Book of Abstracts

BOOK OF ABSTRACTS

THE TENTH IMACS INTERNATIONAL CONFERENCE ON NONLINEAR EVOLUTION EQUATIONS AND WAVE PHENOMENA: COMPUTATION AND THEORY

Athens, Georgia March 29 - April 01, 2017

Sponsored by The International Association for Mathematics and Computers in Simulation (IMACS) The Computer Science Department, U GA

Edited by: Thiab Taha Sponsors Keynote Speakers

International Association for Mathematics Gino Biondini: Singular Asymptotics for and Computers in Simulation (IMACS) Nonlinear Waves" Computer Science Department at UGA Nathan Kutz: Data-driven Discovery of Nonlinear Wave Equations" Thierry Colin: A hierarchy of nonlinear models for tumor growth and clinical applications"

Conference Organization

R. Vichnevetsky (USA), Honorary President of IMACS, Honorary Chair T. Taha (USA), General Chair & Conference Coordinator J. Bona (USA), Co-chair

Scientific Program Committee

Mark Ablowitz (USA) Willy Hereman (USA) Nail Akhmediev(Australia) Alex Himonas (USA) David Amrbrose (USA) Mat Johnson (USA) Stephen Anco(Canada) Pedro Jordan (USA) Andrea Barreiro(USA) Nalini Joshi(Australia) Gino Biondini (USA) Kenji Kajiwara(Japan) Lorena Bociu (USA) Henrik Kalisch (Norway) Jerry Bona (USA) David Kaup (USA) Robert Buckingham (USA) Panayotis Kevrekidis (USA) Annalisa Calini (USA) Alexander Korotkevic (USA) Ricardo Carretero (USA) Gregor Kovacic (USA) Mathieu Colin (France) Stephane Lafortune (USA) Thierry Colin (France) Yuri Latushkin (USA) John Carter (USA) Zhiwu Lin (USA) Min Chen (USA) Yue Liu (USA) Ming Chen (USA) Andrei Ludu (USA) Demetrios Christodoulides (USA) Pavel Lushnikov (USA) Christopher Curtis (USA) K. G. Makris (Greece) Bernard Deconinck (USA) Dionyssis Mantzavions(USA) Vassilios Dougalis (Greece) Peter Miller (USA) Anton Dzhamay (USA) Dimitrios Mitsotakis(USA) Hassan Fathallah (USA) Ziad Musslimani (USA) Bao-feng Feng (USA) Remus Mihai Osan(USA) Thanasis Fokas (UK) Robert Pego (USA) Anna Ghazaryan(USA) Beatrice Pelloni (UK) Unal Goktas(Turkey) Barbara Prinari (USA) Zaher Hani(USA) Zhijun Qiao (George) (USA) Scientific Program Committee (Continue)

Changzheng Qu (China) Thiab Taha (USA) Milena Radnovic (Australia) Michail Todorov(Bulgaria) Xu Runzhang (China) Muhammad Usman (USA) Constance Schober (USA) Jianke Yang (USA) Natalie Sheils (USA) Chongchun Zeng (USA) Israel Michael Sigal (Canada)

Organized Sessions

1. Jerry Bona, Min Chen, Dimitrios Mitsotakis, Shenghao Li, "Nonlinear Waves" 2. Anton Dzhamay, Virgil Pierce, and Chris Ormerod, "Painleve Equations, Integrable Systems, and Random Matrices” 3. Alex Himonas, Curtis Holliman and Dionyssis Mantzavinos, "Evolution Equations and Integrable Systems" 4. David Kaup, Constance Schober, and Thomas Vogel, "Applied Nonlinear Waves" 5. Andrei Ludu and Michail Todorov, "Solitary and Rogue Waves as Solutions of Generalized Schrodinger Equations. Achievements and Challenges" 6. Nalini Joshi, Christopher Lustri, Nobutaka Nakazono, Milena Radnovic and Yang Shi, "Discrete integrable systems" 8. Unal Goktas and Muhammad Usman, “Analytical and computational methods to study nonlinear partial differential equations” 9. Ricardo Carretero and Panos Kevrekidis,"Nonlinear Schrodinger Models and Applications" 11. Yuri Latushkin and Sam Walsh, "Traveling waves and spectral theory" 12. Anna Ghazaryan and Stephane Lafortune, "Wave phenomena in combustion" 13. Pedro Jordan, "Nonlinear Wave Phenomena in Continuum Physics: Some Recent Findings" 14. Baofeng Feng,Kenji Kajiwara, Annalisa Calini, "Integrable systems and the geometry of curves and surfaces" 15. Robert Buckingham and Peter Miller, "Asymptotics and Applied Analysis" 16. John Carter and Henrik Kalish, "Nonlocal and full-dispersion model equations in in fluid mechanics" 17. David Ambrose and Gideon Simpson, "Analysis of numerical methods for dispersive and fluid equations" 18. Alexander Korotkevich and Pavel Lushnikov, "Nonlinear waves, dynamics of singularities, and turbulence in hydrodynamics, physical, and biological systems" 19. Xu Runzhang, "Functional analysis and PDEs" 20. Mathieu Colin and Tatsuya Watanabe, "Stability properties for nonlinear dispersive equations" 21. Vassilis Rothos and Eftathios Charalampidis, "Nonlinear Waves: Mathematical Methods and Applications" 22. Stephen Gustafson, Israel Michael Sigal, Avy Soffer, "Nonlinear evolution equations of quantum physics and their topological solutions" 23. Brad Shadwick, Antoine Cerfon, Bedros Afeyan, "Waves and Instabilities in Vlasov plasmas" 24. Wooyoung Choi,Ricardo Barros, "Surface and Internal Waves and their Interaction" 25. Andrea Barreiro, Katherine Newhall, Remus Osan, Pamela Pyzza , "Nonlinear dynamics in mathematical biology and neuroscience"

