examples being found in contrast. The number of observed solitary waves is consistently oceanography, meteorology, geophysics, nonlinear optics, plasma within 1-2 waves of the prediction, and the amplitude distribution physics and condensed matter physics. Although well-established shows remarkable agreement. Additionally, using Whitham mod- methods, such as the Whitham modulation theory, have proved ulation theory, a universal phase shift formula for the interaction particularly effective for the determination of DSW solutions of of a solitary wave that is initially well-separated from a broad dis- certain nonlinear wave equations, a universal description of these turbance is presented and shown to agree with numerical compu- objects is still lacking. tations. All of the modulation theory predictions are agnostic to The nonlinear Schrodinger¨ (NLS) equation and the Whitham mod- integrable structure of the underlying PDE model. ulation equations both describe slowly varying, locally periodic nonlinear wavetrains, albeit in differing amplitude-frequency do- Dispersive shocks dynamics of phase diagrams mains. Taking advantage of the overlapping asymptotic regime Costanza Benassi and Antonio Moro that applies to both the NLS and Whitham modulation descriptions, we developed a universal analytical description of DSWs Department of Mathematics, Physics and Electrical Engineering generated in Riemann problems for a broad class of integrable and Northumbria University Newcastle, UK non-integrable nonlinear dispersive equations [1]. The proposed [email protected] method extends DSW fitting theory that prescribes the motion of The theory of Nonlinear Conservation Laws arises as a universal a DSW’s edges into the DSW’s interior, that is, this work reveals paradigm for the description of phase transitions, cooperative and the DSW structure. I will present this new method and illustrate its catastrophic behaviours in many body systems at the crossroad of efficacy by considering various physically relevant examples. integrable systems, statistical mechanics and random matrix theory. 1. T. Congy, G. A. El, M. A. Hoefer and M. Shearer, Nonlinear Schrodinger¨ equations and the universal description of dispersive shock In classical magnetic and fluid models the free energy can be ob- wave structure. Stud. Appl. Math. 2018;1–28. tained as a solution of a viscous integrable hierarchy of PDEs and

42 phase transitions are associated to the classical shock dynamics of crystal surfaces and the dendritic crystal growth problems, as spe- order parameters in the space of thermodynamics variables. cial examples. The main challenge of constructing numerical We show that for Hermitian Matrix Models, where the partition schemes with unconditional energy stabilities for these type of mod- function is given by a tau function of the Toda hierarchy, phase els is how to design proper temporal discretizations for the nonlin- transitions are associated to the dispersive shock dynamics of the ear terms with the strong anisotropy. We combine the recently de- continuum limit of the Toda hierarchy. veloped IEQ/SAV approach with the stabilization technique, where some linear stabilization terms are added, which are shown to be crucial to remove the oscillations caused by the anisotropic coefficients, numerically. The novelty of the proposed schemes is that all nonlinear terms can be treated semi-explicitly, and one only needs SESSION 13: “Recent advances in numerical methods of pdes and to solve some coupled/decoupled, but linear equations at each time applications in life science, material science” step. We further prove the unconditional energy stabilities rigor- A second-order fully-discrete linear energy stable ously, and present various 2D and 3D numerical simulations to scheme for a binary compressible viscous fluid model demonstrate the stability and accuracy. Xueping Zhao Efficient and stable numerical methods for a class of Department of Mathematics, University of South Carolina, Columbia, SC stiff reaction-diffusion systems with free boundaries 29208, USA Shuang Liu and Xinfeng Liu [email protected] ∗ Department of Mathematics, University of South Carolina, Columbia, USA Qi Wang [email protected], xfl[email protected] Department of Mathematics, University of South Carolina, Columbia, SC The systems of reaction-diffusion equations coupled with moving 29208, USA boundaries defined by Stefan condition have been widely used to and Beijing Computational Science Research Center, Beijing 100193, China describe the dynamics of spreading population. There are several [email protected] numerical difficulties to efficiently handle such systems. Firstly We present a linear, second order fully discrete numerical scheme extremely small time steps are usually needed due to the stiff- on a staggered grid for a thermodynamically consistent hydrody- ness of the system. Secondly it is always difficult to efficiently namic phase field model of binary compressible fluid flow mix- and accurately handle the moving boundaries. To overcome these tures derived from the generalized Onsager Principle. The hydro- difficulties, we first transform the one-dimensional problem with dynamic model not only possesses the variational structure, but moving boundaries into a system with fixed computational domain, also warrants the mass, linear momentum conservation as well as and then introduce four different temporal schemes: Runge-Kutta, energy dissipation. We first reformulate the model in an equivalent Crank-Nicolsn, implicit integration factor (IIF) and Krylov IIF, for form using the energy quadratization method and then discretize handling such stiff systems. Numerical examples are examined to the reformulated model to obtain a semi-discrete partial differen- illustrate the efficiency, accuracy and consistency for different ap- tial equation system using the Crank-Nicolson method in time. The proaches, and it can be shown that Krylov IIF is superior to other numerical scheme so derived preserves the mass conservation and approaches in terms of stability and efficiency by direct compari- energy dissipation law at the semi-discrete level. Then, we dis- son. cretize the semi-discrete PDE system on a staggered grid in space to arrive at a fully discrete scheme using the 2nd order finite differ- 1. Du, Y., and Lin, Z. (2010). Spreading-vanishing dichotomy in the dif- ence method, which respects a discrete energy dissipation law. We fusive logistic model with a free boundary. SIAM Journal on Mathe- prove the unique solvability of the linear system resulting from the matical Analysis, 42(1), 377-405. fully discrete scheme. Mesh refinements and two numerical exam- 2. Nie, Q., Zhang, Y.-T. and Zhao, R. (2006). Efficient semi-implicit ples on phase separation due to the spinodal decomposition in two schemes for stiff systems. Journal of Computational Physics, 214, 521- polymeric fluids and interface evolution in the gas-liquid mixture 537. are presented to show the convergence property and the usefulness 3. Chen, S. Q. and Zhang, Y. T. (2011). Krylov implicit integration fac- of the new scheme in applications. tor methods for spatial discretization on high dimensional unstructured meshes: application to discontinuous Galerkin methods. Journal of Efficient schemes with unconditionally energy sta- Computational Physics, 230(11), 4336-4352. bilities for anisotropic phase field models: S-IEQ and 4. Piqueras, M.-A., Company, R., Jodar, L. (2017). A front-fixing numer- S-SAV ical method for a free boundary nonlinear diffusion logistic population model. Journal of Computational and Applied Mathematics, 309, 473- Xiaofeng Yang 481. 1523 Greene Street, Columbia, SC, 29208 [email protected] Approximating nonlinear reaction-diffusion We consider numerical approximations for anisotropic phase field problems with multiple solutions models, by taking the anisotropic Cahn-Hilliard/Allen-Cahn equa- Tom Lewis tions with their applications to the faceted pyramids on nanoscale ∗ The University of North Carolina at Greensboro

43 [email protected] incorporates the nonlinear nature of viscoelastic responses as well as the stochastic processes which describe the breaking and reform- In this paper we introduce the class of positone boundary value ing of entanglements in the underlying microscopic network. The problems and the analytic issues that must be addressed when us- model allows a full reconstruction of the microstructure-flow cou- ing an approximation method. In particular, we will consider the pling thereby creating a platform with the ability to investigate how problem of approximating a function that solves the semilinear u microscopic changes affect macroscopic responses. In this talk we elliptic boundary value problem focus on oscillatory flow and show both evolution of stress and species distribution as functions of frequency and strain. ∆u = λ f (u) in Ω − with u > 0 in Ω and u = 0 on ∂Ω, where λ > 0 is a constant; Ω is an open, bounded, convex domain; and f is a postitone operator with sublinear growth, i.e., f satisfies the three conditions f (w) > SESSION 14: “Nonlinear kinetic self-organized plasma dynamics f (w) 0 for all w 0, f is nondecreasing, and limw ∞ = 0. driven by coherent, intense electromagnetic fields” ≥ → w Such problems arise in mathematical biology and the theory of nonlinear heat generation. Under certain conditions, the problems Spectrally accurate methods for kinetic may have multiple positive solutions or even nonexistence of a pos- electron plasma wave dynamics itive solution. We will discuss new analytic techniques for proving Jon Wilkening and Rockford Sison admissibility, stability, and convergence of finite difference meth- Department of Mathematics ods for approximating sublinear positone problems. The admis- University of California sibility and stability results will be based on adapting the method Berkeley, CA 94720-3840 of sub- and supersolutions typically used to analyze the underly- [email protected] and [email protected] ing PDEs. Since most known approximation methods for positone boundary value problems rely upon shooting techniques, they Bedros Afeyan are restricted to one-dimensional problems and/or radial solutions. Polymath Research Inc. The new tools will serve as a foundation for approximating posi- 827 Bonde Court tone boundary value problems in higher dimensions and on more Pleasanton, CA 94566 general domains. [email protected]

A hybrid model for simulating sprouting angiogene- We present two numerical methods for computing solutions of the sis in biofabrication Vlasov-Fokker-Planck-Poisson equations that are spectrally accurate in all three variables (time, space and velocity). The first is a Yi Sun and Qi Wang Chebyshev collocation method for solving the Volterraequation for Department of Mathematics, University of South Carolina the space-time evolution of the plasma density for the linearized, [email protected] and [email protected] collisionless case. This is then used to efficiently represent the velocity distribution function in Case-van Kampen normal modes, We present a 2D hybrid model to study sprouting angiogenesis building on the work of Li and Spies. The second is an arbitrary- of multicellular aggregates during vascularization in biofabrica- order exponential time differencing scheme that makes use of the tion. This model is developed to describe and predict the time evo- Duhamel principle to fold in the effects of collisions and nonlinear- lution of angiogenic sprouting from endothelial spheroids during ity. We investigate the emergence of a continuous spectrum in the tissue or organ maturation in a novel biofabrication technology– collisionless limit and the embedding of Landau’s poles in this gen- bioprinting. Here we employ typically coarse-grained continuum eral setting. We resolve the effects of filamentation, phase mixing, models (reaction-diffusion systems) to describe the dynamics of and Landau and collisional damping to arbitrary order of accuracy, vascular-endothelial-growth-factors, a mechanical model for the focusing on echoes and trapping phenomena. extra-cellular matrix based on the finite element method and cou- ple a cellular Potts model to describe the cellular dynamics. The Improving the performance of plasma kinetic simu- model can reproduce sprouting from endothelial spheroids and net- lations by iteratively learned phase space tiling: work formation from individual cells. variational constrained optimization meet machine learning A parallel approach to kinetic viscoelastic modelling B. Afeyan and R.D. Sydora Paula Vasquez and Erik Palmer ∗ Polymath Research Inc., Pleasanton, California 411 LeConte College [email protected] and [email protected] Columbia, SC 29208, United States [email protected], [email protected] We describe a general method of constrained optimization to or- Viscoelastic materials are characterized by the coupling of micro- ganically change the equations being solved, given prior knowl- structural changes to macroscale deformations. We present an elas- edge on nearby problems (differing via parameter choices, res- tic dumbbell model that leverages the parallel processing power of olution, modeling simplifications, etc.). The general method is High Performance Computing (HPC) Graphics Processing Units called NSCAR: Nearby Skeleton Constrained Accelerated Recom- (GPUs) to create a unique micro-macro scale driven design which puting. We then specialize to plasma kinetic equations and focus

44 on two new methods which improve the performance of PIC codes 2. K. Julien, E. Knobloch and M. Plumley, Impact of domain anisotropy and Vlasov Codes, called BARS and APOSTLE. BARS stands for on the inverse cascade in geostrophic turbulent convection, J. Fluid Bidirectional Adaptive Refinement Scheme and APOSTLE stands Mech., 837, R4 (2018). for Adaptive Particle Orbit Sampling Technique for Lagrangian 3. M. Chertkov, C. Connaughton, I. Kolokolov, V. Lebedev, Dynamics of Evolution. We demonstrate the advantages of these techniques by energy condensation in two-dimensional turbulence, Phys. Rev. Lett., applying them to the learned, sparse (non-uniformly sampled phase 99 (2007), 084501. space) representation of accurate solutions of nonlinear plasma 4. C. Guervilly, D. W. Hughes and C. A. Jones, Large-scale vortices waves in the kinetic/trapping regime. as well as for KEEN waves. in rapidly rotating Rayleigh-Benard´ convection, J. Fluid Mech., 758 Extensions of this method to the multidimensional setting where (2014), 407–435. magnetic fields, the Weibel instability and nonlinear plasma waves 5. B. Favier, L. J. Silvers and M. R. E. Proctor, Inverse cascade and sym- interact inexorably will also be described. metry breaking in rapidly rotating Boussinesq convection, Phys. Fluids, Work supported by a grant from AFOSR and the DOE FES-NNSA 26 (2014), 096605. Joint Program in HEDLP. 6. B. Favier, C. Guervilly and E. Knobloch, Subcritical turbulent condensate in rapidly rotating Rayleigh-Benard´ convection, J. Fluid Mech., in Nonlinear instabilities due to drifting species and press. magnetic fields in high energy density plasmas 7. C. Guervilly and D. W. Hughes, Jets and large-scale vortices in rotating RayleighBenard´ convection, Phys. Rev. Fluids, 2 (2017), 113503. B. A. Shadwick∗ and Alexander Stamm Department of Physics and Astronomy University of Nebraska–Lincoln Impact of cyclotron harmonic wave instabilities on sh [email protected] [email protected] stability of self-organized nonlinear kinetic plasma Relative drifts between particles species are fundamental driving structures forces behind many plasma instabilities. For example, the Bune- R.D. Sydora∗ and B. Afeyan man instability arises due to an election-ions drift. We study the Polymath Research Inc., Pleasanton, California nonlinear evolution of this processes in the presence of externally [email protected] and [email protected] imposed transverse magnetic fields. Our results are primarily drawn from simulations using both Vlasov–Maxwell and macro-particle B.A. Shadwick methods. We compare electrostatically driven modes to full elec- University of Nebraska, Lincoln tromagnetic treatments. Ion to electron mass ratios of 1, 10 and [email protected] 100 will be included. Crossing, intense laser beamsin high energy density plasmas lead Work supported by the DOE NNSA-FES Joint program in HEDLP. to the generation of nonlinear kinetic electron plasma waves (NL- EPW). The usefulness of such plasma structures depends on their Geostrophic turbulence and the formation of large long-time stability. Externally imposed magnetic fields is one scale structure method to confine multidimensional NL-EPW both in the trapping Edgar Knobloch regime and when vortex merger is prevalent. However, magnetic fields introduce cyclotron harmonic waves that may be driven un- Department of Physics, University of California, Berkeley CA 94720 stable (Harris instability) by velocity space anisotropies formed [email protected] through different plasma heating processes along and across the Low Rossby number convection is studied using an asymptotically magnetic field. The Harris instability causes transverse electro- reduced system of equations valid in the limit of strong rotation static perturbations that leads to the escape of trapped particles in [1]. The equations describe four regimes as the Rayleigh number NL-EPW, contributing to their rapid dissipation. In this work we Ra increases: a disordered cellular regime near threshold, a regime assess the importance of the Harris instability on NL-EPW and find of weakly interacting convective Taylor columns at larger Ra, fol- regimes where its impact is minimized. These studies employ self- lowed for yet larger Ra by a breakdown of the convective Taylor consistent particle simulations and the use of reconstructed parti- columns into a disordered plume regime characterized by reduced cle orbit dynamics from the self-consistent electric and magnetic heat transport efficiency, and finally by a new type of turbulence fields. called geostrophic turbulence. Properties of this state will be de- Work supported by a grant from AFOSR and by the DOE NNSA- scribed and illustrated using direct numerical simulations of the FES joint program in HEDLP. reduced equations. These simulations reveal that geostrophic turbulence is unstable to the formation of large scale barotropic vor- Internal wave energy flux from density perturbations tices [1] or jets [2], via a process known as spectral condensation 2,a 1,b [3]. The details of this process are quantified and its implications Frank M. Lee and Michael R. Allshouse a explored. The results are corroborated by direct numerical simula- Department of Physics and Astronomy, University of Nebraska-Lincoln, tions of the Navier-Stokes equations [4]–[6]. Lincoln, NE 68508, USA bDepartment of Mechanical and Industrial Engineering, Northeastern Uni- 1. K. Julien, A. M. Rubio, I. Grooms and E. Knobloch, Statistical and versity, Boston, MA 02115, USA physical balances in low Rossby number Rayleigh-Bnard convection, Geophys. Astrophys. Fluid Dyn., 106 (2012), 392–428.

45 [email protected] and [email protected] points in a modulated honeycomb material weight. We prove that such wave-packet dynamics is governed by the Dirac equation with Harry L. Swinney1 1 a varying mass in a large but finite time. Our analysis provides Center for Nonlinear Dynamics and Department of Physics, University mathematical insights to those topological phenomena in photonic of Texas at Austin, Austin, TX 78712, USA graphene. [email protected] Philip J. Morrison2 1. Mark J. Ablowitz, Christopher W. Curtis and Yi Zhu, On tight-binding approximations in optical lattices, Studies in Applied Mathematics, 2Institute for Fusion Studies and Department of Physics, University of 129 (4) 2012, 362-388. Texas at Austin, Austin, TX 78712, USA [email protected] 2. Charles L. Fefferman and Michael I. Weinstein, Honeycomb lattice potentials and Dirac points, Journal of the American Mathematical Soci- Internal gravity waves arise from buoyancy restoration forces within ety, 25(4) 2012, 1169-1220. a fluid whose density varies with height. The energy of such waves 3. James P. Lee-Thorp, Michael I. Weinstein and Yi Zhu, Elliptic oper- is of interest due to its significant presence in the energy budget ators with honeycomb symmetry: Dirac points, Edge States and Ap- of the ocean, and affects mixing and the thermohaline circulation. plications to Photonic Graphene, Archive for Rational Mechanics and The energy flux of linear internal waves requires the pressure per- Analysis, 2018, 1-63. turbation field, which is at present not an easily measurable quantity in either laboratory or field observations. Unfitted Nitsche’s method for computing edge modes We present a method using Green’s functions that gives the in- in photonic graphene stantaneous energy flux solely from the density perturbation field, which is measurable in the laboratory using synthetic schlieren Hailong Guo [1, 2]. We use simulations of the Navier-Stokes equations to verify School of Mathematics and Statistics, The University of Melbourne the method, which show good agreement, and check the usability [email protected] of the method with laboratory data. We give arguments for the er- Xu Yang ror scaling due to nonlinearity. Using the solution to the linear sys- Department of Mathematics, University of California, Santa Barbara tem as a baseline, it may be possible to use a perturbative method [email protected] to find corrections to the Green’s function and the energy flux for weakly nonlinear waves in future studies. Yi Zhu Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University 1. B. R. Sutherland, S. B. Dalziel, G. O. Hughes and P. F. Linden, Visu- [email protected] alization and measurement of internal waves by ‘synthetic schlieren.’ Part 1. Vertically oscillating cylinder., J. Fluid Mech., 390 (1999), 93- Photonic graphene, a photonic crystal with honeycomb structures, 126. has been intensively studied in both theoretical and applied fields. Similar to graphene which admits Dirac Fermions and topological 2. S. B. Dalziel, G. O. Hughes and B. R. Sutherland, Whole-field density edge states, photonic graphene supports novel and subtle propagat- measurements by ‘synthetic schlieren’, Exp. Fluids, 28 (2000), 322- 335. ing modes (edge modes) of electromagnetic waves. These modes have wide applications in many optical systems. In this paper, we propose a new unfitted Nitsche’s method to computing edge modes in photonic graphene with some defect. The unique feather of the methods is that it can arbitrary handle high contrast with geometric SESSION 15: “Waves in Topological Materials” unfitted meshes. We establish the optimal convergence of methods. Numerical examples are presented to validate the theoretical Wave-packet dynamics in slowly modulated photonic results and to numerically verify the existence of the edge modes. graphene 1. H. Guo, X. Yang, Y.Zhu, Bloch theory-based gradient recovery method Peng Xie and Yi Zhu for computing topological edge modes in photonic graphene, Journal Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Bei- of Computational Physics, 95(2019), 403–420. jing 100084, China 2. H. Guo, X. Yang, Y. Zhu, Unfitted Nitsche’e method for computing [email protected] and [email protected] edge modes in photonic graphene, 2019, preprint. Mathematical analysis on electromagnetic waves in photonic graphene, a photonic topological material which has a honeycomb Topologically protected edge modes in longitudinally structure, is one of the most important current research topics [1]. driven waveguides By modulating the honeycomb structure, numerous topological phe- Mark Ablowitz and Justin Cole nomena have been observed recently [2]. The electromagnetic ∗ Department of Applied Mathematics waves in such a media are generally described by the 2-dimensional University of Colorado, Boulder wave equation. It has been shown that the corresponding ellip- [email protected] tic operator with a honeycomb material weight has Dirac points in its dispersion surfaces [3]. In this article, we study the time evo- A tight-binding approximation is developed for deep longitudinally lution of the wave-packets spectrally concentrated at such Dirac driven photonic lattices. The physical system considered is that of