PROGRAM AT A GLANCE WEDNESDAY, MARCH 29TH, 2017

Mahler Room Q Room R Room J Room F/G Room E Room C 8:00- 8:30 WELCOME 8:30 – 9:30 Keynote 1: Gino Biondini 9:30 – 10:00 COFFEE BREAK 10:00 – 10:50 S3_I/VI S15_I/VI PAPERS S20_I/II S11_I/IV S17_I/II 10:55 – 12:10 S3_II/VI S15_II/VI S6_I/III S17_II/II S21_I/III PAPERS 12:10 – 1:40 LUNCH IN MAGNOLIA BALL ROOM 1:40 – 3:20 S3_III/VI S1_I/II S25_I/V S20_II/II S11_II/IV PAPERS 3:20 – 3:50 COFFEE BREAK 3:50 – 5:55 S3_IV/VI S1_II/II S25_II/V S22_I/II S6_II/III S19_I/I S4_I/I

THURSDAY, MARCH 30TH, 2017 Mahler Room Q Room R Room J Room F/G Room E Room C 8:00- 9:00 Keynote 2: T. Colins 9:10 – 10:00 S21_II/III S15_III/VI S16_I/IV S2_I/III S18_I/V S24_I/II S25_III/V 10:00 – 10:30 COFFEE BREAK 10:30 – 12:10 S3_V/VI S9_I/II S16_II/IV S2_II/III S18_II/V S11_III/IV S25_IV/V 12:10 – 1:40 LUNCH (attendees on their own) 1:40 – 3:20 S3_VI/VI S15_IV/VI S11_IV/IV S2_III/III S18_III/V S24_II/II 3:20 – 3:50 COFFEE BREAK 3:50 – 5:55 S14_I/II S13_I/II S5_I/III S25_V/V S6_III/III S9_II/II S21_III/III 5:00 – 7:00 pm POSTERS Hill Atrium, outside of Mahler 7:00 – 9:00 Conf. Banquet (including student papers awards)

FRIDAY, MARCH 31ST, 2017 Mahler Room Q Room R Room J Room F/G Room E Room C 8:00- 9:00 Keynote 3: Nathan Kutz 9:10 – 10:00 S15_V/VI S18_IV/V S16_III/IV S5_II/III S23_I/IV S12_I/II S08_I/II 10:00 – 10:30 COFFEE BREAK 10:30– 12:10 S15_VI/VI S14_II/II S16_IV/IV S8_I/II S23_II/IV S13_II/II S12_II/II-1 12:10 – 1:40 LUNCH (attendees on their own) 1:40 – 3:20 S18_V/V S5_III/III S22_II/II S23_III/IV 3:20 – 3:50 COFFEE BREAK 3:50 – 5:55 S23_IV/IV

1 PROGRAM

TUESDAY, MARCH 28, 2017 5:00 – 6:00 REGISTRATION 5:00 – 7:00 RECEPTION

WEDNESDAY, MARCH 29, 2017 7:30 – 9:30 REGISTRATION 8:00 – 8:30 WELCOME Thiab Taha/Program Chair and Conference Coordinator Pamela Written/Senior Vice President for Academic Affairs & Provost at UGA. Event Manager/Georgia Center

8:30 – 9:30 KEYNOTE LECTURE I, Mahler: Gino Biodini: Singular Asymptotics for Nonlinear Waves CHAIR: THIAB TAHA

9:30 – 10:00 COFFEE BREAK

10:00 – 10:50 SESSION 3, Mahler: Evolution Equations and Integrable Systems – Part I/VI CHAIRS: HIMONAS, CURTIS HOLLIMAN AND DIONYSSIS MANTZAVINOS, 10:00 – 10:25 Panayotis Kevrekidis: From Discrete Solitons to Discrete Breathers and to Lattice Traveling Waves: A Discussion of Spectral Stability and Associated Criteria 10:25 – 10:50 Nikolaos Tzirakis: Well posedness theory for nonlinear dispersive equations on the half line

10:00 – 10:50 SESSION 15, Room Q: Asymptotics and Applied Analysis – Part I/VI CHAIRS: ROBERT BUCKINGHAM AND PETER MILLER 10:00 – 10:25: Peter Perry: Global Existence and Asymptotics for the Derivative Non-Linear Schr\"{o}dinger Equation in One Dimension 10:25 – 10:50: Anton Dzhamay: Rational Mapping Factorization and Tau-functions of Discrete Painlevé Equations

10:00 – 10:50 PAPERS, Room R: CHAIRS: 10:00 – 10:25 Matthew Tranter and Karima Khusnutdinova: Scattering of nonlinear bulk strain waves in delaminated bars 10:25 – 10:50: Martin Klaus: Eigenvalue asymptotics for Zakharov-Shabat systems with long-range Potentials

10:00 – 10:50 SESSION 20, Room J: Stability properties for nonlinear dispersive equations – Part I/II CHAIRS: MATHIEU COLIN 10:00 – 10:25 Dmitry Pelinovsky: Transverse stability of periodic waves in the Kadomtsev-Petviashvili -II equation 10:25 – 10:50 Tatsuya Watanabe: Standing waves for the nonlinear Schrodinger equation coupled with the Maxwell equation

10:00 – 10:50 SESSION 11, Room F/G: Traveling Waves and Spectral Theory – Part I/IV CHAIRS: YURI LATUSHKIN AND SAM WALSH 10:00 – 10:25 Jared Bronski, Robert Marangell and Mat Johnson: Modulational stability of quasiperiodic solutions of Hamiltonian PDE 10:25 – 10:50 Atanas Stefanov: Traveling waves for the mass-in-mass model of granular chains

10:00 – 10:50 SESSION 17, Room E: Analysis of numerical methods for dispersive and fluid equations – Part I/II CHAIRS: DAVID AMBROSE AND GIDEON SIMPSON 10:00 – 10:25 Gideon Simpson: Adaptive Methods for Derivative Nonlinear Schrödinger Equations 10:25 – 10:50 Leo Rebholz: On conservation laws of Navier-Stokes Galerkin discretizations

10:55 – 12:10 SESSION 3, Mahler: Evolution Equations and Integrable Systems – Part II/VI CHAIRS: ALEX HIMONAS, CURTIS HOLLIMAN AND DIONYSSIS MANTZAVINOS, 10:55 – 11:20 Karen Yagdjian: Integral transform approach to evolution equations in the curved spacetime 11:20 – 11:45 Alex Himonas: Well-posedness of evolutions equations via the unified transform method 11:45 – 12:10 Curtis Holliman: Well-Posedness for a Modified NLS equation

10:55 – 12:10 SESSION 15, Room Q: Asymptotics and Applied Analysis – Part II/VI CHAIRS: ROBERT BUCKINGHAM AND PETER MILLER 10:55 – 11:20 Marco Bertola, Alexander Tovbis: Maximal amplitude formula for the finite gap (quasiperiodic) solutions to the focusin NLS and its applications to large amplitude (rogue) waves 11:20 – 11:45 Christopher Ormerod: Elliptic isomonodromy and the elliptic Painleve equation

10:55 – 12:10 SESSION 6, Room R: Discrete integrable systems – Part I/III CHAIRS: YASUHIRO OHTA 10:55 – 11:20: Nobutaka Nakazono: Elliptic Painlevé equations 11:20 – 11:45: Kazushige Endo: Asymptotic analysis of stochastic cellular automata 11:45 – 12:10: Toshiyuki Mano: Regular flat structures and generalized Okubo systems

10:55 – 12:10 SESSION 17, Room J: Analysis of numerical methods for dispersive and fluid equations – Part II/II Chairs: David Ambrose and Gideon Simpson 10:55 – 11:20 Molei Tao: Explicit high-order symplectic integration of nonseparable Hamiltonians, with a toy NLS example 11:20 – 11:45 Jeremy Marzuola: Trigonometric integrators for quasilinear wave equations 11:45 – 12:10 David Ambrose: Convergence of a boundary integral method for 3D interfacial Darcy flow with surface tension

10:55 – 12:10 SESSION 21, F/G: Nonlinear Waves: Mathematical Methods and Applications- – Part I/II Chairs: Vassilis Rothos and Efstathios Charalampidis 10:55 – 11:20 M. Fararzmand and T. Sapsis: Reduced-order prediction of rogue waves in two-dimensional deep-water waves 11:20 – 11:45 P. Carter and B. Sandstede: Single and double pulses in the FitzHugh--Nagumo system 11:45 – 12:10 O. Wright: Bounded ultra-elliptic solutions of the defocusing nonlinear Schrödinger equation

3 10:55 – 12:10 PAPERS, Room E CHAIRS: Paul Christodoulide 10:55 – 11:20 Paul Christodoulides, Lauranne Pellet, Sarah Donne, Chris Bean and Frederic Dias: Interaction of ocean waves of nearly equal frequencies and the effect on pressure 11:20 – 11:45 Lazaros Aresti, Georgios Florides, Paul Christodoulides and Lazaros Lazari: Groundwater flow and Ground Heat Exchangers 11:45 – 12:10 Katie Oliveras and Christopher Curtis: Instabilities of Two Stratified Fluids Under Linear Shear

12:10 – 1:40 LUNCH in MAGNOLIA BALL ROOM

1:40 – 3:20 SESSION 3, Mahler: Evolution Equations and Integrable Systems-Part III/VI CHAIRS: ALEX HIMONAS, CURTIS HOLLIMAN AND DIONYSSIS MANTZAVINOS, 1:40 – 2:05 Anahit Galstyan: Semilinear Hyperbolic Equation in the de Sitter Spacetime with Hyperbolic Spatial Part 2:05 – 2:30 Dionyssis Mantzavinos: On rigorous aspects of the unified transform method: linear and nonlinear evolution equations on the half-line 2:30 – 2:55 Efstathios Charalampidis: Multi-component nonlinear waves in nonlinear Schr\"odinger (NLS) systems 2:55 – 3:20 John Holmes: A note on the non-periodic compressible Euler equations

1:40 – 3:20 SESSION 01, Room Q: Nonlinear Waves - Part I/II CHAIRS: JERRY BONA, MIN CHEN, DIMITRIOS MITSOTAKIS, Shenghao Li 1:40 – 2:05 Jerry Bona: Higher-order, unidirectional models for surface water waves 2:05 – 2:30 Olivier Goubet and Imen Manoubi: A water wave model with a nonlocal viscous dispersive term 2:30 – 2:55 Shu-Ming Sun: Existence of multi-hump capillary-gravity waves on water of finite depth 2:55 – 3:2 Shenghao Li and Min Chen: Standing waves of two-dimensional Boussinesq systems

1:40 – 3:20 SESSION 25, Room R: Nonlinear dynamics in Mathematical Biology and Neuroscience - Part I/V CHAIRS: ANDREA BARREIRO, KATIE NEWHALL, REMUS OSAN, PAMELA PYZZA 1:40 – 2:05 Igor Belykh: When two wrongs make a right: synchronized neuronal bursting from combined inhibitory and electrical coupling 2:05 – 2:30 Pietra-Luciano Buono: Pattern formation in a nonlocal mathematical model for the multiple roles of the TGF-beta pathway in tumour dynamics