46 a laser-etched waveguide array which is helically-varying in the di- 2. A. Drouot, The bulk-edge correspondence for continuous honeycomb rection of propagation. The lattice is decomposed into sublattices lattices. Preprint available on demand. each of which are allowed move independently of one another. The 3. C. L. Fefferman, J. P. Lee-Thorp and M. I. Weinstein, Edge states in linear Floquet bands are constructed for various rotation patterns honeycomb structures. Ann. PDE 2(2016), no. 2, Art. 12, 80 pp. such as: different radii, different frequency, phase offset and quasi 4. J. P. Lee-Thorp, M. I. Weinstein and Y. Zhu, Elliptic operators with one-dimensional motion. Bulk spectral bands with nonzero Chern honeycomb symmetry: Dirac points, Edge States and Applications to number are calculated and found to support topologically protected Photonic Graphene. To appear in Archives for Rational Mechanics and edge wave envelopes which can propagate scatter-free around de- Analysis; preprint arXiv:1710.03389. fects. Finally, nonlinear soliton modes are found to propagate uni- directionally and scatter-free at lattice defects. Embedded eigenvalues and Fano resonance for metallic structures with small holes Frozen Gaussian approximation for the Dirac equation in semi-classical regime Junshan Lin ∗ Department of Mathematics and Statistics, Auburn University, Auburn, Lihui Chai AL 36849 Sun Yat-sen University, Guangzhou, China [email protected] [email protected] Stephen Shipman Emmanuel Lorin ∗ Department of Mathematics, Louisiana State University, Baton Rouge, LA Carleton University, Ottawa, Canada 70803 [email protected] [email protected] Xu Yang Hai Zhang University of California, Santa Barbabra, US Department of Mathematics, HKUST, Clear Water Bay, Kowloon, Hong [email protected] Kong This work is devoted to the derivation and analysis of the Frozen [email protected] Gaussian Approximation (FGA) for the Dirac equation in the semi- Fano resonance, which was initially discovered in quantum me- classical regime. Unlike the strictly hyperbolic system studied in chanics by Ugo Fano, has been extensively explored in photonics [1], the Dirac equation possesses eigenfunction spaces of multi- since the past decade due to its unique resonant feature of a sharp plicity two, which demands more delicate expansions for deriv- transition from total transmission to total reflection. Mathemati- ing the amplitude equations in FGA. Moreover, we prove that the cally, Fano resonance is related to certain eigenvalues embedded in nonrelativistic limit of the FGA for the Dirac equation is the FGA the continuum spectrum of the underlying differential operator. For of the Schrodinger¨ equation, which shows that the nonrelativis- photonic structures, the quantitative studies of embedded eigenval- tic limit is asymptotically preserved after one applies FGA as the ues mostly rely on numerical approaches. In this talk, based on semiclassical approximation. layer potential technique and asymptotic analysis, I will present quantitative analysis of embedded eigenvalues and their perturba- 1. J. Lu and X. Yang, Convergence of frozen Gaussian approximation for tion as resonances for a periodic array of subwavelength metallic high frequency wave propagation, Comm. Pure Appl. Math., 65 (2012), structure. From a quantitative analysis of the wave field for the 759-789. scattering problem, a rigorous proof of Fano resonance will be given. In addition, the field enhancement at Fano resonance fre- Edge states in near-honeycomb structures quencies will be discussed. Alexis Drouot Linear and nonlinear waves in honeycomb photonic Mathematics Department, Columbia University materials [email protected] Yi Zhu I will study aspects of wave propagation in a continuous honey- Tsinghua University comb structure with a line defect. In a perturbative regime, I will [email protected] give a full description of edge states (time harmonic waves prop- The past few years have witnessed an explosion of researches on agating along the line defect). This shows that all possible edge topological phenomena in different fields. One striking featur is states are adiabatic combinations of Dirac point Bloch modes. This the existence of wave motions that are immune to defects and dis- improves work of Fefferman, Lee-Thorp, Weinstein and Zhu who orders. In this talk, I will introduce our recent progresses on the constructed edge states of this form. analysis of such novel and subtle wave dynamics in topological I will then extend the result outside the perturbative regime. This photonic materials. Specifically, we prove the existence of Dirac amounts to prove topological protection of edge states, a result points in the honeycomb lattices and the existence of topological known as the bulk-edge correspondence. edge modes by introducing a line defect. We then derive the corresponding envelope equations to understand the subtle topological 1. A. Drouot, Characterization of edge states in perturbed honeycomb wave dynamics. Both linear and nonlinear wave dynamics are in- structures. Preprint, arXiv:1811.08218. vestigated.

47 Computing edge spectrum in the presence of disor- 1. M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves der without spectral pollution in the presence of symmetry, I. J. of Funct. Anal. 74 (1987), no. 1, 160–197. Kyle Thicke, Alexander Watson , and Jianfeng Lu ∗ 2. M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves Mathematics Department, Duke University, NC in the presence of symmetry, II. J. of Funct. Anal. 94 (1990), no. 2, [email protected], [email protected], [email protected] 308–348. Edge states, electronic states localized at the edge of a two-dimen- 3. S. De Bievre,` F. Genoud, S. Rota-Nodari, Orbital stability: analysis sional material, are defined mathematically as bound states of a meets geometry, in Nonlinear optical and atomic systems, 2146 (2015), semi-infinite edge Hamiltonian. Accurate numerical computation 147–273. of such states is complicated by the fact that computing using arbitrarily large finite truncations of the Hamiltonian yields spurious Convergence of Petviashvili’s method near periodic edge states localized at the truncation. We present a method which waves in the fractional Korteweg-deVries equation avoids this problem by properly accounting for the effect of the in- Uyen Le and Dmitry E. Pelinovsky finite bulk structure. Using this method we are able to probe com- McMaster University putationally the robustness of edge states of a graphene-like structure, modeled both by a continuum PDE and in the tight-binding [email protected] and [email protected] limit, to a broad class of perturbations. Robustness of such states is of interest for applications because of their potential utility for The fractional Korteweg-De Vries equation is a nonlinear partial wave-guiding [1]. differential equation which has several applications in fluid dynamics. There are many iterative methods to approximate the soli- 1. K. Thicke, A. B. Watson, and J. Lu, Computation of bound states of tary wave solution of this equation. One robust iterative scheme semi-infinite matrix Hamiltonians with applications to edge states of is the classical Petviashvili’s method. However, it has been nu- two-dimensional materials, merically found that the method may not converge in the case of https://arxiv.org/abs/1810.07082 (2018) periodic waves. In this presentation we will explain the failure of the classical Petviashvili’s method in approximating periodic waves in the fractional KdV equation from the spectrum of the generalized eigenvalue problem. We will also show that by modi- fying the method with a mean value shift, we achieve unconditional SESSION 16: “Existence and stability of peaked waves in nonlin- convergence for the Petviashvili’s method. ear evolution equations” Convexity of Whitham’s highest cusped wave Stability of standing waves for a nonlinear Klein- Alberto Enciso and Bruno Vergara Gordon equation with delta potentials ∗ Institute of Mathematical Sciences-ICMAT Elek Csobo [email protected] and [email protected] Delft University of Technology Javier Gomez-Serrano´ [email protected] Princeton University François Genoud [email protected] Ecole Polytechnique Fed´ erale´ de Lausanne Whitham’s model [3] of shallow water waves is a non-local dis- francois.genoud@epfl.ch persive equation that features travelling wave solutions as well as Masahito Ohta singularities. In this talk we will discuss a conjecture of Ehrnstrom¨ Tokyo University of Science and Wahlen´ [1] on the profile of solutions of extreme form to this [email protected] equation and see that there exists a highest, cusped and periodic solution, which is convex between consecutive peaks [2]. Julien Royer Universite´ Paul Sabatier 1. M. Ehrnstrom¨ and E. Wahlen,´ On Whitham’s conjecture of a highest [email protected] cusped wave for a nonlocal shallow water wave equation, Ann. Inst. H. Poincare´ Anal. Non. Lineaire´ , (in press), arXiv:1602.05384. We study the orbital stability of standing wave solutions of a one- 2. A. Enciso, J. Gomez-Serrano´ and B. Vergara, Convexity of Whitham’s dimensional nonlinear Klein-Gordon equation with Dirac poten- highest cusped wave, Submitted, (2018), arXiv:1810.10935. tials. The general theory to study orbital stability of Hamiltonian 3. G.B. Whitham, Variational methods and applications to water waves, systems was initiated by the seminal papers of Grillakis, Shatah, Proc. R. Soc. Lond. Ser. A , 299 (1967), 6-25. and Strauss [1, 2], newly revisited by De Bievre` et. al. in [3]. I present the Hamiltonian structure of the above system and the or- Evolution equations with distinct sectors of peakon- bital stability of the standing wave solutions of the equation. A type solutions major difficulty is to determine the number of negative eigenvalues of the linearized operator around the stationary solution, which we Stephen Anco overcome by a perturbation argument. Department of Mathematics and Statistics Brock University, Canada

48 [email protected] In this talk we outline the main ideas behind proving the existence of large-amplitude solitary wave solutions to the steady Whitham Peakon-type solutions are studied for a family of nonlinear dis- equation cφ + φ2 + K φ = 0 in the absence of surface tension. persive wave equations , mt + f (u, ux)m +(g(u, ux)m)x = 0 The strategy− is to use global∗ bifurcation theory. To construct a . When the nonlinearities or are higher m = u uxx f m (gm)x local curve, we modify and use a center manifold theorem for a than quadratic,− the equation is shown to possess infinitely many class of nonlocal equations. Then, we apply a version of the global distinct sectors of peakon-type solutions. The sectors arise from bifurcation theorem, which gives us an extra alternative related to freedom in how to regularize product of distributions, specifically the loss of compactness to exclude. This issue is dealt with using a a Dirac delta function multiplied by a power of a Heaviside step Hamiltonian identity. function. Only one choice of regularization coincides with the standard notion of a weak solution, but a different choice of regular- Krein signature without eigenfunctions and without ization appears to be necessary to preserve Hamiltonian structures eigenvalues. What is Krein signature and what does and integrability structure when they exist for smooth solutions. A it measure? generalized Camassa-Holm equation, with p-power nonlinearities, is used as an example to illustrate the results. Richard Kollar´ Comenius University, Bratislava, Slovakia 1. S.C. Anco and E. Recio, A general family of multi-peakon equations [email protected] and their properties, Accepted in J. Phys. A: Math. Theor. (2018). arXiv:math-ph/1609.04354 math-ph Krein signature is a frequently used tool to study spectral stability in Hamiltonian problems. Typically it is perceived as a sign of the 2. S.C. Anco and D. Kraus, Hamiltonian structure of peakons as weak linearized (relative) energy of the corresponding eigenstate. We solutions for the modified Camassa-Holm equation, Discrete and Con- tinuous Dynamical Systems (Series A) 38(9), (2018) 4449–4465. present four different ways the Krein signature can be calculated and interpreted without eigenfunctions or even without any corresponding eigenvalue. The different perspectives explain how Krein Regular patterns and defects for the Rayleigh-Benard´ signature relates to robustness of the spectral stability results. One convection of the examples presented is periodic travelling waves for general- Mariana Haragus ized KdV-type equations. Institut FEMTO-ST, Univ. Bourgogne-Franche Comte,´ France A non-local approach to waves of maximal height for [email protected] the Degasperis–Procesi equation We investigate pattern formation in the classical Rayleigh-Benard´ Mathias Nikolai Arnesen convection problem. We focus on regular patterns such as rolls Department of Mathematical Sciences, Norwegian University of Science and squares, and domain walls which are defects arising between and Technology rolls with different orientations. The mathematical problem con- 7491 Trondheim, Norway sists in solving the Navier-Stokes equations for the fluid velocity [email protected] coupled with an additional equation for the deviation of the temperature from the conduction profile in a cylindrical domain. Our We consider the non-local formulation of the Degasperis-Procesi 3 2 analysis relies upon a spatial dynamics formulation of the existence equation ut + uux + L( 2 u )x = 0, where L is the non-local Fourier 2 1 problem and a centre-manifold reduction. In this setting, regular multiplier operator with symbol m(ξ)=(1 + ξ )− . We show patterns and domain walls are found as equilibria and heteroclinic that all L∞, pointwise travelling-wave solutions are bounded above orbits, respectively, of a reduced system of ODEs. A normal form by the wave-speed and that if the maximal height is achieved they transformation allows us to identify a leading-order approximation, are peaked at those points, otherwise they are smooth. For suffi- solutions of which are then shown to persist using transversality ar- ciently small periods we find the highest, peaked, travelling-wave guments. solution as the limiting case at the end of the main bifurcation curve This is a joint work with Gerard´ Iooss (Nice). of P-periodic solutions. The results imply that the Degasperis- Procesi equation does not admit cuspon solutions in L∞. Large-amplitude solitary water waves for the Whitham equation Periodic traveling-wave solutions for regularized dis- Tien Truong persive equations: Sufficient conditions for orbital stability with applications Slvegatan 18, SE-22100 Lund, room: 515 [email protected] Fabio´ Natali Erik Wahln Departament of Mathematics - State University of Maringa´ Avenida Colombo, 5790, Maringa,´ PR, Brazil, CEP 87020-900 Slvegatan 18, SE-22100 Lund, room: 508 [email protected] [email protected] In this talk, we establish a new criterion for the orbital stability Miles H. Wheeler of periodic waves related to a general class of regularized disper- 3.120, Oskar-Morgenstern-Platz 1 sive equations. More specifically, we present sufficient conditions [email protected] for the stability without knowing the positiveness of the associated

49 hessian matrix. As application of our method, we show the orbital The extent of the relation between architectural and functional con- stability for a dispersive fifth-order model. The orbital stability nectivity in the cerebral cortex is a question which has attracted of periodic waves resulting from a minimization of a convenient much attention in recent years. Neuroscientists frequently use the functional is also presented. functional connectivity of neurons, i.e. the measures of causality or correlations between the neuronal activities of certain parts of Waves of maximal height for a class nonlocal equa- a network, to infer the architectural connectivity of the network, tions with homogeneous symbol which indicates the locations of underlying synaptic connections between neurons. Architectural connectivity can be used in the Gabriele Bruell and Raj Narayan Dhara modeling of neuronal processing and in the forming of conjectures Institute for Analysis, Karlsruher Institute of Technology about the nature of the neural code. These two types of connectiv- Department of Mathematics, University of West Bohemia ity are by no means identical, and no one-to-one correspondence or [email protected] and [email protected] mapping exists from one to the other. In particular, certain trivial We discuss the existence and regularity of periodic traveling wave measures of functional connectivity, such as correlations, give rise solutions of a class of nonlocal equations with homogeneous sym- to an undirected network, while synaptic architectural connectivity bol of order r, where r > 1. Based on the properties of the non- is always directed. Nevertheless, architectural connectivity can be local convolution− operator, we apply analytic bifurcation theory inferred from functional connectivity, and this work is one attempt and show that a highest, peaked periodic traveling wave solution to determine how to do so. is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly We begin by reconstructing the entire network using time-delayed Lipschitz. As an application of our analysis, we reformulate the spike-train correlation, and we determine the time required before steady reduced Ostrovsky equation in a nonlocal form in terms of an adequate reconstruction becomes possible and compare this to 2 a Fourier multiplier operator with symbol m(k) = k− . Thereby time spans employed by experimentalists. We then sample the ma- we recover its unique highest 2π-periodic, peaked traveling wave trix randomly and use the tool of matrix completion to fill-in the solution, having the property of being exactly Lipschitz at the crest. rest of the network. To be more experimentally valid, we next examine a small slice or submatrix of the network and determine how 1. L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Oceanol- much information we can deduce about the whole network from ogy, 18 (1978), 119–125. this small piece. An examination of the spectral properties of con- 2. M. Ehrnstrom,¨ M. Johnson, and K. Claasen, Existence of a highest nectivity matrices forms a major part of this analysis. wave in a fully dispersive two-way shallow water model, Arch Rational Mech Anal, (2018). Network microstructure dominates global network connectivity in synchronous event initiation 3. B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation, Princeton Series in Applied Mathematics, Princeton University Press, Duane Nykamp∗ and Brittany Baker Princeton, NJ, 2003. School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA [email protected] and [email protected] Quansheng Liu and Zhijun Qiao School of Mathematical and Statistical Sciences, University of Texas - Rio Using a network model where one can modulate both network mi- Grande Valley crostructure and global features of network connectivity, we examine the effects of both on the initiation of synchronous events. [email protected] The local microstructure is based on the SONET model [1], where one can specify the frequencies of different two-edge motifs in the In this talk, we will talk about some recent developments in inte- network. By combining these local features with global structure grable peakon systems, including the well-known CH, DP, based on an underlying geometry, we investigated the interplay be- FORQ/MCH, NE, and other models. Some high order peakon tween the microstructure and the macrostructure as synchronous models will be reported first time. This is the joint work with Quan- events emerge in the network. We discovered that the microstruc- sheng Liu. ture played the dominate role in determining synchronous event initiation.

1. L. Zhao, B. Beverlin II, T. Netoff and D. Q. Nykamp, Synchroniza- tion from second order network connectivity statistics Frontiers Comp. SESSION 17: “Nonlinear Dynamics of Mathematical Models in Neurosci., 5 (2011), 28. Neuroscience” Idealized models of insect olfaction Network reconstruction: Architectural and functional connectivity in the cerebral cortex Pamela B. Pyzza∗ Ohio Wesleyan University, 61 S. Sandusky Street, Delaware, OH 43015 Paulina Volosov and Gregor Kovacic [email protected] Rensselaer Polytechnic Institute 110 Eighth Street, Troy, NY 12180 Katie Newhall [email protected] and [email protected] University of North Carolina at Chapel Hill, Chapel Hill, NC

50 [email protected] total synchronization, to quenching, or to a non-trivial cluster synchronization state where two distinct oscillating behaviors coexist Douglas Zhou in the network. Shanghai Jiao Tong University, Shanghai, China [email protected] Nonlinear wave equations of shear radial wave prop- Gregor Kovaciˇ cˇ agation in ﬁber-reinforced cylindrically symmetric Rensselaer Polytechnic Institute, Troy, NY media [email protected] Alexei Cheviakov∗ and Caylin Lee David Cai Department of Mathematics and Statistics, University of Saskatchewan Deceased October 21, 2017 Saskatoon, SK, Canada S7N 5E6 [email protected] and [email protected]