1:40 – 3:20 SESSION 20, Room J: Stability properties for nonlinear dispersive equations – Part II/II CHAIRS: TATSUYA WATANABE 1:40 – 2:05: Mats Ehrnstrom : Existence of a highest wave in a fully dispersive two-way shallow water model 2:05 – 2:30: Noriyoshi Fukaya : Instability of solitary waves for a generalized derivative nonlinear Schrödinger equation in a borderline case 2:30 – 2:55: Hiroaki Kikuchi: Global dynamics above the ground state energy for a class of nonlinear Schrodinger equations with critical growth 2:55 – 3:20: Mathieu Colin:Solitons in quadratic media

4 1:40 – 3:20 SESSION 11, Room F/G: Traveling Waves and Spectral Theory – Part II/IV CHAIRS: YURI LATUSHKIN AND SAM WALSH 1:40 – 2:05 Alim Sukhtayev, Kevin Zumbrun, Soyeun Jung and Raghavendra Venkatraman: Diffusive stability of spatially periodic patterns 2:05 – 2:30 Zineb Hassainia, Nader Masmoudi and Miles Wheeler: Global bifurcation of rotating vortex patches 2:30 – 2:55 J. Douglas Wright: Traveling waves in diatomic Fermi-Pasta-Ulam-Tsingou lattices. 2:55 – 3:20 Kristoffer Varholm: Global bifurcation of gravity water waves with multiple critical layers

1:40 – 3:20 PAPERS, Room E CHAIRS: 1:40 – 2:05 Douglas Svensson Seth and Erik Wahlén: Steady three-dimensional ideal flows with nonvanishing vorticity in domains with edges 2:05 – 2:30 Youssef Driss: On the Theory of Nonlinear Shock Waves and Supersonic Flow 2:30 – 2:55 Zhivko S. Athanassov: Evolution Equations in Topological Vector Spaces

3:20 - 3:50 COFFEE BREAK

3:50 – 5:55 SESSION 3, Mahler: Evolution Equations and Integrable Systems-Part IV/VI CHAIRS: ALEX HIMONAS, CURTIS HOLLIMAN AND DIONYSSIS MANTZAVINOS, 3:50 – 4:15 Zhijun Qiao: Short pulse systems produced through the negative WKI hierarchy 4:15 – 4:40 Ryan Thompson: Decay Properties of Solutions to a 4-parameter Family of Wave Equations 4:40 – 5:05 Stephen Anco: Peakons: weak solutions or distributional solutions? 5:05 – 5:30 Erwin Suazo: Soliton solutions for a generalized variable coefficient nonlinear Schrodinger equation 5:30 – 5:55 Axel Schulze-Halberg: Linearization and exact solvability of the Burgers equation with time-dependent coefficients and nonlinear forcing term

3:50 – 5:55 SESSION 01, Room Q: Nonlinear Waves – Part II/II CHAIRS: JERRY BONA, MIN CHEN, DIMITRIOS MITSOTAKIS, Shenghao Li 3:50 – 4:15 John Albert: Well-posedness of the dispersion-managed nonlinear Schrodinger equation and related equations 4:15 – 4:40 Jerry Bona and Min Chen: Singular Solutions of a Boussinesq System for Water Waves 4:40 – 5:05 Maria Bjørnestad and Henrik Kalisch: Shallow Water Waves on a Vertical Shear Flow 5:05 – 5:30 Jerry Bona, Mimi Dai: Norm-inflation results for the BBM equation

3:50 - 5:55 SESSION 25, Room R: Nonlinear dynamics in Mathematical Biology and Neuroscience - Part II/V CHAIRS: ANDREA BARREIRO, KATIE NEWHALL, REMUS OSAN, PAMELA PYZZA 3:50 – 4:15 Victoria Booth: Piecewise smooth maps for the circadian modulation of sleep-wake dynamics 4:15 – 4:40 Tom Dinitz: Stochasticity and the Neural Sleep-Wake Architecture 4:40 – 5:05 Shelby Wilson: Modeling the Dynamics of the Human Sleep/Wake Cycle 5:05 – 5:30 Pamela Pyzza: Modeling the Effects of Temperature on Sleep Patterns

5 3:50 – 5:55 SESSION 22, Room J: Nonlinear evolution equations of quantum physics and their topological solutions – Part I/II CHAIRS: S. GUSTAFSON, I.M. SIGAL, A. SOFFER 3:50 – 4:15 Nicholas Ercolani: PDE Models of Ginzburg-Landau Type for Defect Formation in Pattern-Forming Systems 4:15 – 4:40 Fabio Pusateri: The Nonlinear Schrödinger equation with a potential 4:40 – 5:05 Eric Carlen: Functional Inequalities and Gradient Flow for Quantum Evolution Equations 5:05 – 5:30 Maria Carvalho: Quantum Master Equations in Kinetic Theory 5:30 – 5:55 Pavel Lushnikov: Dynamics of singularities in 2D free surface hydrodynamics

3:50 – 5:55 SESSION 6, Room F/G: Discrete integrable systems – Part II/III CHAIRS: VLADIMIR DRAGOVIC AND HAJIME NAGOYA 3:50 – 4:15: Yasuhiro Ohta: Regular and finite time blowup solutions for discrete integrable equations 4:15 – 4:40: Masato Shinjo: Non-autonomous discrete hungry integrable systems and asymptotic expansions of their determinant solutions 4:40 – 5:05: Takafumi Mase: Spaces of initial conditions for nonautonomous mappings of the plane 5:05 – 5:30: Masataka Kanki: Detecting the integrability of discrete dynamical systems by the co-primeness property 5:30 – 5:55: Nobe Atsushi: Mutations of cluster algebras and discrete integrable systems