When a locust detects an odor, the stimulus triggers a specific se- The framework of nonlinear elasticity can be systematically ap- quence of network dynamics of the neurons in its antennal lobe. plied to model complex materials, including biomembranes [1, 2]. The odor response begins with a series of synchronous oscillations, While the governing equations describe finite material displace- followed by a short quiescent period, with a transition to slow pat- ments without the assumption of their smallness, the mechanical terning of the neuronal firing rates, before the system finally returns properties of specific materials are defined in terms of constitu- to a background level of activity. We begin modeling this behavior tive functions. In this talk, we consider a model of an axially- using an integrate-and-fire neuronal network, composed of exci- symmetric elastic solid undergoing radially-propagating shear dis- tatory and inhibitory neurons, each of which has fast-excitatory, placements. We focus on anisotropic fiber-reinforced materials and fast- and slow-inhibitory conductance responses. We further with two embedded families of interacting, helically-oriented elas- derive a firing-rate model for each (excitatory and inhibitory) neu- tic fibers, commonly found in arterial walls [3]. As a first result, we ronal population, which allows for more detailed analysis of and in- observe that for a wide class of constitutive functions, radial wave sight into the plausible olfaction mechanisms seen in experiments, models considered in a fully non-linear setting lead to linear wave prior models, and our numerical model. We conclude that the tran- equations, which, moreover, do not contain any fiber-related terms. sition of the network dynamics through fast oscillations, a pause The corresponding boundary value problems can be solved exactly. in network activity, and the slow modulation of firing rates can Second, we consider a modified-fiber model, where the fibers have be described by a system which has a limit cycle of the fast vari- a nonzero radial projection. In this case, the shear displacements ables, slowly passes through a saddle-node-on-a-circle bifurcation G(R, t) are shown to satisfy nonlinear wave equations of the form eliminating the oscillations, and, eventually, slowly passes again through the bifurcation point, producing a new limit cycle with a 1 ∂ 2 3 Gtt = R N1GR + N2GR + N3GR + N4 , slower period – a process modulated by the slow variable. R ∂R h i

where Ni are constant material parameters. A further extension of On the dynamics of coupled Morris-Lecar neurons the model, incorporating viscoelastic effects, leads to generalized third-order nonlinear wave equations containing mixed space-time Shelby Wilson∗ Morehouse College derivatives GtR, GtRR. Further analysis of these nonlinear wave Department of Mathematics models is of interest for a better understanding of shear wave prop- 830 Westview Dr. agation in complex ideal and dissipative media, in particular, in Atlanta, GA 30314 biological tissues. [email protected] 1. G. A. Holzapfel and R. W. Ogden, Mechanics of biological tissue, In this work, we study the synchronization dynamics that arise Springer Science & Business Media (2006). from an architecture where Morris-Lecar neurons are globally cou- 2. A. F. Cheviakov and J-F. Ganghoffer, One-dimensional nonlinear elas- pled. We highlight a diverse set of asymptotic behavior for the cou- todynamic models and their local conservation laws with applications pled system, and we analyze these outcomes as a function of the to biological membranes, J. Mech. Behav. Biomed. Mater., 58 (2016), system parameters. We will brieﬂy present the nonlinear dynam- 105-121. ics and bifurcation behavior of Morris-Lecar neurons, thereby evi- 3. G. A. Holzapfel, T. C. Gasser, and R. W. Ogden, A new constitutive dencing Class I and Class II oscillatory behaviors. We also present framework for arterial wall mechanics and a comparative study of ma- the formalism that we use to investigate the globally coupled net- terial models, J. Elast. Phys. Sci. Sol., 61, 1-3 (2000), 1-48. work. We continue by analyzing how the interplay between the coupling strength and the size of the neuronal ensemble determines the asymptotic dynamics of the coupled system. It is found that this collective dynamics strongly depends on the topological nature of the limit- SESSION 18: “Negative ﬂows, peakons, integrable systems, and cycle where the neurons are individually oscillating. Our analysis their applications” shows that near the subcritical bifurcations to or from these limit cycles, the ensemble dynamics can converge to one of three case :

51 Global well-posedness for a nonlocal semilinear with time-dependent amplitudes and speeds; and thirdly, they ex- pseudo-parabolic equation with conical degeneration hibit wave breaking in which certain smooth initial data yields solutions whose gradient blows up in a finite time while stays Huafei Di and Yadong Shang ux u bounded. School of Mathematics and Information Science, Guangzhou University, All of these equations, and their various modified versions and non- Guangdong, Guangzhou 510006, P R China linear generalizations, belong to the general family of nonlinear E-mail:[email protected]; [email protected] dispersive wave equations mt + f (u, ux)m +(g(u, ux)m)x = 0, This paper deals with a class of nonlocal semilinear pseudo- m = u u , where f and g are arbitrary non-singular functions − xx parabolic equation with conical degeneration of u and ux. Remarkably, every equation in this family possesses N-peakon weak solutions [1]. p 1 1 p 1 dx1 In this work, a wide class of nonlinear dispersive wave equations ut But Bu = u − u B u − u dx′, −△ −△ | | − ZB | | x1 are shown to possess a novel type of peakon solution in which | | the amplitude and speed of the peakon are time-dependent. These on a manifold with conical singularity, where B is Fuchsian type novel dynamical peakons exhibit a wide variety of different be- Laplace operator with totally characteristic△ degeneracy on the haviours for their amplitude, speed, and acceleration, including an boundary . By using the modified methods of potential x1 = 0 oscillatory amplitude and constant speed which describes a peakon well with Galerkin approximation and concavity, global existence, breather. Examples are presented of families of nonlinear disper- uniqueness, finite time blow up and asymptotic behavior of solu- sive wave equations that illustrate various interesting behaviours, tions will be discussed at the low initial energy < and crit- J(u0) d such as asymptotic travelling-wave peakons, dissipating/anti- ical initial energy , respectively. Furthermore, we also J(u0) = d dissipating peakons, direction-reversing peakons, runaway and derive the threshold results of global existence and nonexistence blow up peakons, among others. for the sign-changing solutions under some certain conditions. Fi- nally, we investigate the global existence and finite time blow up of > 1. S.C. Anco, E. Recio, A general family of multi-peakon equations and solutions with the high initial energy J(u0) d by the variational their properties. arXiv: 1609.04354 math-ph method.

1. H. Chen, X. Liu, Y. Wei; Cone Sobolev inequality and Dirichlet prob- Instability and uniqueness of the peaked periodic lem for nonlinear elliptic equations on a manifold with conical singu- traveling wave in the reduced Ostrovsky equation larities, Calculus of Variations & Partial Differential Equations, 43(3-4) (2012), 463-484. Dmitry Pelinovsky Department of Mathematics, McMaster University, 2. M. Alimohammady, M.k. Koozehgar, G. Karamali; Global results for Hamilton, ON L8S 4K1, Canada semilinear hyperbolic equations with damping term on manifolds with conical singularity, Mathematical Methods in the Applied Sciences, [email protected] 40(11) (2017), 4160-4178. Anna Geyer∗ Delft Institute of Applied Mathematics, TU Delft, Accelerating dynamical peakons and their behaviour Van Mourik Broekmanweg 6, 2628 XE Delft, The Netherlands [email protected] Stephen C. Anco Department of Mathematics and Statistics, Brock University The existence of peaked periodic waves in the reduced Ostrovksy St. Catharines, ON L2S3A1, Canada equation has been known since the late 1970’s, see [1]. In our [email protected] recent paper we answer the long standing open question whether these solutions are stable and prove linear instability of the peaked Elena Recio∗ periodic waves using semi-group theory and energy estimates. More- Department of Mathematics, Universidad de Cadiz´ over, we prove that the peaked wave is unique in the space of peri- Puerto Real, Cadiz,´ Spain, 11510 odic L2 functions with zero mean and a single minimum per period, [email protected] and that the equation does not admit Hlder continuous solutions, Peakons are peaked travelling waves of the form i.e. there are no cusps. Our analysis relies on Fourier theory and the existence of a first integral, together with sharp estimates of the u(x, t)= a exp( x ct ) solution at the singularity at the peak. −| − | which were first found as weak solutions for the Camassa-Holm 1. L.A. Ostrovsky, Nonlinear internal waves a in rotating ocean, equation. Several other similar peakon equations are well known: Okeanologiya. 18 (1978) 181191. Degasperis-Procesi equation, Novikov equation, modified Camassa- Holm equation (also known as FORQ equation). Much of the interest in these equations is that, firstly, they are integrable systems Qilao Zha, Qiaoyi Hu, and Zhijun Qiao having a Lax pair, bi-Hamiltonian structure, hierarchies of sym- School of Mathematical and Statistical Sciences, University of Texas - Rio metries and conservation laws; secondly, they possess N-peakon Grande Valley weak solutions given by a linear superposition of single peakons [email protected]

52 In this paper, we study a two-component short pulse system, which Liouville correspondences between multi-component was produced through a negative integrable flow associated with integrable hierarchies the WKI hierarchy. The multi-soliton solutions for the two short Jing Kang and Xiaochuan Liu pulse system investigated, in particular, one-, two-, three-loop soliton, and breather soliton solutions are discussed in details with in- School of Mathematics, Northwest University, Xi’an 710069, P.R. China teresting dynamical interactions and shown through figures. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, P.R. China Some analysis results for the U(1)-invariant equation [email protected] and [email protected] Peter J. Olver∗ Stephen Anco and Huijun He∗ School of Mathematics, University of Minnesota, Minneapolis, MN 55455, Address (Department of Mathematics and Statistics, Brock University, USA St. Catharines, Ontario, L2S 3A1, Canada) [email protected] [email protected] and [email protected] Changzheng Qu Zhijun Qiao Department of Mathematics, Ningbo University, Ningbo 315211, P.R. China Address (School of Mathematical and Statistical Sciences, [email protected] University of Texas C Rio Grande Valley (UTRGV), Edinburg, TX, 78539, USA) In this talk, we establish Liouville correspondences for the inte- [email protected] grable two-component Camassa-Holm hierarchy, the two- component Novikov (Geng-Xue) hierarchy, and the two-compo- We study the peakon-like equation (U(1)-invariant equation) [1]: nent dual dispersive water wave hierarchy by means of the related Liouville transformations. This extends previous results on the Re iθ mt + (e (u + ux)(u¯ u¯x))m scalar Camassa-Holm and KdV hierarchies, and the Novikov and − x Sawada-Kotera hierarchies to the multi-component case. i Im(eiθ(u + u )(u¯ u¯ )) m = 0. − x − x (1) By applying the Littlewood-Paley theory and the transport the- Lax algebraic representation for an integrable hier- ory to this complex equation, we can obtain the Local well-posedness archy of U(1)-invariant equation in some certain Besov spaces. Shuxia Li and Zhijun Qiao (2) We study the blow-up phenomenon of the equation according School of Mathematical and Statistical Sciences, University of Texas - Rio 1 to its L conservation laws. Grande Valley (3) We study the analyticity and Gervey regularity of the equation. [email protected] (4) We study the persistence (the asymptotic behavior when the spatial variable x large) of the equation. Using the functional gradient approach of eigenvalues, this talk (5) We study the| orbital| stability of the U(1)-invariant equation. presents a pair of Lenards operators for the Levis vector fields and establishes commutator representations for hierarchies of Levisy 1. S. C. Anco and F. Mobasheramini, Integrable U(1)-invariant peakon equations. The relationship between potential and stationary Levi’s equations from the NLS hierarchy, Physica D, 355 (2017), 1–23. system is discussed in the end.

Some properties of Wronskian solutions of nonlin- A new perspective in anomalous viscoelasticity from ear differential equations the derivative with respect to another function view point Vesselin Vatchev Xiao-Jun Yang University of Texas Rio Grande Valley State Key Laboratory for Geomechanics and Deep Underground Engineer- [email protected] ing, China University of Mining and Technology, Xuzhou 221116, China Wronskian solutions are known for many nonlinear partial dif- [email protected] ferential equations including the well studied KdV and Boussi- Feng Gao nessq Equations. In the talk we present some properties of solu- State Key Laboratory for Geomechanics and Deep Underground Engineer- tions obtained from Wronskian determinants W(φ , φ ,..., φ ) 1 2 N ing, China University of Mining and Technology, Xuzhou 221116, China with generating functions φ (x, t) = cosh γ (x, t) or φ (x, t) = j j j [email protected] sinh γj(x, t) for γj(x, t) = pjx + σj(t), for real x and t and arbitrary functions σj. Hong-Wen Jing By following the Hirota bi-linear method we study the properties State Key Laboratory for Geomechanics and Deep Underground Engineer- of the multi-soliton functions u = (log W)xx, including charac- ing, China University of Mining and Technology, Xuzhou 221116, China terization of the non-singular choices of sinh and cosh. We also [email protected] present an explicit decomposition u = ∑N k ψ2 for k > 0 and j=1 j j j In this article, we address the new perspective in anomalous vis- ψ the eigenfunctions of the eigenvalue operator in the Lax Pair for j coelasticity containing the derivative with respect to another func- KdV and the Boussinessq equations. tion for the ﬁrst time. The Newton-like, Maxwell-like, Kelvin- We also discuss particular non-linear choices of the functions σj.

53 Voigt-like, Burgers-like and Zener-like models via the new deriva- [email protected] tives with respect to another functions are discussed in detail. The results are accurate and efficient in the descriptions of the scale The modified Camassa-Holm (mCH) equation is a bi-Hamiltonian behaviors of the complex materials involving the power law. system possessing N-peakon weak solutions for all N 1 in the setting of an integral formulation which is used in analysis≥ for Some new exact solutions for the extended (3+1)-di- studying local well-posedness, global existence, and wave break- mensional Jimbo-Miwa equation ing for non-peakon solutions. Unlike the original Camassa-Holm equation [1], the two Hamiltonians of the mCH equation do not a b a, Wenhao Liu , Binlu Feng , Yufeng Zhang ∗ reduce to conserved integrals (constants of motion) for 2-peakon a School of Mathematics, China University of Mining and Technology, weak solutions. Xuzhou, Jiangsu, 221116, Peoples Republic of China In this talk, we address this perplexing situation by finding an b School of Mathematics and Information Sciences, Weifang University, explicit conserved integral for N-peakon weak solutions for all Weifang, Shandong, 261061, Peoples Republic of China N 2. When N is even, the conserved integral is shown to provide≥ a Hamiltonian structure with the use of a natural Poisson n this paper, firstly, the bilinear form of the extended (3+1)-dimen- bracket that arises from reduction of one of the Hamiltonian struc- sional Jimbo-Miwa equation is provided, and its transformation of tures of the mCH equation. But when N is odd, the Hamiltonian dependent variable also is given. Secondly, we derived different so- equations of motion arising from the conserved integral using this lutions of the equation by using the homoclinic test approach, the Poisson bracket are found to differ from the dynamical equations three-wave method and the multiple exp-function method, respec- for the mCH N-peakon weak solutions. tively. Finally, all these solutions are presented via 3-dimensional Moreover, we show that the lack of conservation of the two Hamil- plots with choices some special parameters to show the dynamic tonians of the mCH equation when they are reduced to 2-peakon characteristics. weak solutions extends to N-peakon weak solutions for all N 2, and we discuss the connection between this loss of integrability≥ New integrable peakon equations from a modified structure and related work by Chang and Szmigielski on the Lax AKNS scheme pair for the mCH equation [2]. Evans Boadi, Sicheng Zhao, and Stephen Anco Department of Mathematics and Statistics 1. R. Camassa and D. D. Holm, An integrable shallow water equation Brock University, Canada with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664 2. X. Chang and J. Szmigielski, Lax integrability and the peakon problem The standard AKNS scheme for generating integrable evolution for the modified Camassa-Holm equation, Commun. Math. Phys., 358 systems is modified to obtain integrable peakon systems. In the (2018), 295-341. simplest situation given by sl(2,R) matrices, the modified scheme in the 1-component case yields the well-known Camassa-Holm equation, the modified Camassa-Holm (FORQ) equation, and a quadratic peakon equation on Novikov’s list. Large families of integrable peakon equations which contain arbitrary functions of SESSION 19: “Network dynamics” the dynamical variables are obtained in the 2-component case. A reduction of a family yields the U(1)-invariant integrable peakon Configurational stability for the Kuramoto-Sakagu- equations found recently [1, 2] by the tri-Hamiltonian splitting chi model method. Jared Bronski and Lee DeVille Recent results on work in progress for su(2), sl(2,C), sl(3,R), su(3), su(2,1) will be presented as well. Department of Mathematics University of Illinois at Urbana-Champaign, IL, 61801 1. S.C. Anco and F. Mobasheramini, Integrable U(1)-invariant peakon [email protected] and [email protected] equations from the NLS hierarchy, Physica D 355 (2017), 1–23. Thomas Carty∗ 2. S.C. Anco, X. Chang, J. Szmigielski, The dynamics of conservative Department of Mathematics peakons in the NLS hierarchy, Studies in Applied Math. (2018), 1–34. Bradley University, Peoria, IL, 61625 [email protected] Hamiltonian structure of peakons as weak solutions The Kuramoto–Sakaguchi model is a generalization of the well- for the modified Camassa-Holm equation known Kuramoto model that adds a phase-lag parameter, or “frus- Stephen Anco tration” to a network of phase-coupled oscillators. The Kuramoto model is a flow of gradient type, but adding a phase-lag breaks Department of Mathematics and Statistics the gradient structure, significantly complicating the analysis of Brock University the model. We present several results determining the stability of [email protected] phase-locked configurations: the first of these gives a sufficient Daniel Kraus* condition for stability, and the second a sufficient condition for in- Department of Mathematics stability. In fact, the instability criterion gives a count, modulo 2, SUNY Oswego

54 of the dimension of the unstable manifold to a fixed point and hav- 3. Sun-Ho Choi and Seung-Yeal Ha, Quantum synchronization of the ing an odd count is a sufficient condition for instability of the fixed schrodinger lohe model, Journal of Physics A 47 (35), 355104 (2014). point. http://iopscience.iop.org/article/10.1088/1751- 8113/47/35/355104/meta 1. J.C. Bronski, T. Carty, and L. DeVille, Configurational stability for the 4. Lee DeVille, Synchronization and stability for quantum kuramoto, Kuramoto-Sakaguchi model, Chaos 28, 103109 (2018), 16 99. Journal of Statistical Physics (2018). https://doi.org/10.1007/s10955-018-2168-9 Adaptive zero determinant strategies in the iterated prisoners dilemma tournament The universal covariant representation and amenability Emmanuel Estrada and Dashiell Fryer ∗ Mamoon Ahmed Department of Mathematics Amman, 11941, Jordan San Jose´ State University, CA, 95192 [email protected] [email protected] and [email protected] Let be a quasi-lattice ordered group. In this paper we give a We have created an adaptive zero determinant strategy that changes (G, P) modified proof of Laca and Raeburn’s theorem about the covariant its parameters using the outcome of the last round as input. We then isometric representations of amenable quasi-lattice ordered groups ran this adaptive zero determinant strategy against a tournament of [1, Theorem 3.7]. First, we construct a universal covariant repre- other zero determinant strategies. We observed that the adaptive sentation for a given quasi-lattice ordered group and show strategy had a higher average score than the other zero determi- (G, P) that it is unique. Then we show if is amenable, true rep- nant strategies when we ran the tournament for a large amount of (G, P) resentations of generate -algebras that are isomorphic to rounds. (G, P) C∗ the universal object.

A matrix valued Kuramoto model 1. M. Laca and I. Raeburn, Semigroup crossed products and the Toeplitz Jared Bronski algebras of nonabelian groups, J. Funct. Anal. 139 (1996), 415–440. Department of Mathematics 2. M. Ahmed and A. Pryde, The structure Theorem and the Commutator University of Illinois at Urbana-Champaign, IL, 61801 Ideal of Toeplitz Algebras, To appear Glasg. Math. J. [email protected] Bistability in the Kuramoto model Thomas Carty and Sarah Simpson∗ Department of Mathematics Timothy Ferguson∗ Bradley University, Peoria, IL, 61625 Department of Mathematics [email protected] and [email protected] Arizona State University, AZ, 85281 [email protected] A need to better approximate quantum mechanical oscillatory phenomena motivated Lohe [1], [2] and others [3], [4] to derive non- The Kuramoto model is a general model for the behavior of net- Abelian generalizations of the Kuramoto model for phase-locking. work coupled oscillators. For such a system stable phase-locked Here we propose and analyze a purely real-valued model of this solutions are of critical importance to the global long-time behavior type in which we consider a collection of symmetric matrix-valued of the system. In particular, we consider the question of bistability, variables. This is a gradient ﬂow where the matrices evolve to min- namely, when two such stable solutions exist simultaneously. In imize energy in such a way as to try to align their eigenframes. The this regard, we give a generic condition for a bistability forming phase-locked state is one where the eigenframes all align, and thus bifurcation to occur in ring networks with positive coupling, and the matrices all commute. We analyze the stability of the phase- apply it to produce examples of bistability. Furthermore, we derive locked state for n n matrices, and show that it is stable. We a necessary condition for this bifurcation in terms of the phase- also show that in the× case of 2 2 matrices the model reduces to angles and numerically demonstrate that this condition is closely × a form of the Kuramoto model with dynamic coupling. Addition- related to the order parameter for N = 3 oscillators. ally, we show that in the case of 2 2 matrices, the model has a dynamically unstable set of ﬁxed points× analogous to the twist states arising in the standard Kuramoto model. SESSION 20: “Dynamical Systems and integrability” 1. M.A. Lohe, Non-Abelian Kuramoto models and synchronization, Journal of Physics A 42 (39), 395101 (2009). Hidden solutions of discrete systems http://iopscience.iop.org/article/10.1088/1751- Nalini Joshi 8113/42/39/395101/meta School of Mathematics and Statistics F07, 2. M.A. Lohe, Quantum synchronization over quantum networks, Journal The University of Sydney, NSW 2006, Australia of Physics A 43 (46), 465301 (2010). [email protected] http://iopscience.iop.org/article/10.1088/1751- 8113/43/46/465301/meta Christopher J. Lustri Department of Mathematics, Macquarie University, NSW 2109, Australia

55 [email protected] 2. K. W. Chow, S. C. Tsang and C. C. Mak, Another exact solution for two-dimensional, inviscid sinh Poisson vortex arrays, Phys. Fluids, 15 Steven Luu (2003), 2437-2440. School of Mathematics and Statistics F07, The University of Sydney, NSW 2006, Australia [email protected] Ellipsoidal billiards and Chebyshev-type polynomials Hidden solutions are well known in irregular singular limits of dif- Vladimir Dragovic´ ferential equations. Such solutions are not able to be identified ∗ uniquely through conventional analysis, because free parameters The University of Texas at Dallas, Richardson, TX identifying the solution lie hidden beyond all orders of a diver- [email protected] gent asymptotic expansion. We identify such solutions of discrete Milena Radnovic´ Painleve´ equations, specifically q-PI, d-PI, and d-PII, in the limits University of Sydney, Sydney, Australia where their independent variable goes to infinity and extend the in- [email protected] vestigation to further solutions and to partial difference equations. Through such analysis, we determine regions of the complex plane A comprehensive study of periodic trajectories of the billiards within in which the asymptotic behaviour is described by a power series ellipsoids in the d-dimensional Euclidean space is presented. The expression, and find that the behaviour of these asymptotic solu- novelty of the approach is based on a relationship established be- tions shares a number of features with the tronquee´ and tri-tronquee´ tween the periodic billiard trajectories and the extremal polynomi- solutions of corresponding differential Painleve´ equation. als of the Chebyshev type on the systems of d intervals on the real line. As a byproduct, for d = 2 a new proof of the monotonicity of JL15. N. Joshi and C.J. Lustri. Stokes phenomena in discrete Painleve´ I. the rotation number is obtained and the result is generalized for any d. The case study of trajectories of small periods T, d T 2d is Proceedings of the Royal Society of London A: Mathematical, Physi- ≤ ≤ cal and Engineering Sciences, 471(2177):20140874, 2015. given. In particular, it is proven that all d-periodic trajectories are JLL17. N. Josh and C.J. Lustri and S. Luu. Stokes phenomena in discrete contained in a coordinate-hyperplane and that for a given ellipsoid, Painleve´ II. Proc. R. Soc. A, 473(2198):20160539, 2017. there is a unique set of caustics which generates d + 1-periodic trajectories. A complete catalog of billiard trajectories with small JLL18. N. Joshi, C.J. Lustri, and S. Luu. Nonlinear q-Stokes phenomena periods is provided for d = 2 [2] and d = 3 [1]. for q-Painleve´ I. arXiv:1807.00450 [math-ph], 2018. Surprisingly enough, the Cayley type conditions for d = 2 appear JL18. N. Joshi, C.J. Lustri. Generalized Solitary Waves in a Finite- to be connected to the so-called discriminantly separable polyno- Difference Korteweg-de Vries Equation. arXiv:1808.09654 [math- mials, a class of polynomials introduced by the first author in his ph], 2018. study [3] of the classical Kowalevski integration of the Kowalevski top. Two dimensional stationary vorticity distribution and integrable system 1. V. Dragovic,´ M. Radnovic,´ Periodic ellipsoidal billiard trajectories and extremal polynomials, arXiv 1804.02515 Yasuhiro Ohta 2. V. Dragovic,´ M. Radnovic,´ Caustics of Poncelet polygons and classical Department of Mathematics, Kobe University extremal polynomials, arXiv 1812.02907 Regular and Chaotic Dynam- Rokko, Kobe 657-8501, Japan ics, Vol. 24, 2019. [email protected] 3. V. Dragovic,´ Geometrization and Generalization of the Kowalevski top, In two dimensional inviscid incompressible fluid, stationary flows Communications in Mathematical Physics, Vol. 298, no. 1, p. 37-64, are described by solutions of the nonlinear Klein-Gordon equation 2010. for stream function. It is well-known that in the two dimensional Toda lattice hierarchy there are some integrable systems of the A discrete analogue of the Toda hierarchy and its form of nonlinear Klein-Gordon equation, namely Liouville equa- some properties tion, sine-Gordon equation, sinh-Gordon equation, Tzitzeica equa- Masato Shinjo and Koichi Kondo tion. Many solutions for these equations and stationary vorticity distributions have been widely and deeply investigated in the con- Faculty of Science and Engineering, Doshisha University, text of fluid dynamics. See for example [1], [2]. 1-3 Tatara miyakodani, Kyotanabe, Kyoto 610-0394, Japan [email protected] [email protected] We study a class of solutions of integrable system which are related and with two dimensional vorticity distributions similar to the Stuart The Toda equation describing motions governed by nonlinear vortex street. The solutions correspond to deformed vortex streets springs is well-known as famous soliton equation in the study of and some of them have singularities which are regarded as an array integrable systems. Flaschka’s variables [1] lead to Lax dynamics of point vortices. Relevance of such solutions as steady fluid flow of the Toda equation with tridiagonal matrix. In [2], a skillful dis- is also discussed. cretization of the Toda equation is presented. The discrete Toda equation contributes to computing eigenvalues of tridiagonal ma- 1. M. C. Haslam, C. J. Smith, G. Alobaidi and R. Mallier, Some nonlinear trices. vortex solutions, Int. J. Diff. Eq., 2012 (2012), 929626. One of extensions of the Toda equation with associated tridiagonal matrix is called the Toda hierarchy [3]. In this talk, we propose

56 a discrete analogue of the Toda hierarchy, which corresponds to a 3. P. J. Olver. Applications of Lie Groups to Differential Equations. generalization of the discrete Toda equation in [2]. We comprehen- Springer-Verlag, Berlin, 1986. sively clarify Lax dynamics and solutions to both of the continuous 4. E. T. Whittaker. A Treatise on the Analytical Dynamics of Particles and discrete equations. and Rigid Bodies. Cambridge University Press, Cambridge, 1999.

1. H. Flaschka, The Toda lattice. II. Existence of integrals, Phys. Rev. B, 9 (1974), 1924–1925. 2. R. Hirota, Conserved quantities of “random-time Toda equation”, J. Phys. Soc. Jpn., 66 (1997), 283–284. SESSION 21: “Stochastic Dynamics in Nonlinear Systems” 3. J. Moser, Finitely many mass points on the line under the influence of an exponential potential-An integrable system, Dynamic Systems, A network of transition pathways in a model granu- Theory and Applications, Lecture Notes in Phys., 38 (1975), 467–497. lar system Katie Newhall In this talk I shall introduce the idea of a quasi-pfaffian, this is the UNC Chapel Hill pfaffian equavalent to a quasi-determinant, these quasi-pfaffians [email protected] have identities analogous to quasi-determinant identities and in the Many intriguing dynamical properties of complex systems, such commutative case, they reduce to ratios of pfaffians. We will look as metastability or resistance to applied forces, emerge from the at some quasi-pfaffian identities and look at the connection be- underlying energy landscape. High-dimensional systems can have tween these quasi-pfaffians and noncommutative integrable sys- complex energy landscapes with numerous energy-minimizing states. tems. Especially in randomly packed granular materials for which knowing the single global energy minimizing state is unimportant, un- On the inverse problem of the discrete calculus of derstanding the interconnectivity of minimums via transition paths variations through saddles allows for extracting the dominant features of the G. Gubbiotti system. The energy landscape of a jammed 2D packing of bidis- School of Mathematics and Statistics,The University of Sydney, Carslaw perse disks is modeled as a collection of overlapping circles, defin- Building, F07, 2006, Sydney (NSW), Australia ing an energy penalty based on the amount of overlap. I propose a [email protected] systematic approach to mapping out the transition pathways from energy minimizer to saddle point to minimizer forming a network One of the most powerful tools in Mathematical Physics since Eu- of transition pathways. This computational method is based on the ler and Lagrange is the calculus of variations. The variational for- climbing string method of W. Ren and E. Vanden-Eijnden that has mulation of mechanics where the equations of motion arise as the been successfully applied to problems in chemistry. The ultimate minimum of an action functional (the so-called Hamilton’s princi- goal is to relate observable phenomena like a granular material’s ple), is fundamental in the development of theoretical mechanics rearrangements preceding failure events to dynamics on the net- and its foundations are present in each textbook on this subject work representation of the energy landscape of the system. [1, 2, 4]. Beside this, the application of calculus of variations goes beyond mechanics as many important mathematical problems, e.g. Limiting behaviors of high dimensional stochastic the isoperimetrical problem and the catenary, can be formulated in spin ensemble terms of calculus of variations. Y. Gao , J. Marzuola and K. Newhall An important problem regarding the calculus of variations is to de- ∗ termine which system of differential equations are Euler–Lagrange Department of Mathematics, University of North Carolina at Chapel Hill equations for some variational problem. This problem has a long [email protected], [email protected] and and interesting history, see e.g. [3]. The general case of this prob- [email protected] lem remains unsolved, whereas several important results for par- K. Kirkpatrick ticular cases were presented during the 20th century. Department of Mathematics, University of Illinois at Urbana-Champaign In this talk we present some conditions on the existence of a La- [email protected] grangian in the discrete scalar setting. We will introduce a set of differential operators called annihilation operators. We will use J. Mattingly these operators to reduce the functional equation governing of ex- Department of Mathematics, Duke University istence of a Lagrangian for a scalar difference equation of arbitrary [email protected] > even order 2k, with k 1 to the solution of a system of linear par- Lattice spin models in statistical physics are used to understand tial differential equations. Solving such differential equations one magnetism. Their Hamiltonians are a discrete form of a version can either find the Lagrangian or conclude that it does not exist. of a Dirichlet energy, signifying a relationship to the Harmonic map heat flow equation. The Gibbs distribution, defined with this 1. H. Goldstein, C. Poole, and J. Safko. Classical Mechanics. Pearson Hamiltonian, is used in the Metropolis-Hastings (M-H) algorithm Education, 2002. to generate dynamics tending towards an equilibrium state. In the 2. L. D. Landau and E. M. Lifshitz. Mechanics. Course of Theoretical limiting situation when the inverse temperature is large, we estab- Physics. Elsevier Science, 1982. lish the relationship between the discrete M-H dynamics and the

57 continuous Harmonic map heat flow associated with the Hamil- ensembles operate in the nanoscale, directly observing an attach- tonian. We show the convergence of the M-H dynamics to the ment state is difficult, and creates a need for developing models Harmonic map heat flow equation in two steps: First, with fixed which provide statistics for multiple motor ensembles based on lattice size and proper choice of proposal size in one M-H step, known parameters from one motor systems. the M-H dynamics acts as gradient descent and will be shown to converge to a system of Langevin stochastic differential equations The effect of moderate noise on a limit cycle oscilla- (SDE). Second, with proper scaling of the inverse temperature in tor: counterrotation and bistability the Gibbs distribution and taking the lattice size to infinity, it will Jay Newby be shown that this SDE system converges to the deterministic Har- Department of Mathematical and Statistical Sciences, University of Al- monic map heat flow equation. Our results are not unexpected, but berta, Edmonton, Canada show remarkable connections between the M-H steps and the SDE [email protected] Stratonovich formulation, as well as reveal trajectory-wise out of equilibrium dynamics to be related to a canonical PDE system with The effects of noise on the dynamics of nonlinear systems is known geometric constraints. We are currently working on introducing to lead to many counterintuitive behaviors. Using simple planar spatially correlated noise to obtain the convergence to a stochastic limit cycle oscillators, we show that the addition of moderate noise PDE. leads to qualitatively different dynamics. In particular, the system can appear bistable, rotate in the opposite direction of the deter- Improving sampling accuracy of SG-MCMC meth- ministic limit cycle, or cease oscillating altogether. Utilizing stan- ods via non-uniform subsampling of gradients dard techniques from stochastic calculus and recently developed stochastic phase reduction methods, we elucidate the mechanisms Ruilin Li, Xin Wang, Hongyuan Zha, Molei Tao ∗ underlying the different dynamics and verify our analysis with the Georgia Institute of Technology use of numerical simulations. Last, we show that similar bistable [email protected] behavior is found when moderate noise is applied to the FitzHugh- In the training of neural networks or Bayesian inferences with big Nagumo model, which is more commonly used in biological appli- data, additive gradients that sum a large amount of terms need to cations. be repeatedly evaluated. To reduce the computational cost of such evaluations, the machine learning community relied on Stochastic- Stochastic parameterization of subgrid-scales in one- Gradient-MCMC methods, which approximate gradients by stochas- dimensional shallow water equations tic ones via uniformly subsampled data points. This, however, in- Matthias Zacharuk, Stamen Dolaptchiev, Ulrich Achatz troduces extra variance artificially. How to design scalable algo- Johann Wolfgang Goethe-Universitt Frankfurt/Main rithms that correctly sample the target distribution is an outstand- [email protected], [email protected], ing challenge. [email protected] This talk will describe a heuristic step towards this challenge. The core idea is to use exponentially weighted stochastic gradients Ilya Timofeyev∗ (EWSG) to replace uniform ones. A demonstration based on sec- University of Houston ond-order Langevin equation coupled with a Metropolis chain will [email protected] be provided. The improved performance will be discussed through We address the question of parameterizing the subgrid scales in both theoretical evidence and numerical experiments on multiple simulations of geophysical flows by applying stochastic mode re- learning tasks. While statistical accuracy has improved, the speed duction to the one-dimensional stochastically forced shallow water of convergence was empirically observed to be at least comparable equations. The problem is formulated in physical space by defin- to the uniform version. ing resolved variables as local spatial averages over finite-volume cells and unresolved variables as corresponding residuals. Based Averaging for systems of nonidentical molecular mo- on the assumption of a time-scale separation between the slow spa- tors tial averages and the fast residuals, the stochastic mode reduction Joseph Klobusicky and Peter Kramer procedure is used to obtain a low-resolution model for the spatial Department of Mathematical Sciences averages alone with local stochastic subgrid-scale parameterization Rensselaer Polytechnic Institute coupling each resolved variable only to a few neighboring cells. [email protected] and [email protected] The closure improves the results of the low-resolution model and outperforms two purely empirical stochastic parameterizations. It John Fricks is shown that the largest benefit is in the representation of the en- School of Mathematical and Statistical Sciences ergy spectrum. By adjusting only a single coefficient (the strength Arizona State University of the noise) we observe that there is a potential for improving [email protected] the performance of the parameterization, if additional tuning of the coefficients is performed. In addition, the scale-awareness of the The shuttling of molecular cargo across a cell is aided by the di- parameterizations is studied. rected transport of molecular motors on a microtubule network. A cargo may be attached to several motors which can attach and de- 1. M. Zacharuk, S. I. Dolaptchiev, U. Achatz, I. Timofeyev, ”Stochas- tach from a microtubule during a typical procession. Since motors tic subgrid-scale parameterization for one-dimensional shallow water

58 1 dynamics using stochastic mode reduction”, Q.J.R. Meteorol. Soc., KdV is wellposed in H− 144(715), (2018), 1975-1990. Rowan Killip∗ and Monica Visan Department of Mathematics, UCLA Coupling for Hamiltonian Monte Carlo [email protected] and [email protected] Nawaf Bou-Rabee I will describe a proof of the well-posedness of the Korteweg–de Department of Mathematical Sciences 1 Vries equation in the Sobolev space H− that works both on the Rutgers University Camden line and on the circle. On the line, this result was previously un- 311 North Fifth Street known; on the circle it was proved by Kappeler and Topalov. This Camden, NJ 08102 is joint work [1] with Monica Visan. [email protected] 1 We present a new coupling approach to study the convergence of 1. R. Killip and M. Visan, KdV is wellposed in H− . Preprint the Hamiltonian Monte Carlo (HMC) method. Specifically, we arXiv:1802.04851. prove that the transition step of HMC is contractive w.r.t. a care- fully designed Kantorovich (L1 Wasserstein) distance. The lower The construction and evaluation of shock wave so- bound for the contraction rate is explicit. Global convexity of the lutions to the KdV equation and a linear KdV-like potential is not required, and thus multimodal target distributions equation are included. Explicit quantitative bounds for the number of steps required to approximate the stationary distribution up to a given er- Thomas Trogdon∗ ror ǫ are a direct consequence of contractivity. These bounds show University of California, Irvine that HMC can overcome diffusive behavior if the duration of the [email protected] Hamiltonian dynamics is adjusted appropriately. This talk is based We consider the problem of computing the inverse scattering trans- on joint work with Andreas Eberle and Raphael Zimmer. form for the KdV equation on R when the initial data q0(x) satis- N. Bou-Rabee, A. Eberle, and R. Zimmer, Coupling and convergence fies limx +∞ q0(x) = limx ∞ q0(x). We build on the work for Hamiltonian Monte Carlo, arXiv preprint arXiv:1805.00452, 2018. of Cohen→ and Kappeler (1985)→− [2] and Andreiev et al. (2016) [1]. In particular, we demonstrate how the use of both left and right reflection coefficients is necessary, in contrast to decaying initial data. Properties of this solution motivate a linearization that shares SESSION 22: “Modern methods for dispersive wave equations” non-trivial structure with its nonlinear counterpart. This is joint work with Deniz Bilman, Dave Smith and Vishal Vasan. Singular limits of certain Hilbert-Schmidt integral operators and applications to tomography 1. K Andreiev, I Egorova, T L Lange, and G Teschl. Rarefaction waves of the Korteweg–de Vries equation via nonlinear steepest descent. J. Marco Bertola Differ. Equ., 261(10):5371–5410, 2016. Department of Mathematics & Statistics, Concordia University 2. A Cohen and T Kappeler. Scattering and inverse scattering for steplike Montreal, Quebec H3G 1M8 Canada potentials in the Schrodinger equation. Indiana Univ. Math. J., 34:127– [email protected] 180, 1985.

Elliot Blackstone∗, Alexander Katsevich and Alexander Tovbis Department of Mathematics, University of Central Florida Long-time asymptotics for the massive Orlando, Florida 32816-1364 U.S.A. Thirring model [email protected], [email protected] and Alexan- Aaron Saalmann [email protected] Weyertal 86-90 In this talk we discuss the asymptotics of the spectrum of self- 50939 Cologne, Germany adjoint Hilbert-Schmidt integral operators with the so-called inte- [email protected] grable kernels in a certain singular limit, where the limiting operator is still bounded but has a continuous spectral component. Such From the analytical point of view, the massive Thirring model operators appear when studying stability of the interior problem of (MTM), i(u + u )+ v + u v 2 = 0, tomography. They are related to Finite Hilbert Transform (FHT) t x | | i(v v )+ u + u 2v = 0, on several intervals, when neighboring intervals are touching each t − x | | other. The case of separate intervals, when the corresponding inte- is of special interest, because it has a representation in terms of a gral operators are of Hilbert-Schmidt class, was studied in [1]. Our Lax pair, consisting of two linear operators L and A. Thanks to the work is based on the method of Riemann-Hilbert problems. Lax pair, the MTM admits an exact solution by the inverse scattering transform (IST), see [1]. 1. M. Bertola, A. Katsevich and A. Tovbis, Singular Value Decomposi- As it is also known from other nonlinear dispersive equations one tion of a Finite Hilbert Transform Deﬁned on Several Intervals and the can create solitons for the MTM. These special solutions are waves Interior Problem of Tomography: The Riemann-Hilbert Problem Ap- that move at constant speed and do not change in shape. They can proach. Communications on Pure and Applied Mathematics, 2016.