3:50 – 5:55 SESSION 19, Room E: Functional analysis and PDEs –Part I/I CHAIR: XU RUNZHANG 3:50 – 4:15 Yanbing Yang: Global well-posedness of solutions for a class of fourth-order strongly damped nonlinear wave equations 4:15 – 4:40 Wei Lian: Global non-existence for nonlinear wave equations with conical degeneration with low initial energy 4:40 – 5:05 Yongbing Luo: Global existence and nonexistence for strongly damping wave equations with conical degeneration 5:05 – 5:30 Yuxuan Chen: Global existence and blow up of solution for semi-linear edge-degenerate parabolic equations 5:30 – 5:55 Salik Ahmed: Global existence and blow up of solution for semi-linear hyperbolic equation with logarithmic nonlinearity

3:50 – 4:15 SESSION 4, Room C: Applied Nonlinear Waves – Part I/I CHAIRS: DAVID KAUP, C.ONSTANCE SCHOBER, AND TOM VOGEL 3:50 – 4:15 A. Calini and C. Schober: Rogue waves over non-constant backgrounds 4:15 – 4:40 M. Russo: Breathers and rogue waves on a vortex filament with nontrivial axial flow 4:40 – 5:05 T. Vogel: Internally driven oceanic surface waves 5:05 – 5:30 A. Bhatt and B. Moore: Structure preserving exponential integrators for damped-driven nonlinear waves 5:30 – 5:55 Z. Shuai: Modeling cholera spread in a stream environment

6 THURSDAY, MARCH 30, 2017 7:30 – 9:30 REGISTRATION

8:00 – 9:00 KEYNOTE LECTURE 2, Mahler T. Colin: A hierarchy of nonlinear models for tumor growth and clinical applications CHAIR: JERRY BONA

9:10 – 10:00 SESSION 15, Room Q: Asymptotics and Applied Analysis – Part III/VI CHAIRS: ROBERT BUCKINGHAM AND PETER MILLER 9:10 – 9:35 Richard Kollar: Spectral stability in reduced systems 9:35 – 10:00 Thomas Bothner: On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo-Miwa-Ueno differential

9:10 – 10:00 SESSION 16, Room R: Nonlocal and full-dispersion model equations in in fluid mechanics – Part I/IV CHAIRS: HENRIK KALISCH AND JOHN CARTER 9:10 – 9:35 John Carter: Comparisons between experiments and various versions of the Whitham equation 9:35 – 10:00 Filippo Remonato, Mats Ehrnstrom, Henrik Kalisch and Mat Johnson: Two-dimensional bifurcation in the Whitham Equation with surface tension

9:10 – 10:00 SESSION 2, Room J: Painleve Equations, Integrable Systems, and Random Matrices – Part I/III CHAIRS: CHRIS ORMEROD, ANTON DZHAMAY, AND VIRGIL PIERCE 9:10 – 9:35 Akane Nakamura: Three facets of the theta divisor associated with the autonomous Garnier system of type 9/2 9:35 – 10:00 Alessandro Arsie and Paolo Lorenzoni: Complex reflection groups, bi-flat F-manifolds and Painlevé equations

9:10 – 10:00 SESSION 18 I/V, Room F/G: Nonlinear waves, dynamics of singularities, and turbulence in hydrodynamics, physical, and biological systems – Part I/V CHAIRS: ALEXANDER KOROTKEVICH AND PAVEL LUSHNIKOV 9:10 – 9:35 Stephen Gustafson: Stability of periodic waves of 1D nonlinear Schrödinger equations 9:35 – 10:00 Tobias Schaefer, Instantons and the stochastic Burgers equation

9:10 – 10:00 SESSION 24, Room E: Nonlinear internal waves and their Interaction with surface waves –Part I/II CHAIRS: WOOYOUNG CHOI 9:10 – 9:35 Ricardo Barros, Wooyoung Choi and Paul Milewski, Large amplitude internal waves in three-layer flows 9:35 – 10:00 Yuri Lvov and Esteban Tabak, Internal Waves in the Ocean in the Presence of Shear: wave turbulence perspective.

9:10 - 10:00 SESSION 25, Room C: Nonlinear dynamics in Mathematical Biology and Neuroscience - Part III/V CHAIRS: ANDREA BARREIRO, KATIE NEWHALL, REMUS OSAN, PAMELA PYZZA 9:10 - 9:35 Remus Osan: Traveling waves in one-dimensional Hodgkin Huxley neuronal networks 9:35 - 10:00 Christina Lee: Wave Patterns in an Excitable Neuronal Network

10:00-10:30 COFFEE BREAK

10:30 – 12:10 SESSION 3, Mahler: Evolution Equations and Integrable Systems-Part V/VI CHAIRS: ALEX HIMONAS, CURTIS HOLLIMAN AND DIONYSSIS MANTZAVINOS, 10:30 – 10:55 Jean-Claude Saut: On the Whitham and related equations 10:55 – 11:20 Henrik Kalisch: Particle Trajectories and Wave Breaking in the KdV Approximation 11:20 – 11:45 Hongqiu Chen: Stability of Solitary Wave Solutions to a coupled System 11:45 – 12:10 Mathias Arnesen: Non-uniform dependence on initial data for equations of Whitham type

10:30 – 12:10 SESSION 9, Room Q: Nonlinear Schrödinger Models and Applications - Part I/II CHAIRS: RICARDO CARRETERO AND PANOS KEVREKIDIS 10:30 – 10:55 Sergej Flach: Intermittent many-body dynamics at equilibrium 10:55 – 11:20 Christopher Chong: Nonlinear Excitations in Lattices with Long Range Interactions 11:20 – 11:45 David Kaup: Surface Breathers in Graphene 11:45 – 12:10 Igor Barashenkov: A PT-symmetric necklace of optical waveguides with a gain and loss ombré