59 refuse to disperse only because of the presence of the nonlinear- [email protected] and [email protected] ity in the equation. It is relatively simple to characterize solitons, ¨ based on their scattering data. Using suitable Riemann-Hilbert The defocusing nonlinear Schrodinger equation has a family of ra- techniques it is possible to analyse the interaction of two (or more) tional solutions that can be expressed in terms of generalized Her- mite polynomials. These special polynomials have strong ties to solitons. Furthermore, it can be shown precisely that each soli- rational solutions of the fourth Painleve´ equation. The family of ton will eventually enter the light cone t > x . Using the solutions to the NLS equation can be expressed in terms of or- ∂–method (nonlinear steepest descent) one{| can| show| |} that outside the light cone any solution (not only solitons) converges to zero thogonal polynomials. Using this, we apply the Deift-Zhou nonlinear steepest-decent method to asymptotically analyze the limit with a rate of t 3/4. Inside the light-cone there are basically − ∞ two different possibilities.| | Assuming that the initial data is free n , where n indexes the rational solutions to the nonlinear Schr→odinger¨ equation. of solitons one can use the ∂–method and some model Riemann– Hilbert problems to show that the solution of the MTM scatters to a linear solution modulo phase correction. This linear solution can 1. R. Buckingham,Large-degree asymptotics of rational Painleve-IV´ functions associated to generalized Hermite polynomials, Int. Math. be computed explicitly from the scattering data and its amplitude 1/2 Resea. Notic., (2018) decays with a rate of t − . The second possibility is that the initial data contains finitely| | many solitons. Then, one can prove 2. P. Clarkson, The fourth Painleve´ equation and associated special poly- that any solution breaks up into finitely many single solitons that nomials, J. Math. Phys., 65 (2003), 5350–5374 travel at different speeds and thus, diverge. The remainder term is 1/2 O( t − ). Long-time behavior of solutions to the modified KdV | | In the talk it will be explained how the MTM can be rewritten in equation in weighted sobolev space terms of a Riemann-Hilbert problem and the main aspects of the Gong Chen and Jiaqi Liu Riemann–Hilbert analysis will be discussed. University of Toronto 1. Dmitry E. Pelinovsky and Aaron Saalmann. Inverse Scattering for the [email protected], [email protected] Massive Thirring Model. Fields Institute Communications, (2019), (ac- he long time behavior of solutions to the defocusing modified Korte- cepted). weg-de vries (MKDV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach uses the inverse Semiclassical soliton ensembles and the three-wave scattering transform and the nonlinear steepest descent method of resonant interaction (TWRI) equations Deift and Zhou and its reformulation by Dieng, Miller and Robert Buckingham McLaughlin through ∂-method. University of Cincinnati [email protected] Soliton resolution for the derivative NLS Robert Jenkins Robert Jenkins∗ Colorado State University Department of Mathematics, Colorado State University [email protected] Fort Collins, Colorado 80523-1801, U. S. A. [email protected] Peter Miller University of Michigan Jiaqi Liu [email protected] Department of Mathematics, University of Toronto Toronto, Ontario, Canada M5S 2E4 I’ll discuss some of our ongoing work [1] on the the TWRI equa- [email protected] tions, a universal model of the first stage of nonlinear behavior in weakly nonlinear systems which support resonant triads. This Peter A. Perry∗ system is integrable with a third order Lax-Pair. The higher or- Department of Mathematics, University of Kentucky der nature of the system complicates the scattering theory for the Lexington, Kentucky 40506–0027, U. S. A. Lax operator. I’ll present a scheme we’ve introduced to study the [email protected] system using a soliton ensemble approach, some numerical exper- Catherine Sulem iments, and analytic results. Department of Mathematics, University of Toronto 1. R. Buckingham, R. Jenkins and P. Miller, Semiclassical soliton ensem- Toronto, Ontario, Canada M5S 2E4 bles for the three wave resonant interaction equations, Comm. Math. [email protected] Phys., 354 (2017), 1015-1100. Kaup and Newell [4] showed that the Derivative Nonlinear Schro-¨ dinger equation Asymptotics of rational solutions of the defocusing nonlinear Schrodinger¨ equation 2 iut + uxx iε u u = 0, (4) − | | x Robert J. Buckingham and Donatius DeMarco∗ Department of Mathematical Sciences. University of Cincinnati which describes the propagation of nonlinear Alfven´ waves in plas- PO Box 210025 Cincinnati, OH 45221. mas [5], is completely integrable. Here we’ll report on joint work

60 with Robert Jenkins, Jiaqi Liu, and Catherine Sulem [3] which ex- for the physically motivated scenarios including classical Landau ploits the complete integrability to show that, for generic decaying damping, and growth of longitudinal and transverse plasma insta- initial data, the soliton resolution conjecture holds for this equa- bilities in single and multiple species plasmas. tion. That is, we show that the solution u(x, t) of the initial value problem for (4) resolves into the sum of ﬁnitely many soliton so- 1. J. W. BANKSAND A.GIANESINI ODUAND R.L.BERGERAND lutions and a radiation term. A consequence of our analysis is the T. CHAPMANAND W. T. ARRIGHIAND S.BRUNNER, High-Order asymptotic stability of soliton solutions. Accurate Conservative Finite Difference Methods for Vlasov Equations To obtain the asymptotics we use the ∂-steepest descent methods in 2D+2V, SIAM J. Sci. Comput. (submitted). pioneered by Dieng and McLaughlin [2] and further developed by 2. J. W. BANKSAND J. A. F. HITTINGER, A new class of non-linear, Borghese, Jenkins, and McLaughlin [1] to prove soliton resolution ﬁnite-volume methods for Vlasov simulation, IEEE T. Plasma. Sci., 38 for the cubic focusing NLS. (2010), pp. 2198–2207.

1. Michael Borghese, Robert Jenkins, Kenneth D. T.-R. McLaughlin. Non-canonical Hamiltonian structure and integra- Long-time asymptotic behavior of the focusing nonlinear Schrodinger¨ bility for 2D fluid surface dynamics equation. Ann. Inst. H. Poincare´ Anal. Non Lineaire´ 35 (2018), no. 4, 1 2 3, 887–920. A. I. Dyachenko , S. A. Dyachenko , P. M. Lushnikov ∗, V. E. 1,4 5,6 2. Momar Dieng, Kenneth D. T.-R. McLaughlin. Long-time Asymptotics Zakharov , and N. M. Zubarev for the NLS equation via ∂-methods. arXiv:0805.2807. 1Landau Institute For Theoretical Physics, Russia, 2Department of Mathe- 3 3. Robert Jenkins, Jiaqi Liu, Peter Perry, Catherine Sulem. Soliton reso- matics, University of Illinois at Urbana-Champaign, USA, Department of lution for the derivative nonlinear Schrodinger¨ equation. Comm. Math. Mathematics and Statistics, University of New Mexico, USA, 4Department Phys. 363 (2018), no. 3, 1003–1049. of Mathematics, University of Arizona, USA, 5Institute for Electrophysics, 6 4. David Kaup, Alan Newell. An exact solution for a derivative nonlinear Yekaterinburg, Russia, Lebedev Physical Institute, Moscow, Russia Schrodinger¨ equation. J. Mathematical Phys. 19 (1978), no. 4, 798– [email protected] 801. We consider 2D fluid surface dynamics. A time-dependent con- 5. Einar Mjolhus. Modulational instability of hydromagnetic waves par- formal transformation maps a fluid domain into the lower complex allel to magnetic field. J. Plasma Physics 16 (1976), 321–334. half-plane of a new spatial variable [1]. The fluid dynamics is fully characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. Both a single ideal fluid dynamics (corresponds e.g. to oceanic waves dynamics) and a dynamics of superfluid Helium 4 with two fluid com- SESSION 23: “Nonlinear waves in optics, fluids and plasma” ponents are considered. A superfluid Helium case is shown to be completely integrable for the zero gravity and surface tension limit High-order accurate conservative finite differences with the exact reduction to the Laplace growth equation which is for Vlasov equations in 2D+2V completely integrable through the connection to the dispersionless J. W. Banks∗ and A. Gianesini Odu limit of the integrable Toda hierarchy and existence of the infinite Rensselaer Polytechnic Institute set of complex pole solutions [2]. A single fluid case with nonzero 110 8th street gravity and surface tension turns more complicated with the infinite Troy, NY USA set of new moving poles solutions found [3,4] which are however [email protected] and [email protected] unavoidably coupled with the emerging moving branch points in the upper half-plane. Residues of poles are the constants of motion. n this talk, we discuss numerical simulation for the Vlasov-Poisson These constants commute with each other in the sense of underly- and Vlasov-Maxwell systems in phase space using high-order ac- ing non-canonical Hamiltonian dynamics [5]. It suggests that the curate conservative finite difference algorithms. One significant existence of these extra constants of motion provides an argument challenge confronting direct kinetic simulation is the significant in support of the conjecture of complete Hamiltonian integrability computational cost associated with high-dimensional phase space of 2D free surface hydrodynamics [4,6]. descriptions. Here, we advocate the use of high-order accurate numerical schemes as a means to reduce the computational cost re- [1] P.M. Lushnikov, S.A. Dyachenko and D.A. Silantyev, Proc. Roy. Soc. quired to deliver a given level of error in the computed solution. A 473, 20170198 (2017). [2] P.M. Lushnikov and N.M. Zubarev, Phys. Rev. Lett. 120, 204504 We pursue a discretely conservative finite difference formulation (2018). of the governing equations, and discuss fourth- and sixth-order [3] A. I. Dyachenko and V. E. Zakharov, Free-Surface Hydrodynamics in accurate schemes. In addition, we employ a minimally dissipa- the conformal variables, arXiv:1206.2046. tive nonlinear scheme based on the well-known WENO approach. [4] A. I. Dyachenko, S. A. Dyachenko, P. M. Lushnikov and V. E. Za- These algorithms represent the core of the Eulerian-based kinetic kharov, Dynamics of Poles in 2D Hydrodynamics with Free Surface: New code LOKI [1, 2], which simulates solutions to Vlasov systems Constants of Motion. arXiv:1809.09584 in 2+2-dimensional phase space. To leverage large computational [5] A. I. Dyachenko, P. M. Lushnikov and V. E. Zakharov, Non-Canonical resources, LOKI uses MPI parallelism, details of which are dis- Hamiltonian Structure and Poisson Bracket for 2D Hydrodynamics with cussed here. Results of code verification studies using the method Free Surface, Submitted to Journal of Fluid Mechanics (2018). of manufactured solutions are presented. Results are also presented arXiv:1711.02841

61 [6] A. I. Dyachenko and V. E. Zakharov, Phys. Lett. A 190, 144-148 Colorado, USA (1994). [email protected] Well-balanced discontinuous Galerkin methods for Ferromagnetic materials have been known to humanity for over blood flow simulation with moving equilibrium 4 millennia, yet its properties continue to challenge our physical understanding. Part of the difficulty (and beauty) of magnetism Jolene Britton and Yulong Xing is that quantum mechanical effects at sub-nanometer scales man- Department of Mathematics, University of California, Riverside ifest at macroscopic scales. Microscopic magnetization dynam- Department of Mathematics, The Ohio State University ics are described by a vectorial partial differential equation known [email protected] and [email protected] as the Landau-Lifshitz equation (LLE). While numerical methods The simulation of blood flow in arteries can be modeled by a sys- are regularly utilized to solve the LLE, analytical approaches are tem of conservation laws and have a range of applications in med- typically limited to linearized or weakly nonlinear regimes. In ical contexts. This system of partial differential equations is in this talk, I will present a hydrodynamic formulation for the LLE the same vein as the shallow water equations. We present well- equation that is amenable to analytical study in the nonlinear, dis- balanced discontinuous Galerkin methods for the blood flow model persive regime[1]. I will discuss the paradigm of interpreting a which preserve the general moving equilibrium. Schemes for sys- solid-state material in the context of a fluid and its relation to well- tems with zero-velocity have been recently been addressed, how- known systems such as Bose-Einstein condensates and other fa- ever we focus on the development of schemes that consider general miliar concepts such as sub and supersonic flow [2]. In the context moving equilibrium. Recovery of well-balanced states via appro- of effectively defocusing media, I will present a matched asymp- priate source term approximations and approximations of the nu- totic solution for a spin channel with arbitrary injection strength merical fluxes are the key ideas. Numerical examples will be pre- that sustain nonlinear waves in effectively one-dimensional chan- sented to verify the well-balanced property, high order accuracy, nels [3, 4]. For weak, subsonic injection strength, the solution ex- and good resolution for both smooth and discontinuous solutions. hibits an algebraic spatial profile. At large, supersonic injection strength, a stationary soliton is formed within a narrow, bound- Invariant conserving local discontinuous Galerkin ary layer near the injection site that is asymptotically matched to methods for the modified Camassa-Holm equation an algebraic profile. The soliton effectively reduces the efficiency Zheng Sun and Yulong Xing of spin transport in the channel, a dispersive, coherent counterpart to the onset of turbulence in pipe flow of a viscous fluid at high Department of Mathematics, Ohio State University, Columbus OH 43210 Reynolds numbers. [email protected] and [email protected]

In this presentation, we design, analyze, and numerically test an 1. E. Iacocca, T. J. Silva and M. A. Hoefer, Breaking of Galilean Invari- invariant preserving local discontinuous Galerkin method for solv- ance in the Hydrodynamic Formulation of Ferromagnetic Thin Films, ing the nonlinear modified Camassa-Holm equation. This model Phys. Rev. Lett., 118 (2017), 017203. is integrable and admits peakon solitons. The proposed numerical 2. E. Iacocca and M. A. Hoefer, Vortex-antivortex proliferation from an method is high order accurate, and preserves two invariants, mo- obstacle in thin film ferromagnets, Phys. Rev. B, 95 (2017), 134409. mentum and energy, of this nonlinear equation. The L2-stability 3. E. Iacocca, T. J. Silva and M. A. Hoefer, Symmetry-broken dissipa- of the scheme for general solutions is a consequence of the energy tive exchange flows in thin-film ferromagnets with in-plane anisotropy, preserving property. The numerical simulation results for differ- Phys. Rev. B, 96 (2017), 134434. ent types of solutions of the modified Camassa-Holm equation are 4. E. Iacocca and M. A. Hoefer, Hydrodynamic description of long- provided to illustrate the optimal convergence rate, energy conser- distance spin transport through noncollinear magnetization states: the vation and other capability of the proposed method. role of dispersion, nonlinearity, and damping, arXiv:1812.10438 Non-uniqueness of Leray-Hopf weak solutions for the 3D Hall-MHD system Weak solutions for the 3D Navier-Stokes equations Mimi Dai with discontinuous energy University of Illinois at Chicago Alexey Cheskidov and Xiaoyutao Luo [email protected] Department of Mathematics, Statistics and Computer Science, We will talk about the non-uniqueness of weak solutions in Leray- University of Illinois At Chicago, Chicago, Illinois 60607 Hopf space for the three dimensional magneto-hydrodynamics with [email protected] and [email protected] Hall effect. We adapt the widely appreciated convex integration Since the classical work of Leray it is known that for any diver- framework developed in a recent work of Buckmaster and Vicol gence free initial data with finite energy there exists a weak solu- for the Navier-Stokes equation, and with deep roots in a sequence tion to the 3D Navier-Stokes equations. We construct finite energy of breakthrough papers for the Euler equation. wild solutions with various properties. First, we show that there A hydrodynamic formulation for solid-state ferro- exists a weak solution whose jump discontinuities of the energy magnetism profile are dense and of positive Lebesgue measure in time. The proof relies on a family of approximate solutions to the stationary Ezio Iacocca* Navier-Stokes equations and a new convex integration scheme. As University of Colorado at Boulder, Department of Applied Mathematics,

62 a byproduct, we also obtain finite energy nontrivial stationary weak Rigorous derivation of nonlinear Dirac equations for solutions to the unforced 3D Navier-Stokes equations. wave propagation in honeycomb structures Jack Arbunich and Christof Sparber Inverse cascade of gravity waves in the presence of condensate: numerical results and analytical expla- Department of Mathematics, Statistics, and Computer Science, M/C 249, nation University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607, USA Alexander O. Korotkevich [email protected] and [email protected] Department of Mathematics and Statistics, We show how to rigorously obtain nonlinear equations of Dirac University of New Mexico, Albuquerque, NM, USA type as an effective description for slowly modulated, weakly and Landau Institute for Theoretical Physics, Moscow, Russia. nonlinear waves in honeycomb lattices. Both, local and nonlocal [email protected] Hartree-nonlinearities are discussed and connections to closely re- We consider direct numerical simulation of isotropic turbulence of lated earlier results in semiclassical analysis are pointed out. Our surface gravity waves in the framework of the primordial dynam- results have recently been published in [1]. ical equations. We use approximation of a potential flow of ideal incompressible fluid. System is described in terms of weakly non- 1. J. Arbunich and C. Sparber, Rigorous derivation of nonlinear Dirac linear equations [1] for surface elevation η(~r, t) and velocity po- equations for wave propagation in honeycomb structures, J. Math. Phys., 59 (2018), 011509, 19pp. tential at the surface ψ(~r, t) (~r = (−−−→x, y))

η˙ = kˆψ ( (η ψ)) kˆ[ηkˆψ]+ kˆ(ηkˆ[ηkˆψ]) On the excited states of the interacting boson sys- − ∇ ∇ − tem: A non-Hermitian view 1 2 1 2 1 + ∆[η kˆψ]+ kˆ[η ∆ψ]+ F− [γ η ], 2 2 k k Dionisios Margetis and Stephen Sorokanich 1 b Department of Mathematics, University of Maryland, College Park, MD ˙ 2 ˆ 2 ψ = gη ( ψ) (kψ) 20742 USA − − 2 h ∇ − i 1 [email protected] and [email protected] [kˆψ]kˆ[ηkˆψ] [ηkˆψ]∆ψ + F− [γ ψ ]+ P~ . − − k k r In this talk, we focus on modeling aspects of the weakly interacting b ˆ Here dot means time-derivative, ∆ — Laplace operator, k is a lin- Boson system under periodic boundary conditions, as well as in ear integral operator kˆ = √ ∆ , F 1 is an inverse Fourier trans- the presence of a trapping external potential. The main goal is to − − provide an analytical description of the excited many-body states form, γk is a dissipation rate, P~r isb the driving term which simulates pumping on small scales. These equations were derived as a of this system. A central theme is a non-unitary transformation results of Hamiltonian expansion in terms of kˆη up to the fourth of the system Hamiltonian, which results in investigating a non- order terms. Hermitian operator. This view is examined as an alternative to the Like in works [2, 3] formation of long waves background (con- Bogoliubov transformation. densate) and inverse cascade was observed. This time all inver- tial interval (range of scales where there is no pumping or damp- Nonlinear Schrodinger¨ equations with a potential in ing, only nonlinear interaction of waves) in the inverse cascade re- dimension 3 gion. Currently observed slopes of the inverse cascade are close to Avraham Soffer∗ n k 3.15, which differ signiﬁcantly from theoretically predicted k − Rutgers University n ∼ k 23/6 k 3.83. In our work we propose some analytical k − − [email protected] analysis∼ of results,≃ which is in part based on recent works [4, 5]. Fabio Pusateri 1. V. E. Zakharov, V. S. Lvov, and G. Falkovich, Kolmogorov Spectra of University of Toronto Turbulence I (Springer-Verlag, Berlin, 1992). [email protected] 2. A. O. Korotkevich, Phys. Rev. Lett., 101, 074504 (2008), 0805.0445. We present recent results and ongoing work on the long-time dy- 3. A. O. Korotkevich, Math. Comput. Simul., 82, 1228 (2012), namics of small solutions of nonlinear Schrodinger¨ equations with 0911.0741. potentials in 3 dimensions. 4. A. O. Korotkevich, JETP Lett., 97, 3 (2013), 126-130. Inspired by problems related to the stability of (topological) soli- 5. A. O. Korotkevich and V. E. Zakharov, Nonlin. Process. Geophys., 22, tons, our general goal is to understand the global dynamics of dis- (2015), 325-335. persive and wave equations of the form

i∂ u + L( )u + V(x)u = N(u, u¯), u(t = 0)= u , t |∇| 0 for an unknown u : (t, x) R Rd C with small initial ∈ × −→ SESSION 24: “Mathematical perspectives in quantum mechanics condition u0, where L is the linear dispersion relation, V is a real and quantum chemistry” potential, and N is a nonlinear function vanishing quadratically when u = 0.