10:30 – 12:10 SESSION 16, Room R: Nonlocal and full-dispersion model equations in in fluid mechanics – Part II/IV Chairs: Henrik Kalisch and John Carter 10:30 – 10:55 Gabriele Bruell, Mats Ehrnström and Long Pei: Symmetry and decay of traveling waves to a nonlocal shallow water model 10:55 – 11:20 Gabriele Bruell, Mats Ehrnström and Long Pei: On the symmetry of traveling-wave solutions to the Whitham equation 11:20 – 11:45 Vincent Duchêne, Dag Nilsson and Erik Wahlén: Solitary waves for a class of nonlocal Green-Naghdi systems 11:45 – 12:10 Evgueni Dinvay, Henrik Kalisch, Daulet Moldabayev, Denys Dutykh and Emilian Parau: The Whitham equation with capillarity 10:30 – 12:10 SESSION 2, Room J: Painleve Equations, Integrable Systems, and Random Matrices – Part II/III CHAIRS: ANTON DZHAMAY, VIRGIL PIERCE, AND CHRIS ORMEROD 10:30 – 10:55 Peter Miller: Rational Solutions of Painlevé Equations 10:55 – 11:20 Robert Buckingham, Robert Jenkins and Peter Miller: Semiclassical dynamics of the three-wave resonant interaction equations 11:20 – 11:45 Seung-Yeop Lee and Meng Yang: Discontinuity in the asymptotic behavior of planar orthogonal polynomials under a perturbation of the Gaussian weight 11:45 – 12:10 Dylan Murphy: Algebraic geometry of scattering theory for orthogonal polynomials

10:30 – 12:10 SESSION 18, Room F/G: Nonlinear waves, dynamics of singularities, and turbulence in hydrodynamics, physical, and biological systems – Part II/V CHAIRS: ALEXANDER KOROTKEVICH AND PAVEL LUSHNIKOV 10:30 – 10:55 Avy Soffer: Global Existence, Blowup and Scattering for large data Supercritical and other wave equations 10:55 – 11:20 Daniel Appeloe: An Energy Based Discontinuous Galerkin Method for Hamiltonian Systems 11:20 – 11:45 Benno Rumpf: Ensemble dynamics and the emergence of correlations in wave turbulence in one and two dimensions 11:45 – 12:10 Anastassiya Semenova: Hamiltonian Integration Method for Nonlinear Schrödinger Equation

10:30 – 12:10 SESSION 11, Room E: Traveling Waves and Spectral Theory – Part III/IV CHAIRS: YURI LATUSHKIN AND SAM WALSH

8 10:30 – 10:55 Peter Howard, Yuri Latushkin and Alim Sukhtayev: The Maslov and Morse indices for Schrodinger operators on R 10:55 – 11:20 Paul Cornwell and Christopher Jones: The Maslov index and the stability of traveling waves 11:20 – 11:45 Robert Marangell: Absolute instability for travelling waves in a chemotaxis model 11:45 – 12:10 Yuri Latushkin and Selim Sukhtaiev: The Maslov index and the spectra of second order elliptic operators

10:30 - 12:10 SESSION 25, Room C: Nonlinear dynamics in Mathematical Biology and Neuroscience - Part IV/V CHAIRS: ANDREA BARREIRO, KATIE NEWHALL, REMUS OSAN, PAMELA PYZZA 10:30 - 10:55 Duane Nykamp: Edge-correlations and synchrony in neuronal networks 10:55 - 11:20 Jennifer Crodelle: Synchronizing cortical dynamics via electrotonic junctions between excitatory neurons 11:20 - 11:45 Cheng Ly: Firing Rate Heterogeneity and Consequences for Coding in Feedforward Circuits 11:45 - 12:10 Deena Schmidt: Dimension reduction for stochastic conductance based neural models with time scale separation

12:10 – 1:40 Lunch on your own

1:40 – 3:20 SESSION 3, Mahler: Evolution Equations and Integrable Systems-Part VI/VI CHAIRS: ALEX HIMONAS, CURTIS HOLLIMAN AND DIONYSSIS MANTZAVINOS, 1:40 – 2:05 Jacek Szmigielski: Lax integrability and the peakon problem for the modified Camassa-Holm equation: Part I and Part II 2:05 – 2:30 Fangchi Yan: The unified transform method and well-posedness of the general NLS on the half line 2:30 – 2:55 Luiz Farah: Nonlinear Profile Decomposition and the Concentration Phenomenon for the Supercritical gKdV Equation

1:40 – 3:20 SESSION 15, Room Q: Asymptotics and Applied Analysis-Part IV/VI CHAIRS: ROBERT BUCKINGHAM AND PETER MILLER 1:40 – 2:05 Karl Liechty: Free fermions at finite temperature and the MNS matrix model 2:05 – 2:30 Guilherme Silva: Asymptotics for the normal matrix model and the mother body problem 2:30 – 2:55 David Smith: Nonlocal problems for linear evolution equations 2:55 – 3:20 Bingying Lu: The semi-classical sine-Gordon equation, universality at phase transition and the gradient catastrophe

1:40 – 3:20 SESSION 11, Room R: Traveling Waves and Spectral Theory - Part IV/IV CHAIRS: YURI LATUSHKIN AND SAM WALSH 1:40 – 2:05 Kyle Claassen and Mathew Johnson: Nondegeneracy of antiperiodic standing waves for fractional nonlinear Schrodinger equations 2:05 – 2:30 Graham Cox: A dynamical approach to semilinear elliptic equations 2:30 – 2:55 Ming Chen, Lili Fan, Hongjun Gao and Yue Liu: Break waves and solitary waves to the rotation-two-component Camassa-Holm system 2:55 – 3:20 Stephane Lafortune and Thomas Ivey: Spectral stability of solutions to the Vortex Filament Hierarchy

1:40 – 3:20 SESSION 2, Room J: Painleve Equations, Integrable Systems, and Random Matrices – Part III/III CHAIRS: VIRGIL PIERCE, CHRIS ORMEROD, AND ANTON DZHAMAY 1:40 – 2:05 Plamen Iliev: The generic quantum superintegrable system on the sphere and Racah operators