63 In this talk we will give a global existence and pointwise decay Edge states in honeycomb structures result in the case of the Schrodinger¨ equation, = ∆, in dimen- L Michael I Weinstein sion 3 with a sufficiently smooth and decaying potential− V with Department of Applied Physics and Applied Mathematics no bound states, and a nonlinearity N = u2. Despite its apparent simplicity, this model presents several difficulties since a quadratic and Department of Mathematics Columbia University nonlinearity in 3d is critical with respect to the Strauss exponent; New York, NY moreover, even in the case V = 0, the nonlinearity u2 creates non- [email protected] trivial fully coherent interactions (unlike the case of N = u¯2, see [1]). Using the Fourier transform adapted to the Schrodinger¨ opera- This talk concerns recent progress on the mathematical theory of tor ∆ + V, we are able to prove integrable-in-time decay through graphene and its artificial analogues with a focus on edge states, the − a distorted Fourier analogue of weighted estimates. A key aspect localization of energy about spatially extended line-defects. Two of our analysis is the development of novel multilinear harmonic types of line-defects are discussed: a) the interpolation between analysis techniques in this setting, which rely on a precise under- deformed honeycomb media across a domain wall and b) honey- standing of the “nonlinear spectral measure” and its singularities, comb media sharply terminated and interfaced with a vacuum. I’ll and extend the more manageable 1d analysis of [2]. discuss the roles played by the spectral properties of the single electron model for the bulk honeycomb structure, and the orientation 1. P. Germain, Z. Hani and S. Walsh. Nonlinear resonances with a poten- of the line-defect. Collaborations with A Drouot, CL Fefferman, tial: multilinear estimates and an application to NLS. Int. Math. Res. JP Lee-Thorp, J Lu, A Watson and Y Zhu. Not., IMRN (2015), 8484-8544. 2. P. Germain, F. Pusateri and F. Rousset. The nonlinear Schrodinger¨ 1. A. Drouot , C. L. Fefferman and M. I. Weinstein, equation with a potential in dimension 1. Ann. Inst. H. Poincare´ C, Defect modes for dislocated periodic media, (2018), 1477-1530. https://arxiv.org/abs/1810.05875 (2018) 2. C. L. Fefferman, J. P. Lee-Thorp, and M. I. Weinstein, Bifurcations of edge states – topologically protected and non-protected A central limit theorem for integrals of random waves – in continuous 2D honeycomb structures, 2D Materials, 3 014008 Matthew de Courcy-Ireland (2015) Department of Mathematics 3. C. L. Fefferman, J. P. Lee-Thorp and M. I. Weinstein, Edge states in honeycomb structures, Annals of PDE, 2 #12 (2016) Princeton and EPF Lausanne [email protected] 4. C. L. Fefferman, J. P. Lee-Thorp and M. I. Weinstein, Honeycomb Schroedinger operators in the strong-binding regime, Comm. Pure Marius Lemm∗ Appl. Math., 71 #6 (2017) Department of Mathematics 5. C. L. Fefferman and M. I. Weinstein, Edge States of continuum Harvard University Schroedinger operators for sharply terminated honeycomb structures, [email protected] https://arxiv.org/abs/1810.03497 (2018) It is known from work of Han and Tacy that the mean-square of 6. J. P. Lee-Thorp, M.I. Weinstein and Y. Zhu, Elliptic operators with honeycomb symmetry; Dirac points, edge states and applications to random waves on Riemannian manifolds converges to a constant in photonic graphene, the high-frequency limit over shrinking balls. We establish a cen- Arch. Rational Mech. Anal., https://doi.org/10.1007/s00205-018- tral limit theorem for the appropriately normalized mean-square. 1315-4 (2018) 7. J. Lu, A. Watson and M. I. Weinstein, Dirac operators and domain Concentration properties of Majorana spinors in the walls, https://arxiv.org/abs/1808.01378 (2018) Jackiw–Rossi theory Akos´ Nagy Boltzmann equations via Wigner transform and dis- Department of Mathematics persive methods Duke University Thomas Chen∗, Ryan Denlinger, Natasaˇ Pavlovic´ [email protected] Department of Mathematics Following the works of Jackiw et al. on the plane [?, ?], I will in- University of Texas at Austin troduce an Abelian gauge theory on Riemann surfaces. Physically, [email protected], [email protected], the theory describes the surface excitations of a TI-SC interface. [email protected] Solutions of the corresponding variational equations are Majorana In this talk, we present some of our recent work on the analysis spinors over Ginzburg–Landau vortices. of Boltzmann equations with tools of nonlinear dispersive PDEs. I will present my results on closed surfaces. The solutions posses The starting point of our approach is to map the Boltzmann equa- an interesting “concentration” property, which is in accordance tion, by use of the Wigner transform, to an equation similar to a with the physical expectations. Using this concentration property I Schrodinger¨ equation in density matrix formulation with a nonlin- will describe the solutions in the large coupling limit. ear self-interaction. We prove local well-posedness, propagation of moments, and small data global well-posedness in spaces defined by weighted space-time norms of Sobolev type.

64 1. T. Chen, R. Denlinger, N. Pavlovic,´ Local well-posedness for Boltz- [email protected] mann’s equation and the Boltzmann hierarchy via Wigner transform, Commun. Math. Phys., to appear. In this talk we consider a post-processing of planewave approxima- https://arxiv.org/abs/1703.00751 tions for nonlinear Schrodinger¨ equations by considering the exact 2. T. Chen, R. Denlinger, N. Pavlovic,´ Moments and Regularity for a solution as a perturbation of the discrete, computable solution. Ap- Boltzmann Equation via Wigner Transform, submitted. plying then Katos perturbation theory leads to computable correc- https://arxiv.org/abs/1804.04019 tions with a provable increase of the convergence rate in the asymptotic range for a very little computational overhead. We illustrate New developments in quantum chemistry on a quan- the key-features of this post-processing for the Gross-Pitaevskii tum computer equation that serves as a toy problem for DFT Kohn-Sham models. Finally some numerical illustrations in the context of DFT Artur F. Izmaylov∗ Kohn-Sham models are presented. Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada 1. E. Cances,` G. Dusson, Y. Maday, B. Stamm, M. Vohral180k,´ Post- [email protected] processing of the planewave approximation of Schrodinger¨ equations. Part I: linear operators, submitted, HAL preprint hal-01908039 Quantum computers are an emerging technology intended to ad- ´ dress computational problems that are exponentially hard for clas- 2. E. Cances,` G. Dusson, Y. Maday, B. Stamm, M. Vohral180k, A perturbation-method-based post-processing for the planewave dis- sical computers. The electronic structure problem of quantum chem- cretization of Kohn-Sham models, J. Comput. Phys., Vol. 307, pp. istry is one of such problems. One of the most practical approaches 446459 (2016) to engaging currently available universal-gate quantum comput- 3. E. Cances,` G. Dusson, Y. Maday, B. Stamm, M. Vohral180k,´ A ers to this problem is the variational quantum eigensolver (VQE) perturbation-method-based a posteriori estimator for the planewave method. In this talk I will discuss two recent improvements of the discretization of nonlinear Schrodinger¨ equations, C. R. Acad. Sci. VQE method: 1) introducing symmetry constraints and 2) improv- Paris., Vol. 352, No. 11, pp. 941-946 (2014) ing projective measurement process. To create a robust and computationally efficient VQE approach that would be able to access any electronic state of interest it is essential Spinning Landau-Lifshitz solitons - a quantum me- to introduce symmetry constraints. Two approaches to introducing chanical analogy symmetry constraints were considered: 1) the penalty functions Christof Melcher [1] and 2) constructing projectors on irreducible representations RWTH Aachen of symmetry operators. It was found that even though the lat- [email protected] ter approach is more rigorous, its hardware resource requirements make it practically infeasible. On the other hand, constrained VQE In this talk we shall discuss dynamic excitations of topological through application of penalty functions can obtain electronic states solitons in two-dimensional ferromagnets. We shall focus on sys- with a certain number of electrons and spin without significant ad- tems without individual rotational symmetry in spin and coordinate ditional quantum resources. space, respectively. Examples include chiral skyrmions in magnets Current implementations of the VQE technique involve splitting without inversion symmetry and curvature stabilized vortices on a the system qubit Hamiltonian into parts whose elements commute spherical shell. As a consequence of reduced rotational symmetry, within their single qubit subspaces. The number of such parts the Hamiltonian dynamics governed by the Landau-Lifshitz equa- rapidly grows with the size of the molecule, this increases the un- tion lacks conservation of individual angular momenta, which may certainty in the measurement of the energy expectation value be- be interpreted as an emerging spin-orbit phenomenon generating cause elements from different parts need to be measured indepen- joint rotations in spin and coordinate space. We shall examine vari- dently. To address this problem we introduce a more efficient par- ational formulations and existence of spinning solitons on spherical titioning of the qubit Hamiltonian using fewer parts that need to be shells by means of concentration-compactness methods combining measured separately [2]. joint work with S. Komineas and Z. N. Sakellaris, respectively.

1. I.G. Ryabinkin, S.N. Genin, and A.F. Izmaylov, Constrained variational quantum eigensolver: Quantum computer search engine in the Fock space, J. Chem. Theory Comp., 15 (2019), 249-255. SESSION 25: “Nonlinear waves, singularities, vortices, and turbu- 2. A.F. Izmaylov, T.C. Yen, and I.G. Ryabinkin, Revising measurement lence in hydrodynamics, physical, and biological systems” process in the variational quantum eigensolver: Is it possible to reduce the number of separately measured operators? arXiv preprint, arXiv:1810.11602 Powerful conformal maps for adaptive resolving of the complex singularities of Stokes wave

A perturbation-method-based post-processing of Denis A. Silantyev∗ planewave approximations for nonlinear Courant Institute, University of New York, New York, NY Schrodinger¨ equations [email protected] Benjamin Stamm Pavel M. Lushnikov MathCCES, Schinkelstr. 2, 52062 Aachen, Germany University of New Mexico, Albuquerque, NM

65 smoothing procedures and Poincare-Birkhoff´ normal form transformations; (4) a normal form identification argument that allows A new highly efficient method is developed for computation of us to handle Benajamin-Feir resonances by comparing with the for- traveling periodic waves (Stokes waves) on the free surface of deep mal computations of [3] and Craig-Worfolk [2] Craig-Sulem [1]. water. The convergence rate of the numerical approximation is determined by the complex singularities of the travelling wave in the 1. W. Craig and C. Sulem. Mapping properties of normal forms transfor- complex plane above the free surface [1]. An auxiliary conformal mations for water waves. Boll. Unione Mat. Ital., 9 (2016), 289-318. mapping is introduced which moves the singularities away from the free surface thus dramatically speeding up Fourier series con- 2. W. Craig and P. Worfolk. An integrable normal form for water waves in infinite depth. Phys. D, 84 (1995), 3-4, 513-531. vergence of the solution by adapting the numerical grid for resolving singularities [2]. Three options for the auxiliary conformal map 3. V.E. Zakharov and A.I. Dyachenko. Is free-surface hydrodynamics an are described with their advantages and disadvantages for numer- integrable system? Physics Letters A, 190 (1994), 144-148. ics. Their efficiency is demonstrated for computing Stokes waves near the limiting Stokes wave (the wave of the greatest height) with Stability and noise in frequency combs: 100-digit precision. Drastically improved convergence rate signif- harnessing the music of the spheres icantly expands the family of numerically accessible solutions and allowing to study the oscillatory approach of these solutions to the Curtis R. Menyuk, Zhen Qi, and Shaokang Wang limiting wave in great detail. CSEE Dept., University of Maryland Baltimore County 1000 Hilltop Circle, Baltimore, MD 21250 1. Sergey A. Dyachenko, Pavel M. Lushnikov, Aleksander O. Korotke- [email protected] vich, The complex singularity of a Stokes wave, Pis’ma v ZhETF, vol. Frequency combs have revolutionized the measurement of time 98, iss. 11, pp. 767-771 (2013). and frequency and impacted a wide range of applications spanning 2. Pavel M. Lushnikov, Sergey A. Dyachenko, Denis A. Silantyev, New basic physics, astrophysics, medicine, and defense. Frequency conformal mapping for adaptive resolving of the complex singularities combs are modeled mathematically at lowest order by the nonlin- of Stokes wave, Proc. Roy. Soc. A, vol. 473, 2202, (2017). ear Schrodinger¨ equation (NLSE), as is the case for many other physical systems. Although the NLSE can yield important qualita- The Zakharov-Dyachenko conjecture on the integra- tive insights, it is too simplistic to be useful for quantitative mod- bility of gravity water waves eling. The key theoretical issues in understanding and designing frequency Massimiliano Berti combs are finding regions in the adjustable parameter space where SISSA combs operate stably, determining their noise performance, and [email protected] optimizing them for high power, low noise, and/or large band- Roberto Feola width. Similar issues arise in many of the physical systems that are Laboratoire de Mathematiques´ Jean Leray, Universite´ de Nantes modeled at lowest order by the NLSE. To date, these issues have [email protected] been studied either by using brute-force evolutionary simulations or by using dynamical systems methods in nearly-analytical limits, Fabio Pusateri∗ where the equations are too simplified to model the experimental Mathematics Department, University of Toronto systems accurately. [email protected] In recent work, we have shown that these issues can be efficiently We consider the gravity water waves system with a periodic one- and accurately addressed by combining 400-year-old dynamical dimensional interface in infinite depth, and prove a rigorous reduc- systems methods with modern computational techniques. Our com- tion of these equations to Birkhoff normal form up to degree four. putational tools are 3–5 orders of magnitude faster than standard This proves a conjecture of Zakharov-Dyachenko [3] based on the evolutionary methods and provide important physical insight. We formal Birkhoff integrability of the water waves Hamiltonian trun- have applied these tools to frequency combs from passively mod- cated at order four. As a consequence, we also obtain a long-time elocked lasers with fast and with slow saturable absorbers and to stability result: periodic perturbations of a flat interface that are of frequency combs from microresonators. Our methods predict im- size ǫ in a sufficiently smooth Sobolev space lead to solutions that proved operating regimes for combs that are produced from both 3 the passively modelocked lasers and the microresonators. remain regular and small up to times of order ǫ− . This is the first such long-time existence result for quasilinear PDEs in the absence Despite our progress to date, there is much that remains to be done of external parameters. to put the computational tools that we have developed on a firm Some of the main difficulties in the proof are the quasilinear na- theoretical foundation and to make them sufficiently robust so that ture of the equations, the presence of small divisors arising from they can be used on a broad range of modern-day experimental near-resonances, and non-trivial resonant four-waves interactions, frequency comb systems. We discuss the open questions, as well the so-called Benjamin-Feir resonances. The main ingredients that as our progress. we use are: (1) various reductions to constant coefficient operators through flow conjugation techniques; (2) the verification of Higher-order Runge–Kutta-type schemes based on key algebraic properties of the gravity water waves system which imply the integrability of the equations at non-negative orders; (3)

66 the method of characteristics for hyperbolic equa- may open a new road to the study of noise and randomness in non- tions with crossing characteristics linear wave equations. We apply our numerical approach to predict the emergence of phase randomness in the Nonlinear Schro- Taras I. Lakoba and Jeffrey S. Jewell dinger equation (NLS) [2], random solitary waves interactions, the Department of Mathematics and Statistics, University of Vermont, Burling- emergence of polarization randomness in the coupled NLS [3], and ton, VT 05401 shock formation in the Burgers equation. [email protected] The numerical Method of Characteristics (MoC) is widely used to 1. A. Ditkowski, G. Fibich, and A. Sagiv A spline-based approach solve hyperbolic evolution equations. For example, for a system to uncertainty-quantiﬁcation and density estimation. arXiv preprint, arXiv:1803:10991 (2018). w + c w = f (w , w ), w c w = f (w , w ), 1, t 1, x 1 1 2 2, t − 2, x 2 1 2 2. A. Sagiv, A. Ditkowski, and G. Fibich. Loss of phase and universality of stochastic interactions between laser beams. Opt. Exp., 25:24387– a change of independent variables: (x, t) (ξ , t) for the ith → i 24399, 2017. equation (i = 1, 2), where ξ1 = x c t and ξ2 = x + c t, reduces these partial differential equations− to ordinary differential 3. G. Patwardhan, X. Gao, A. Sagiv, A. Dutt, J. Ginsberg, A. Ditkowski, G. Fibich, and A. Gaeta. Loss of polarization in collapsing beams. equations (ODEs) along characteristics: arXiv preprint, arXiv:1808.07019 (2018).

w1, t = f1(w1, w2) along ξ1 = const, (5a) w2, t = f2(w1, w2) along ξ2 = const. (5b) On density functional theory Each of the ODEs is then solved by an ODE numerical solver. One Israel Michael Sigal of the main advantages of the MoC is that it preserves the linear Dept. of Mathematics dispersion relation of the hyperbolic equations (), while allowing to University of Toronto specify arbitrary (i.e., not only periodic) boundary conditions. One [email protected] of the main disadvantages of the MoC so far has been the fact that In this talk I will review some recent results in the density func- among explicit ODE solvers of (5), only first- and second-order tional theory including the time-dependent one and the one cou- accurate ones have been known. In this talk I will explain how one pled to the electro-magnetic field. I will also formulate some open can construct MoC schemes based on higher-order Runge–Kutta problems. The talk is based on the joint results with Ilias Chenn. (RK)-type ODE solvers. To begin, I will show how the standard RK solver can be modified Rogue waves in the nonlocal PT-symmetric nonlin- for a system like (5), where each equation is solved along its own ear Schrodinger¨ equation characteristic. However, it turns out that such a modified algorithm can become strongly numerically unstable. To overcome this insta- Bo Yang and Jianke Yang bility, I will explain how the above modification can be applied to Department of Mathematics and Statistics, University of Vermont, Burling- a so-called pseudo-RK solver, which has not been found to suffer ton, VT 05405, USA from the instability problem. (A pseudo-RK solver is a hybrid be- [email protected]; [email protected] tween an RK and a multi-step solver.) Finally, I will explain how non-periodic boundary conditions can be implemented for an MoC Rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger¨ scheme based on a higher-order pseudo-RK solver. (NLS) equation are studied. These waves are derived by the Dar- boux transformation and bilinear KP reduction methods, and ex- Efficient numerical methods for nonlinear dynamics pressed as determinants in terms of Schur polynomials. Unlike with random parameters rogue waves in the local NLS equation, the present rogue waves show a much wider variety. For instance, the polynomial degrees Adi Ditkowski, Gadi Fibich, and Amir Sagiv∗ of their denominators can be not only n(n + 1), but also n(n Department of Applied Mathematics, Tel Aviv University, Tel Aviv, Israel 1)+ 1, n2 and other integer values, where n is an arbitrary posi-− [email protected], fi[email protected], and [email protected] tive integer. Dynamics of these rogue waves is also examined. It is shown that these rogue waves can be bounded for all space and We present a novel numerical approach for the study of nonlinear time or develop collapsing singularities, depending on their types PDEs with random initial conditions or parameters. The naive ap- as well as values of their free parameters. In addition, the solution proach to compute the statistics of these random dynamics, e.g., the dynamics exhibits rich patterns, most of which have no counter- Monte-Carlo and histogram methods, might be prohibitively inef- parts in the local NLS equation. ficient. This problem has spurred the growth in recent years of the field of uncertainty quantification. Specifically, the Polynomial- Chaos Expansion (gPC), a spectrally-accurate algorithm for the 1. B. Yang and J. Yang, “Rogue waves in the PT-symmetric nonlinear Schrodinger¨ equation”, Lett. Math. Phys. DOI: 10.1007/s11005-018- computation of statistical moments, has become widely popular. 1133-5 (2018). Nevertheless, and perhaps surprisingly, we show that the gPC approach might fail to compute efficiently the probability density 2. B. Yang and J. Yang, “On general rogue waves in the parity-time- function (PDF) of the model output. symmetric nonlinear Schrodinger¨ equation”, preprint. Our newly developed spline-based method offer a good approximation of PDF, with theoretical guarantees [1]. Therefore, the method

67 Family of potentials with power-law kink tails We explore the singularities in the analytic continuation of the velocity potential to the exterior of the fluid domain enclosed under Avadh Saxena the free boundary. We demonstrate that certain classes of singu- Los Alamos National Lab, USA larities are persistent under the evolution in Euler equations [1]. [email protected] Moreover, these singularities are associated with new, previously Avinash Khare∗ undiscovered nontrivial constants of motion. Some of these motion Savitribai Phule Pune University, India constants have been shown to commute under the Poisson bracket, [email protected] and suggest that free–surface hydrodynamics may have more hidden structure then previously discovered. We provide examples of a large class of one dimensional higher We demonstrate the results of the numerical simulations and illus- order field theories with kink solutions which asymptotically have trate with reconstruction of analytical structure of the fluid poten- a power-law tail either at one end or at both ends. We provide tial outside of fluid. analytic solutions for the kinks in a few cases but mostly provide implicit solutions. We also provide examples of a family of poten- 1. A. I. Dyachenko, S. A. Dyachenko, P. M. Lushnikov, V. E. Zakharov, tials with two kinks, both of which have power law tails either at Dynamics of Poles in 2D Hydrodynamics with Free Surface: New Con- both ends or at one end. In addition, we show that for kinks with a stants of Motion, JFM submitted (2018) power law tail at one end or both the ends, there is no gap between the zero mode and the continuum of the corresponding stability equation. This is in contrast to the kinks with exponential tail at Nonlinear waves acting like linear waves in NLS both the ends in which case there is always a gap between the zero Katelyn (Plaisier) Leisman∗ mode and the continuum [1]. University of Illinois Dept. of Mathematics, Altgeld Hall, 1409 Green Street, Urbana, IL 61801 1. A. Khare and A. Saxena, Family of potentials with power-law kink [email protected] tails, arXiv:1810.12907 Gregor Kovaciˇ cˇ Dynamical problems arising in blood flow: Rensselaer Polytechnic Institute nonlinear waves on trees 110 8th Street, Amos Eaton, Troy, NY 12180 [email protected] Jerry Bona∗ David Cai Address: Department of Mathematics, Statistics and Computer Science The University of Illinois at Chicago Shanghai Jiao Tong University, China [email protected] Courant Institute of Mathematical Sciences, New York University, USA [email protected] Pulmonary arterial hypertension is a pernicious disease whose only curative treatment at the moment is lung or heart-lung transplant. The linear part of the Nonlinear Schrodinger¨ Equation (NLS) (iqt = 2 One of the characteristics of this disease is the right-ventricle re- qxx) has dispersion relation ω = k . We don’t expect solutions to modeling that occurs because the heart is asked to work harder due the fully nonlinear equation to behave nicely or have any kind of to the pressure overload imposed by the pulmonary vasculature. effective dispersion relation like this. However, I have seen that so- In this lecture, we will discuss an ongoing project aimed at ob- lutions to the NLS are actually weakly coupled and are often nearly taining a better understanding of this disease. Mathematically, this sinusoidal in time with a dominant frequency, often behaving sim- comes down to a large coupled system of nonlinear wave equations ilarly to modulated plane waves. whose spatial domain is a rooted tree. Preliminary analysis of the system is put forth and some comparisons with real data provided. Instantons and fluctuations in complex systems Tobias Schafer¨ Singularities in the 2D fluids with free surface City University of New York [email protected] Sergey Dyachenko∗ Department of Mathematics, University of Illinois at Urbana–Champaign After a short overview of path integral techniques and their rela- [email protected] tionship to large deviation theory, I will present recently devel- Alexander Dyachenko oped methods to compute instantons (minimizers of the Freidlin- Landau Institute for Theoretical Physics Wentzell functional) in complex stochastic systems. The stochas- [email protected] tically driven Burgers equation [1] and the stochastic nonlinear Schrodinger¨ equation [2] will serve as examples. In addition to Pavel Lushnikov the instanton, it is often desirable to also take into account fluc- Department of Mathematics and Statistics, University of New Mexico tuations in order to compute the prefactor. I will discuss recently [email protected] developed computational methods involving the solution of the as- Vladimir Zakharov sociated matrix-Riccati equation. Department of Mathematics, University of Arizona [email protected]

68 1. T. Grafke, R. Grauer, T.Schafer,¨ and E. Vanden-Eijnden, Relevance Stable blow-up dynamics in the critical and super- of instantons in Burgers turbulence. Eurphysics Letters, 109 (2015) critical NLS and Hartree equations 34003. Svetlana Roudenko∗ and Kai Yang 2. G. Poppe and T.Schafer:¨ Computation of minimum action paths of the Department of Mathematics and Statistics, DM430 stochastic nonlinear Schrodinger¨ equation with dissipation. J. Phys. A: Math. Theor., 51, (2018) 335102. Florida International University, Miami, FL 33199 sroudenko@fiu.edu and yangk@fiu.edu Clebsch variables for stratified compressible fluids Yanxiang Zhao Department of Mathematics Benno Rumpf George Washington University, Washington DC 20052 Mathematics Department, Southern Methodist University, Dallas, Texas [email protected] [email protected] We study stable blow-up dynamics in the nonlinear Schrodinger Clebsch variables provide a canonical Hamiltonian representation (NLS) equation and generalized Hartree equation with radial sym- of the Euler equation. While this is desirable from a theoretical per- metry in the L2-critical and supercritical regimes. The NLS equa- 2σ spective, Clebsch variables have practical disadvantages: Firstly, tion is with pure power nonlinearity iut + ∆u + u u = 0, and it is often difficult to compute the initial conditions for the Cleb- the Hartree equation is a Schrodinger-type¨ equation| | with a non- sch variables from the initial conditions of the velocity field. Sec- local, convolution-type nonlinearity in dimension d: iut + ∆u + ondly, Clebsch variables usually show ’non-physical’ divergences (d 2) p p 2 x − − u u − u = 0, p 2. that pose difficulties on perturbation expansions. In my talk, I will | | ∗| | | | ≥ 2 discuss strategies to overcome these difficulties. First, we consider the L -critical case for the NLS equation in dimensions 4 d 12 and for the Hartree in dimensions d = ≤ ≤ 1. R. Salmon, Hamiltonian fluid mechanics, Ann. Rev. Fluid Mech., 20 3, 4, 5, 6, 7. We show that a generic blow-up in both equations ex- 1 (1988), 225-256. hibits not only the rate u(t) L2 (T t)− 2 , but also the “log-log” correction, thus,k∇ behavingk similarly∼ − to the stable collapse in the lower dimensional NLS (such as the 2d cubic NLS). In this Appearance of Stokes waves in deep water setting we also study blow-up profiles and show that generic blow- Anastassiya Semenova∗, Alexander Korotkevich, Pavel Lush- up solutions converge to the rescaled Q, the ground state solution nikov of the elliptic equations, which is well-known in the NLS case: 2σ+1 Department of Mathematics and Statistics, ∆Q + Q Q = 0, and for the Hartree it is ∆Q + Q − (d 2) − p p 2 − − University of New Mexico x − − Q Q − Q = 0. [email protected], [email protected], | | ∗| | | | Next, we examine the L2-supercritical cases for both equations. [email protected] For the self-similar blow-up solutions we study the profile equa- We study evolution of a finite amplitude monochromatic wave in tions and discuss the existence and local uniqueness theory of the deep ocean taking into account gravity but not capillary effects. We solutions. We then show that our numerical simulations indicate simulate one period of Stokes wave to allow for superharmonics, that the solutions Q to such profile equations exhibit a multi-bump and avoiding modulational instability at wavelengths longer than structure, and thus, in a sense, not unique. Direct numerical sim- the initial spatial period of monochromatic wave. We investigate ulations of the NLS and generalized Hartree equations by the dy- the possibility of generation of Stokes waves in Euler equations in namic rescaling method indicate that only one of those multi-bump the long time limit. profile solutions serves as the stable blow-up profile. We also investigate the rate of the blow-up and obtain the square root blow-up Chiral magnetic skyrmions for 2D Landau-Lifshitz rate without any corrections. Our findings indicate that the nonlin- equations earity type in the Schrodinger-type equations is not essential for the stable collapse formation. Stephen Gustafson∗ University of British Columbia Slow light propagation in two-level active media [email protected] Katelyn Plaisier-Leisman Landau-Lifshitz equations are the basic dynamical equations in a University of Illinois at Urbana Champaign micromagnetic description of a ferromagnet. They are naturally [email protected] viewed as geometric evolution PDE of dispersive (“Schrodinger¨ map”) or mixed dispersive-diffusive type, which scale critically Gino Biondini with respect to the physical energy in two dimensions. We de- University at Buffalo scribe recent results on existence and stability of important topo- [email protected] logical soliton solutions known as “chiral magnetic skyrmions”. Gregor Kovaciˇ cˇ Joint work with Li Wang. ∗ Rensselaer Polytechnic Institute [email protected]

69 In ruby crystals, slow light pulses were observed, and described [email protected] using two-level Maxwell-Bloch equations with high polarizability Yuri V Lvov damping. We compute that two regimes exist, depending on the ∗ ratio of medium-polarizability and level-inversion damping. When Rensselaer Polytechnic Institute Troy NY 12203 this ratio is moderate, soliton-like pulses exist. Damping decreases [email protected] their amplitudes and speed. A precursor of radiation coexists, and We investigate the celebrated β-Fermi-Pasta-Ulam-Tsingou dominates for strong damping and large damping ratio. Starting (FPUT) chain and establish numerically and theoretically the ex- slowly, it accelerates to the speed of light. istence of the second order anomalous correlator. The anomalous correlator manifests in the frequency-wave number Fourier spec- Expansion of the strongly interacting superfluid trum as a presence of “ghost” waves with negative frequency, in Fermi gas: symmetry and self-similar regimes addition to the waves with positive frequencies predicted by the linear dispersion relationship. We explain theoretically the exis- E.A. Kuznetsov(a),(b), M.Yu. Kagan(c) and A.V. Turlapov(d) tence of anomalous correlator and the ghost waves by nonlinear (a) P.N. Lebedev Physical Institute RAS, Moscow, Russia interactions between waves. Namely, we generalize the classical (b) L.D. Landau Institute for Theoretical Physics RAS, Chernogolovka, Wick’s decomposition by including the second order anomalous Moscow region, Russia correlator and show that the latter is responsible for the presence (c) P.L. Kapitza Institute of Physical Problems RAS, Moscow, Russia of such “ghost” waves. From a physical point of view, the develop- (d) Institute of Applied Physics RAS, Nizhnii Novgorod, Russia ment of the anomalous correlator is related to formation of nonlinear standing waves. Indeed, we show numerically in the nonlinear We consider an expansion of the strongly interacting superfluid regime a transition from pure travelling waves to standing waves. Fermi gas in the vacuum in the so-called unitary regime when We predict that similar phenomenon might occur in nonlinear sys- 2 2/3 the chemical potential µ ∝ h¯ /mn− where n is the density of tem dominated by nonlinear interactions, including surface gravity the Bose-Einstein condensate of Cooper pairs of fermionic atoms. waves. Such expansion can be described in the framework of the Gross- Pitaevskii equation (GPE) [1]. Because of the chemical potential Excitation of interfacial waves via near-resonant sur- dependence on the density n 2/3 the GPE has additional sym- face — interfacial wave interactions ∼ − metries resulting in existence of the virial theorem connected the Joseph Zaleski mean size of the gas blob and its Hamiltonian. It leads asymptot- ∗ Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180 ically at t ∞ to the ballistic expansion of the gas. We care- [email protected] fully study→ such asymptotics and reveal a perfect matching between the quasi-classical self-similar solution and the ballistic ex- Philip Zaleski pansion of the non-interacting gas. This matching is governed by New Jersey Institute of Technology, 323 Dr Martin Luther King Jr Blvd, the virial theorem derived in [2] utilizing the Talanov transforma- Newark, NJ 07102 tion [3] which was first obtained for the stationary self-focusing [email protected] of light in the media with cubic nonlinearity due to the Kerr ef- Yuri Lvov fect. In the quasi-classical limit the equations of motion coincide with 3D hydrodynamics for the perfect gas with γ = 5/3. Their Rensselaer Polytechnic Institute,110 8th Street, Troy, NY 12180 self-similar solution describes, on the background of the gas ex- [email protected] pansion, the angular oscillations of the gas shape in the framework The term “ocean waves” typically evokes images of surface waves of the Ermakov-Ray-Reid type system. shaking ships during storms in the open ocean, or breaking rhyth- mically near the shore. However, much of the ocean wave action [1] L.P.Pitaevskii, Superfluid Fermi liquid in a unitary regime, Physics takes place far underneath the surface, and consists of surfaces of Uspekhi , v.51, pp.603-608, 2008. constant density being disturbed and modulated. The relationship [2] E.A.Kuznetsov, S.K. Turitsyn, Talanov transformation in self-focusing between surface and interfacial waves provides a mechanism for problems and instability of stationary waveguides, Phys.Lett., v.112 A, pp. coupling of the atmosphere and the ocean—wind creates surface 273-276, 1985. waves, which in turn distribute energy to the lower bulk of the [3] V.I. Talanov, On the self-focusing of light in the cubic media, Pis’ma ocean. Zh.Eksp.Teor.Fiz., v.11, p.303, 1970. We consider interactions between surface and interfacial waves in Anomalous correlators, ghost waves and nonlinear a two layer system, based on the novel Hamiltonian discovered by standing waves in the β-FPUT system Choi [2]. Our approach includes the general procedure for diag- onalization of the quadratic part of the Hamiltonian. This allows Joseph Zaleski us to derive from first principles the coupled kinetic equations de- Rensselaer Polytechnic Institute Troy NY 12203 scribing spectral energy transfers in this system and analyze the [email protected] interaction crossection between surface and interfacial waves. No- tably, interactions are not limited to resonant wavenumbers. The Miguel Onorato kinetic equations include the effects of “near”—resonant interac- Dip. di Fisica, Universit di Torino and INFN, Sezione di Torino, Via P. tions, physically motivated by observed changes in the shape of Giuria, 1, Torino, 10125, Italy

70 the spectra along nonresonant wavenumbers. We find that the en- We study the linear stability of the Peregrine breather both nu- ergy transfers are dominated by the generalization of the class III merically and with analytical arguments based on its derivation resonances described in Alam [1]. We apply our formalism to cal- as the singular limit of a single-mode spatially periodic breather culate the rate of growth of interfacial waves for different values of as the spatial period becomes infinite. By constructing solutions the wind velocity and simulate the system of kinetic equations for of the linearization of the nonlinear Schrodinger¨ equation in terms the case describing the evolution of coupled 1-D spectra. of quadratic products of components of the eigenfunctions of the Zakharov-Shabat system, we show that the Peregrine breather is 1. Mohammad-Reza Alam, Journal of fluid mechanics, 691 (2012), 267- linearly unstable. A numerical study employing a highly accurate 278. Chebychev pseudo-spectral integrator confirms exponential growth 2. Wooyoung Choi, private communications. of random initial perturbations of the Peregrine breather.

1. A. Calini, C.M. Schober and M. Strawn, Linear Instability Optical phase modulated nonlinear waves in a of the Peregrine Breather: Numerical and Analytical Investi- graphene waveguide gations, APNUM, online publication (2018). DOI information 10.1016/j.apnum.2018.11.005 G. T. Adamashvili Technical University of Georgia, Kostava str.77, Tbilisi, 0179, Georgia. Oblique dispersive shock waves in steady shallow wa- [email protected] ter ﬂows

D. J. Kaup∗ Adam Binswanger∗, Patrick Sprenger Department of Mathematics & Institute for Simulation and Training Department of Applied Mathematics 526 UCB, University of Colorado, University of Central Florida, Boulder, CO 80309-0526, USA Orlando, Florida, 32816-1364, USA. [email protected], [email protected] [email protected] Mark Hoefer The different mechanisms that bring about the creation of opti- Department of Applied Mathematics 526 UCB, University of Colorado, cal nonlinear waves in a waveguide containing a graphene mono- Boulder, CO 80309-0526, USA layer (or graphene-like two-dimensional material) are studied in [email protected] the general case when resonant and nonresonant nonlinearities are Steady shallow water flows are studied for a boundary value prob- simultaneously included. The conditions for the formation of op- lem that corresponds to the deflection of a supercritical flow of a tical hybrid, nonresonant and resonant phase modulated breathers thin sheet of water past a slender wedge. Due to surface wave in graphene, for waveguide TE-modes, are presented. It is shown dispersion, the ensuing steady structure is a spatially extended, os- that the characteristic parameters of these optical nonlinear waves, cillatory pattern referred to as an oblique dispersive shock wave depends on the graphene Kerr-type third-order susceptibility, the (DSW), which can be approximated as a modulated nonlinear wave- graphene conductivity, the reciprocal of Beers absorption length, train limiting to an oblique solitary wave at one edge and small as well as the initial values of the ensemble of the atomic system amplitude harmonic waves at the other edge. This corner wedge and/or the semiconductor quantum dots that are embedded in the boundary value problem is modeled by a weakly nonlinear model transition layer. In the case of the amplifier (active atomic system) of KdV-type that incorporates higher order dispersion. Asymptotic transition layer, the conditions for the existence of a dark (topo- analysis, numerical simulations, and an in-house shallow water ex- logical) breather, as well as the conditions when a nonlinear wave periment demonstrate evidence of a bifurcation in the flow pattern cannot be formed, are determined and given. An explicit analyti- as a control parameter (the wedge angle) is varied. The Bond num- cal expression for the profile of an optical nonlinear wave is also ber, B, measuring the effects of surface tension relative to grav- presented. ity, characterizes the bifurcation and is controlled by appropriate variation of water depth. The bifurcation, a result of higher order dispersion, occurs near B = 1/3, corresponding to a fluid depth of approximately 5 mm, and is a transition between classical and SESSION 26: “Physical applied mathematics” non-classical DSW profiles. They are differentiated by the monotonicity or lack thereof of the solitary wave edge as well as the Linear instability of the Peregrine breather: structure of the modulated nonlinear wavetrain that ensues. Numerical and analytical investigations Solitons and pseudo-solitons in the Korteweg-de Constance Schober∗ and Maria Strawn Vries equation with step-up boundary conditions Dept. of Mathematics, University of Central Florida, FL 1 2 1, [email protected] and [email protected] Mark Ablowitz , Xu-Dan Luo , Justin Cole ∗ 1 Department of Applied Mathematics Anna Calini University of Colorado, Boulder Dept. of Mathematics, College of Charleston, SC 2 Department of Mathematics [email protected] State University of New York at Buffalo [email protected]

71 The Korteweg-deVries (KdV) equation with step-up boundary con- incorporated by the simulator at every iteration, helping the scheme ditions is considered, with an emphasis on soliton dynamics. When mimic the behaviour of the original IBVP. an initial soliton is of sufficient size, it can propagate through the The present work aims at studying the application of the spectral step; in this case, the phase shift is calculated via the inverse scat- re-normalization method to the proto-typical dissipative, Burger’s tering transform. On the other hand, when the amplitude is not equation subject to periodic boundary conditions. Having repro- large enough, the wave becomes “trapped” inside the ramp region. duced the numerical experiments for the case presented in [1], the In the trapped case, the transmission coefficient of the associated work explores the incorporation of higher order Cauchy-Filon in- linear Schrodinger¨ equation can become large at a point exponen- tegration methods into the fixed point iteration. The rationale for tially close to the continuous spectrum. This point is referred to as the incorporation is to test the robustness of the algorithm (con- a pseudo-embedded eigenvalue. Employing the inverse problem, it vergence of the iteration with the use of large time steps), coupled is shown that the continuous spectrum associated with a branch cut to high accuracy. Comparisons between the different integration in the neighborhood of the pseudo-embedded eigenvalue plays the strategies will be laid out, in order to explore the potential of the role of discrete spectra, which in turn leads to a trapped soliton or algorithm. “pseudo-solitons” in the KdV equation. 1. M. J. Ablowitz and Z. H. Musslimani, Spectral renormalization method Various dynamical regimes, and transitions from ho- for computing self-localized solutions to nonlinear systems, OPTICS mogeneous to inhomogeneous steady states in oscil- LETTERS, Vol. 30, No.16 (2005), 2140-2142. lators with delays and diverse couplings 2 Ryan Roopnarain and S. Roy Choudhury 2. J. T. Cole and Z. H. Musslimani, Time-dependent spectral renormalization method, arXiv:1702.06851v2, (2 Aug 2017), 2140-2142. Department of Mathematics University of Central Florida [email protected] and [email protected] Time-dependent spectral renormalization method applied to conservative PDEs This talk will involve coupled oscillators with multiple delays, and dynamic phenomena including synchronization at large coupling, Abdullah Aurko∗ and Ziad H. Musslimani and a variety of behaviors in other parameter ranges including tran- Department of Mathematics, Florida State University, Tallahassee, FL, sitions between Amplitude Death and Oscillation Death. Both an- 32306-4510 alytic multiple scale and energy methods, as well as numerical re- [email protected] and [email protected] sults will be presented. Behaviors in both limit cycle and chaotic oscillators will be compared for various couplings. Finally, the ef- The time-dependent spectral renormalization method was first in- fects of distributed delays will be considered for systems already troduced by Cole and Musslimani as a numerical means to simu- treated using discrete delays, including bifurcation theory results late linear and nonlinear evolution equations [1]. The essence of not available in the latter case. the method is to convert the underlying evolution equation from its partial or ordinary differential form (using Duhamel’s principle) Spectral renormalization algorithm applied to solv- into an integral equation. The solution sought is then viewed as a ing initial-boundary value problems fixed point in both space and time. The resulting integral equation is then numerically solved using a simple renormalized fixed-point Sathyanarayanan Chandramouli and Ziad Musslimani iteration method. Convergence is achieved by introducing a time- MCH 221 (c), Florida State University and LOV 218, Florida State Uni- dependent renormalization factor which is numerically computed versity from the physical properties of the governing evolution equation. [email protected] and [email protected] The most profound feature of the method is that it has the ability to incorporate physics into the simulations in the form of conser- The Spectral Renormalization algorithm was developed as a novel vation laws. numerical scheme for soliton solutions. [1] The main theme of In this paper, we apply this novel scheme on the classical nonlinear the work was to transform the equation governing the soliton into Schrodinger (NLS) equation- a benchmark evolution equation, and Fourier space and determine a non-linear, nonlocal integral equa- a prototypical example of a conservative dynamical system. We tion coupled to an algebraic equation. The coupling was seen to consider the classical NLS equation as a test bed for the perfor- enforce the convergence of the constructed fixed point iteration mance of the time-dependent spectral renormalization scheme be- scheme. The method was envisioned to have wide applications cause: 1) It has wide physical applications, such as in optics, con- in diverse areas, including Bose-Einstein condensation and fluid densed matter physics, and fluid mechanics (deep water waves). 2) mechanics. The work was extended to the time domain in or- The classical NLS is an integrable evolution equation that admits der to solve Initial-Boundary value problems (IBVP) using a time- an infinite number of conserved quantities. The second property dependent spectral renormalization algorithm.[1] Here, a conver- is what we aim to explore. For the NLS equation, we have the gent fixed point iteration scheme was constructed by introducing a following three conserved quantities: power, momentum, and the time-dependent renormalization factor. This renormalization fac- Hamiltonian (energy). We first incorporate each conserved phys- tor is computed either from equation(s) expressing conservation ical quantity separately, using the method, as outlined in [1], but of a physically relevant quantity, or physically relevant dissipation using a higher order integration technique for evaluating the time rate equation(s). Thus, besides facilitating the convergence of the integral. After that, we proceed to successfully incorporate more fixed point iteration scheme, the physics underlying the problem is