9 2:05 – 2:30 Megan Mccormick Stone: Eigenvalue densities for the Hermitian two-matrix model 2:30 – 2:55 Sevak Mkrtchyan: The birth and death of a random matrix

1:40 – 3:20 SESSION 18, Room F/G: Nonlinear waves, dynamics of singularities, and turbulence in

hydrodynamics, physical, and biological systems – Part III/V CHAIRS: ALEXANDER KOROTKEVICH AND PAVEL LUSHNIKOV 1:40 – 2:05 Israel Michael Sigal, On the Bogolubov-de Gennes Equations 2:05 – 2:30 Sergey Dyachenko: Instability of steep ocean waves and whitecapping 2:30 – 2:55 Zhen Qi: Cnoidal wave solutions to the Lugiato-Lefever Equations with applications to microresonators 2:55 – 3:20 Denis Silantyev: Langmuir wave filamentation in the kinetic regime and multidimensional Vlasov simulations

1:40 – 3:20 SESSION 24, Room E: Nonlinear internal waves and their Interaction with surface waves – Part II/II CHAIRS: RICARDO BARROS 1:40 – 2:05 Sergey Gavrilyuk: Valery Liapidevskii and Alexander Chesnokov: A mathematical model for spilling breakers 2:05 – 2:30 William Batson and Wooyoung Choi: Nonlinear evolution of Faraday waves in a rectangular container 2:30 – 2:55 Sunao Murashige: Long wave approximation with hodograph transformation for periodic internal waves in a two-fluid system 2:55 – 3:20 Wooyoung Choi: On modeling nonlinear surface and internal waves

3:20-3:50 COFFEE BREAK

3:50 – 5:55 SESSION 14, Mahler: Integrable systems and the geometry of curves and surfaces – Part CHAIRS: ANNALISA CALINI, BAOFENG FENG, KENJI KAJIWARA 3:50 – 4:15 Hsiao-Fan Liu: Some Examples of Integrable Geometric Curve Flows 4:15 – 4:40 Annalisa Calini, Stephane Lafortune and Brenton Lemesurier: On the stability of the Hasimoto Filament 4:40 – 5:05 Sampei Hirose, Jun-Ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura and Yasuhiro Ohta: dNLS flow on discrete space curves 5:05 – 5:30 Baofeng Feng: The geometric interpretation of the complex short pulse equation 5:30 – 5:55 Gloria Mari Beffa and Annalisa Calini: Hamiltonian structures for lattice Wn algebras and centro-affine geometry

3:50 – 5:55 SESSION 13, Room Q: Nonlinear wave phenomena in continuum physics: some recent findings – Part I/II CHAIR: PEDRO M. JORDAN 3:50 – 4:15 Sandra Carillo: KdV-type nonlinear evolution equations: non-Abelian versus Abelian Bäcklund charts 4:15 – 4:40 Len Margolin: The reality of artificial viscosity

10 4:40 – 5:05 Vinesh Nishawala, Martin Ostoja-Starzewski: Waves in random media with fractal and Hurst characteristics 5:05 – 5:30 Mads Peter Sørensen, Yuri B. Gaididei, Anders Roenne, Peter Leth Christiansen: Oscillating shock waves in nonlinear acoustics 5:30 – 5:55 Ronald E. Mickens, ꞌKale Oyedeji: Analysis of the traveling wave solutions to a modified diffusionless Fisher equation

3:50 – 5:55 SESSION 5, Room R: Solitary and Rogue Waves as Solutions of Generalized Schrodinger Equations. Achievements and Challenges - Part I/III CHAIRS: ANDREI LUDU, MICHAIL TODOROV 3:50 – 4:15 Efstathios Charalampidis, Panayotis Kevrekidis, Jingsong He, R. Babu Mareeswaran, T Kanna, Dimitri Frantzeskakis and Jesus Cuevas- Maraver: Formation of rogue waves in NLS systems: Theory and Computation 4:15 – 4:40 Andreas Mieritz, Mads Peter Soerensen, Allan Peter Engsig-Karup, Rasmus Dybbro Engelsholm, Ivan Bravo Gonzalo and Ole Bang: Design of supercontinuum optical sources aided by high performance computing 4:40 –-5:05 Andrus Salupere and Mart Ratas: On the application of 2D spectal analysis in case of the KP equation 5:05 – 5:30 Kert Tamm, Tanel Peets and Jüri Engelbrecht: On numerical modelling of solitary waves in lipid bilayers and complexity 5:30 – 5:55 Tanel Peets, Kert Tamm and Jüri Engelbrecht: On solitonic solutions of a Boussinesq-type equation modelling mechanical waves in biomembranes

3:50 - 5:55 SESSION 25, Room J: Nonlinear dynamics in Mathematical Biology and Neuroscience - Part V/V Chairs: Andrea Barreiro, Katie Newhall, Remus Osan, Pamela Pyzza 3:50 - 4:15 Andrea Barreiro: Constraining Neural Networks with Spiking Statistics 4:15 - 4:40 Joe Klobusicky: Effective Dynamics of Multiple Molecular Motors 4:40 - 5:05 Jay Newby: An artificial neural network approach to automated particle tracking analysis of 2D and 3D microscopy videos 5:05 - 5:30 John Fricks: Stochastic Modeling of Motor-driven DNA Origami 5:30 - 5:55 Gennady Cymbalyuk: Paw-shake as a transient response of a multi-functional central pattern generator