72 than one physically conserved quantity simultaneously, using the I will discuss laboratory experiments on randomized surface waves time-dependent spectral renormalization method. Future work in- propagating over variable bathymetry. The experiments show that volves repeating the same procedure for the Korteweg-de Vries an abrupt depth change can qualitatively alter wave statistics, trans- (KdV) equation. forming an initially Gaussian wave field into a highly skewed one. The altered wave field conforms closely to a gamma distribution, 1. J. T. Cole, and Z. H. Musslimani, Time-dependent spectral renormal- which offers a simple way to estimate statistical quantities such as ization method, Physica D: Nonlinear Phenomena, 358 (2017), 15-24. skewness or kurtosis. Compared to Gaussian, the relatively slow decay of the gamma distribution indicates an elevated level of extreme events, i.e. rogue waves. In our experiments, the probability On N-soliton interactions: Effects of local and nonlocal potentials of a rogue wave can be up to 50 times greater than would be expected from normal statistics. V.S. Gerdjikov1 and M.D. Todorov2 1Institute of Mathematics and Informatics and Institute for Nuclear Re- search and Nuclear Energy, BAS, Sofia, Bulgaria 2Dept of Applied Mathematics and Computer Science, Technical Univer- SESSION 27: Canceled sity of Sofia, 1000 Sofia, Bulgaria [email protected] and mtod@tu-sofia.bg We study the dynamical behavior of the N-soliton trains of nonlinear Schrodinger¨ equation (NLSE) perturbed by local and nonlocal SESSION 28: “Recent advances in analytical and computational potential terms: methods for nonlinear partial differential equations”

∂u 1 ∂2u i + + u 2u(x, t)+ V(x)u(x, t) Optimal control of HPV infection and cervical can- ∂t 2 ∂x2 | | cer with HPV vaccine ∞ + gu(x, t) R( x y ) u(y, t) 2dy = 0. (6) Kinza Mumtaz, Mudassar Imran, Adnan Khan Z | − | | | ∞ Lahore University of Management Sciences − [email protected], [email protected] The effects of several types of local potentials V(x) have been analyzed earlier both for the NLSE and the Manakov model, see [1, 2]. In this paper, we develop an HPV epidemic model and transmission dynamics from susceptible population infected by Human Pa- Recently Salerno and Baizakov pointed out [3] that specific nonlo- pilloma Virus into cervical cancer. For ideal control under vaccina- cal potentials with R(z)=(1/(√2πw)) exp( z2/(2w2)) may tion program, we have utilized one type of vaccination: a bivalent lead to formations of bound state of solitons with− molecular-like vaccine that objectives two HPV composes (16 and 18). To portray interactions, i.e., attractive at long distances and repulsive at short the cooperation of vaccinated and the other four classes (suscepti- distances. Our aim is to check whether their results are compatible ble, infected, precancerous and cancerous), we built up a system of with the adiabatic approximation. We derive perturbed complex five ODEs. Under constant vaccination controls, the basic repro- Toda chain like in [1, 2] with additional terms accounting for the duction number R0 and the disease-free equilibrium for the given nonlocal potential R(z). We show that the soliton interactions dy- model are calculated in terms of related parameters. Also the sta- namic compares favorably to full numerical results of the original bility of the disease-free equilibrium of the given model in terms NLSE, Eq. (6). of R0 is established which is locally asymptotically stable when R < 1 and unstable when R > 1 and globally stability occurs 1. M. D. Todorov, V. S. Gerdjikov and A. V. Kyuldjiev, Multi-soliton in- 0 0 when R 1. Using PRCC technique sensitivity analysis is ad- teractions for the Manakov system under composite external potentials, 0 ≤ Proceedings of the Estonian Academy of Sciences, Phys.-Math. Series, ditionally investigated to review the influence of model parameters 64, No. 3 (2015), 368-378. on the Human Papilloma Virus infection widespread. Expecting infection predominance below the consistent control, ideal control 2. V. S. Gerdjikov and M. D. Todorov, Manakov model with hypothesis is utilized to detail vaccination methodologies for the gain/loss terms and N-soliton interactions: Effects of periodic potentials, Journal Applied Numerical Mathematics, given model once the vaccination rate is performed of your time. https://doi.org/10.1016/j.apnum.2018.05.015, arXiv:1801.04897v1 The result of those techniques on the infected population and there- [nlin.SI]. fore the accrued price is assessed and contrasted with the consistent control case. 3. M. Salerno and B. B. Baizakov, Normal mode oscillations of a nonlocal composite matter wave soliton, Phys. Rev. E, 98 (2018), 062220. Keywords: HPV; Vaccine; Mathematical Model; Stability Analy- sis; Sensitivity Analysis; Optimal Control

Anomalous waves induced by abrupt changes in to- Applications of ﬁxed point theorems to integral and pography differential equations Nick Moore Muhammad Arshad Zia Florida State University International Islamic University, Islamabad Pakistan [email protected] [email protected]

73 The fixed point theory is one of the most rapidly growing topic of living organisms, as witnessed by different discordant data in the nonlinear functional analysis. It is a vast and interdisciplinary sub- literature. ject whose study belongs to several mathematical domains such We consider a variational data assimilation approach for the es- as: classical analysis, functional analysis, operator theory, topol- timation of the cardiac conductivity parameters able to combine ogy and algebraic topology, etc. This topic has grown very rapidly available patient-specific measures with mathematical models. In perhaps due to its interesting applications in various fields within particular, it relies on the least-square minimization of the misfit and out side the mathematics such as: integral equations, initial between experiments and simulations, constrained by the underly- and boundary value problems for ordinary and partial differential ing mathematical model. Operating on the conductivity tensors as equations, many existence theorems for the solution of differential control variables of the minimization, we obtain a parameter es- equations are proved by means of fixed point theorems. timation procedure. The methodology significantly improves the Inspired by the fact that the famous Banach contraction princi- numerical approaches present in literature. Moreover, we present ple has a lot of applications in theory of integral and differential an extensive numerical simulation campaign reproducing experi- equations and looking into the applications of fixed point theory mental and realistic settings in presence of noisy data [1]. We will in various domains, we have introduced a new concept of Fixed discuss the interplay between the estimation of Monodomain and point Theory to solve the Differential and Integral Equations. Us- Bidomain conductivities as well as experimental validation with ing fixed point theory, we have verified the existence and unique- ex-vivo animal tissues. ness of solutions for differential and integral equation. We have This work has been supported by the NSF under grant number also focused ourselves to establish a new fixed point theorem for DMS 1412973/1413037. generalized contraction mappings in complete metric spaces. We have illustrated examples to advocate the usability of our results. 1. A. Barone, F. Fenton and A. Veneziani, Numerical sensitivity analysis of a variational data assimilation procedure for cardiac conductivities, A collocation method for a class of a nonlinear par- Chaos, 27(9), 2017, 093930. tial differential equations Muhammad Usman Effective integration of some integrable NLS equa- University of Dayton, 300 College Park, Dayton OH 45469-2316, USA tions [email protected] Otis C. Wright, III Collocation methods have attracted the attention of computational Department of Science and Mathematics mathematicians during the last decade. In this talk, we will discuss Cedarville University some analytical results on an initial and boundary value problems 251 N. Main St. of the Korteweg-de Vries type equation. Numerical results are pre- Cedarville, OH 45314 sented to show the verification of analytical results using sinc col- [email protected] location methods. Some recent results are presented for the effective integration of finite-gap solutions of integrable nonlinear Schrodinger¨ equations [1, 2]. In particular, simple formulas are derived for critical values of the amplitude of the solution. CONTRIBUTED PAPERS

Cardiac conductivity estimation by a variational data 1. Wright, III, O.C., Effective integration of ultra-elliptic solutions of the focusing nonlinear Schrodinger¨ equation, Physica D, 321-322 (2016) assimilation procedure: Analysis and validation 16-38. Alessandro Barone and Alessandro Veneziani 2. Wright, III, O.C., Bounded ultra-elliptic solutions of the defocusing Emory University, Department of Mathematics nonlinear Schrodinger¨ equation, Physica D, 360 (2017) 1-16. 400 Dowman Dr, Atlanta, GA 30322 USA [email protected] and [email protected] Advanced dispersion engineering for wideband on- Flavio Fenton chip optical frequency comb generation School of Physics, Georgia Institute of Technology Ali Eshaghian Dorche, Ali Asghar Eftekhar, Ali Adibi 837 State St NW, Atlanta, GA 30332 USA School of Electrical and Computer Engineering, Georgia Institute of Tech- fl[email protected] nology, Alessio Gizzi 778 Atlantic Drive NW, Atlanta, GA 30332, USA Department of Engineering, University Campus Biomedico of Rome [email protected], [email protected], [email protected] Via Alvaro del Portillo, 21, 00128 Roma RM, Italy Optical frequency combs, which are the equidistant narrow- [email protected] linewidth optical signals in the frequency domain, provide a unique An accurate patient-specific parameter estimation is crucial for ex- platform for a variety of applications ranging from precise mea- tending computational tools from medical research to clinical prac- surements to enhanced optical signal processing and wideband in- tice. In cardiac electrophysiology, critical parameters are the con- terconnection. To generate wideband optical frequency combs ductivity tensors and their quantification is quite troublesome in through efficient power transfer from a pump signal to other comb

74 lines in an optical microresonator anomalous dispersion is required at the initial stage of the development of integrable turbulence, and to balance the Kerr nonlinearity dispersion with the cold cavity dis- our theoretical predictions exhibit a good agreement with the nu- persion. Considering the versatile application of this technology, merical simulations. it is of much interest to make these optical signals in a miniatur- ized chip-scale platform; however, having anomalous dispersion in 1. V. E. Zakharov. Turbulence in integrable systems. Stud. Appl. Math., this platform is more challenging due to limitations imposed by 122(3):219–234, 2009. both materials and fabrication processes. Silicon nitride (SiN) is 2. M. Onorato, D. Proment, G. El, S. Randoux, and P. Suret. On the origin the dominant CMOS-compatible material platform for on-chip op- of heavy-tail statistics in equations of the nonlinear Schrodinger¨ type. tical frequency comb generation. However, achieving anomalous Physics Letters A, 380(39):173–3177, 2016. dispersion in the SiN-on-oxide (SiO2) requires complicated fabrication processes to ameliorate cracks formed at SiN thicknesses Spectral stability of ideal-gas shock layers in the strong above 450 nm which is necessary in conventional dispersion engi- shock limit neering techniques. Bryn Balls-Barker∗ and Blake Barker Here we report a new dispersion-engineering approach to achieve Department of Mathematics, Brigham Young the necessary anomalous dispersion based on optimized coupled University, Provo, UT 84602, USA optical microresonators formed by bending an optimized air-clad, [email protected] and [email protected] over-etched, dispersion-engineered thin-film SiN waveguide. This session will focus on advanced dispersion engineered for efficient Olivier Lafitte optical frequency comb generation on a chip, including mathemat- LAGA, Institut Galilee, Universite Paris 13, 93 430 ical modeling to extract the eigenmodes of our proposed structure, Villetaneuse and CEA Saclay, DM2S/DIR, 91 191 numerical approach to solve generalized Lugiato-Lefever equation Gif sur Yvette Cedex, France solving the nonlinear dynamic of optical signal propagating inside lafi[email protected] a microresonator. An open question in gas dynamics is the stability of viscous shock layers, or traveling-wave solutions of the compressible Navier- Early stage of integrable turbulence in 1D NLS equa- Stokes equations. In general, the Evans function, which is typi- tion: the semi-classical approach to statistics cally computed numerically, plays a key role in determining the Giacomo Roberti and Gennady El stability of these traveling wave solutions. Northumbria University, NE1 8ST - Department of Mathematics, Physics The goal of this research is to analytically describe the spectral sta- and Electrical Engeneering, Newcastle upon Tyne, UK bility of ideal-gas shock layers in the strong shock limit using the [email protected] and [email protected] Evans function. The numerical stability of this system has been previously demonstrated [1] and we seek to make this stability Stephane´ Randoux and Pierre Suret more rigorous with an analytic proof. We do this by analytically Univ. Lille, CNRS, UMR 8523 - Physique des Lasers Atomes et Molecules´ solving for a basis of the unstable and stable manifolds and then by (PHLAM), using these solutions to create the Evans function. Due to numeri- F-59000 Lille, France cal instability in the Evans system associated with the compressible [email protected] and [email protected] Navier-Stokes equations, we utilize the compound matrix method The concept of integrable turbulence introduced by Zakharov [1] and a change of variables to find the bases. With the resulting an- has been recently recognised as a novel theoretical paradigm of ma- alytic approximation to the Evans function, we are able to study jor importance for a broad range of physical applications from pho- meaningful bounds on the stability of the shock layers. tonics to oceanography. One of the applications of the integrable 1. J. Humpherys, G. Lyng and K Zumbrun, Spectral stability of ideal-gas turbulence theory is the statistical description of the appearance of shock layers, Arch Rational Mech Anal, 194 (2009), 1029-1079. rogue waves. We consider the evolution of an initial partially coherent wave Frequency downshift in the ocean field with Gaussian statistics in the framework of the 1D Nonlinear Schrodinger¨ equation (1D-NLSE), and we analyse the normalized Camille R. Zaug∗ and John D. Carter fourth order moment of the field’s amplitude, which characterises Mathematics Department the “tailedness” of the probability density function (PDF) of the Seattle University field. The relation between this statistical quantiity and the spec- [email protected] and [email protected] tral width of the field has been recently provided in Onorato et al. Frequency downshift occurs when a measure of a waves frequency [2], however, it requires the spectral width knowledge at each step (typically its spectral peak or spectral mean) decreases monotoni- in time. In our work, thanks to the combination of tools from the cally. Carter and Govan (2016) derived a viscous generalization of wave turbulence theory and the semi-classical theory of 1D-NLSE, the Dysthe equation that successfully models frequency downshift we derive for the first time an analytical formula for the short time in wave tank experiments for certain initial conditions. The classi- evolution of the fourth order moment as a function of the statistical paper by Snodgrass et al. (1966) shows evidence that narrow- cal characteristics of the initial condition. This formula provides a banded swell traveling across the Pacific Ocean also display fre- quantitative description of the appearance of the ”heavy” (”low”) quency downshift. In this work, we test the viscous Dysthe equa- tail of the PDF in the focusing (defocusing) regime of the 1D-NLS tion against the Dysthe equation, nonlinear Schrodinger equation,

75 and the dissipative nonlinear Schrodinger equation to see which Stability of traveling waves in compressible Navier- generalization best models the ocean data reported in Snodgrass et Stokes al. We do so by comparing the Fourier amplitudes, the change in Taylor Paskett and Blake Barker the spectral peak and spectral mean, and conserved quantities representing mass and momentum between the ocean measurements Brigham Young University and numerical simulations. Provo, UT [email protected] 1. J. D. Carter, A. Govan. Frequency downshift in a viscous fluid. Euro- We develop a method for proving stability of traveling waves in pean Journal of Mechanics - B/Fluids, 59 (2016), 177-185. compressible Navier-Stokes using rigorous numerical verification. We use interval arithmetic to obtain complete error bounds on all 2. J. D. Carter, D. Henderson, and I. Butterfield. A comparison of fre- computations, including machine truncation error. We explain sev- quency downshift models of wave trains on deep water. Physics of Flu- ids, 31 (2019), 013103. eral novel methods that we employed to reduce numerical error in the computer-assisted computations. 3. F. E. Snodgrass, K. F. Hasselmann, G. R. Miller, W. H. Munk, W. H. Powers, Propagation of ocean swell across the Pacific, Philo- sophical Transactions of the Royal Society of London. Series A, Math- The narrow-capture problem in a unit sphere: ematical and Physical Sciences, 259 (1966), 431-497. Global optimization of volume trap arrangements

Alexei Cheviakov∗ and Jason Gilbert Department of Mathematics and Statistics, University of Saskatchewan Saskatoon, SK, Canada S7N 5E6 [email protected] and [email protected] POSTERS The determination of statistical characteristics for particles under- Shock formation in finite time for the 1D compress- going Brownian motion in constrained domains have multiple ap- ible Euler equations plications in various areas of research. This work presents a first attempt to systematically compute globally optimal configurations of Lucas Schauer and Geng Chen traps inside a three-dimensional domain that minimize the average University of Kansas mean first passage (MFPT) time for the narrow capture problem – Lawrence, KS 66049 the average time it takes a particle to be captured by any trap. [email protected] For a given domain, the mean first passage time satisfies a linear The majority of physical models in science and engineering are for- Poisson problem with Dirichlet-Neumann boundary conditions. mulated as partial differential equations (PDEs). My research fo- While no closed-form general solution of such problems is known, cuses on the analysis of fundamental properties on many important approximate asymptotic MFPT expressions for small traps in a unit nonlinear PDE models, especially existence, uniqueness, and sta- sphere have been found. These solutions explicitly depend on trap bility of solutions. This also includes singularity formation, such parameters, including locations, through a pairwise potential func- as the shock wave in gas dynamics. These solutions exhibiting tion. singularities give way to very important, exciting, and challeng- After probing the applicability limits of asymptotic formulas ing research topics. The study on the formation and propagation through comparisons with numerical and available exact solutions of these singularities, which is notoriously difficult due to the lack of the narrow capture problem, full three-dimensional global opti- of regularity, is one of the central topics in the field of nonlinear mization was performed to find optimal trap positions in the unit PDEs. sphere for 2 N 100 identical traps. The interaction en- Compressible Euler equations, governing compressible inviscid ergy values and≤ geometrical≤ features of the putative optimal trap flow, have been widely used for gas dynamics and engineering such arrangements are presented. as aircraft designs, and are one of the most fundamental PDE systems. The Euler equations were first found by Leonhard Euler in 1. J. Gilbert and A. Cheviakov, Globally optimal volume-trap arrange- 1757, and then were studied by many great mathematicians, in- ments for the narrow-capture problem inside a unit sphere, Phys. Rev. cluding Riemann, Lagrange, Stokes, Courant, Von Neumann, Lax, E 99 (2019), 012109. etc. This system is a natural model to capture the formation and 2. A. Cheviakov and M. J. Ward, Optimizing the principal eigenvalue of propagation of shock waves in the gas. the Laplacian in a sphere with interior traps, Math. Comp. Mod. 53 For the isentropic 1-D solutions for Euler equations, I will discuss a (2011), 1394-1409. research project on the shock formation theory for the Euler equations with damping. One key thing I will reference is Peter Lax’s celebrated work on shock formation in 1964. Honing his clever Theory and observation of interacting linear waves technique as it pertains to this system, I can show the existence of and nonlinear mean flows in a viscous fluid conduit an optimal density lower bound. Hence, showing existence of a Ryan Marizza, Jessica Harris, Michelle Maiden, and blow up in finite time follows from this bound. Mark A. Hoefer Department of Applied Mathematics, University of Colorado, Boulder [email protected]

76 A theoretical and experimental analysis is described for the interactions of linear, small amplitude, dispersive waves with evolving, nonlinear mean flows that include oscillatory, compressive dispersive shock waves and smooth expansion waves in a viscous fluid conduit. Analysis of such interactions has been developed for waves described by the Kortweg-de Vries (KdV) equation in the context of shallow water waves [1]. In this poster, a similar analysis is applied to linear wave-mean flow interactions for the conduit equation that models a viscous fluid conduit—the cylindrical, free interface between two miscible, Stokes fluids with high viscosity contrast. A condition on the linear wave’s wave-number pre and post interaction determines whether the linear wave will be trans- mitted through or be trapped by the mean flow. This analysis is complemented by direct numerical solutions of the conduit equation and preliminary experimental results.

[1] T. Congy, G. A. El, and M. A. Hoefer, Interaction of linear modulated waves with unsteady dispersive hydrodynamic states, arXiv:1812.06593 (2018).