3:50 – 5:55 SESSION 6, Room F/G: Discrete integrable systems – Part III/III CHAIRS: TOSHIYUKI MANO AND NOBE ATSUSHI 3:50 – 4:15: Vladimir Dragovic: Discriminantly Separable Polynomials and Integrability 4:15 – 4:40: Shuhei Kamioka: Plane partitions and the discrete two-dimensional Toda molecule 4:40 – 5:05: Giorgio Gubbiotti: Darboux integrability of the trapezoidal H4 and H6 equations 5:05 – 5:30: Takeshi Morita: On a bilateral series solution of the Hahn-Exton q-Bessel type equation 5:30 – 5:55: Hajime Nagoya: Irregular conformal blocks and Painlevé tau functions

3:50 – 5:55 SESSION 9, Room E: Nonlinear Schrödinger Models and Applications – Part II/II CHAIRS: RICARDO CARRETERO AND PANOS KEVREKIDIS 3:50 – 4:15 Boris Malomed: Two-dimensional dipolar gap solitons in free space with spin-orbit coupling 4:15 – 4:40 Hidetsugu Sakaguchi: Vortex solitons in spin-orbit coupled Bose Einstein condensates 4:40 – 5:05 Jessica Taylor: Enhanced fractal dynamics of a BEC induced by dipolar interactions 5:05 – 5:30 Gershon Kurizki: Nonlinear quantum optics via highly nonlocal interactions

11 5:05 – 5:55 Ricardo Carretero: Vortex Rings in Bose-Einstein Condensates

3:50 – 5:55 SESSION 21, Room C: Nonlinear Waves: Mathematical Methods and Applications– Part II/II Chairs: Vassilis Rothos and Efstathios Charalampidis 3:50 – 4:15 V. Rothos: Second Order Maxwell-Bloch equation as an infinite dimensional dynamical system 4:15 – 4:40 Q. Wang, G. Biondini and M. Ablowitz: Whitham modulation theory for the Kadomtsev-Petviashvili and two dimensional Benjamin-Ono equations 4:40 – 5:05 G. Mylonas: Stability of gap solitons in the presence of a weak nonlocality

5:00 – 7:00 POSTERS: Hill Atrium, outside of Mahler • Isabelle Butterfield and John Carter: Comparisons Between Mathematical Models and Experiments of Waves on Deep Water • Hung Le: Second-order elliptic equations with Wentzel and transmission boundary conditions and applications • Daniel Ferguson, Katie Oliveras and Vishal Vasan: A new perspective on steady flow over bathymetry • Kyle Pounder, Robert Jenkins and Kenneth T.-R. McLaughlin: Asymptotics of the finite Toda lattice • Xin Yang, Bernard Deconinck and Tom Trogdon: Numerical inverse scattering for the sine-Gordon equation • Kelsey Dipietro and Alan E. Lindsay: Efficient Moving mesh simulation of fourth order PDES in 2D: Modeling of elastic-electrostatic deflections • Bernard Deconinck and Jeremy Upsal: On the integrability of long and short wave interaction models • Timothy Ferguson: Volume Bounds for the Synchronization Region in the Kuramoto Model

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

FRIDAY, MARCH 31, 2017 7:30 – 9:30 REGISTRATION 8:00 – 9:00 KEYNOTE LECTURE 3, Mahler Nathan Kutz: Data-driven Discovery of Nonlinear Wave Equations Chair: Alex Himonas

12 9:10 – 10:00 SESSION 15, Mahler: Asymptotics and Applied Analysis – Part V/VI CHAIRS: ROBERT BUCKINGHAM AND PETER MILLER 9:10 – 9:35 Ramon Plaza: Spectral and nonlinear stability of traveling fronts for a hyperbolic Allen-Cahn model with relaxation 9:35 – 10:00 Deniz Bilman: Numerical inverse scattering for the Toda lattice

9:10 – 10:00 SESSION 18, Room Q: Nonlinear waves, dynamics of singularities, and turbulence in hydrodynamics, physical, and biological systems – Part IV/V CHAIRS: ALEXANDER KOROTKEVICH AND PAVEL LUSHNIKOV 9:10 – 9:35 Gregor Kovacic: Nonlinear Schroedinger and Maxwell-Bloch systems with non-zero boundary conditions 9:35 – 10:00 Katie Newhall: Metastability of the Nonlinear Wave Equation

9:10 – 10:00 SESSION 16, Room R: Nonlocal and full-dispersion model equations in in fluid mechanics – Part III/IV CHAIRS: HENRIK KALISCH AND JOHN CARTER 9:10 – 9:35 Mark Groves: Solitary wave solutions to the full dispersion Kadomtsev-Petviashvili equation 9:35 – 10:00 Rosa Maria Vargas Magana, Antonmaria Minzoni and Panayotis Panayotaros: Whitham-Boussinesq model for variable depth topography

9:10 – 10:00 SESSION 5, Room J: Solitary and Rogue Waves as Solutions of Generalized Schrodinger Equations. Achievements and Challenges – Part II/III CHAIRS: MICHAIL TODOROV 9:10 – 9:35 Andrei Ludu: Rotating Hollow Patterns in Fluids 9:35 – 10:00 Alexey Sukhinin: Self-focusing and Spatio-Temporal Dynamics of Nonresonant co-Filaments in air

9:10 – 10:00 SESSION 23, Room F/G: Waves and Instabilities in Vlasov plasmas – Part I/IV CHAIR: BRAD SHADWICK 9:10 – 9:35 Jacob Bedrossian: Nonlinear echoes and landau damping 9:35 – 10:00 Maxime Perin: Hamiltonian fluid reductions of kinetic equations in plasma physics

9:10 – 10:00 SESSION 12, Room E: Wave Phenomena in Combustion – Part I/II CHAIRS: ANNA GHAZARYAN AND STEPHANE LAFORTUNE 9:10 – 9:35 Peter Gordon: Gelfand type problem for laminar co-flow jets 9:35 – 10:00 Gregory Lyng: Stability of viscous detonation waves

9:10 – 10:00 SESSION 8, Room C: Analytical and computational methods to study nonlinear partial differential equations – Part I/II CHAIRS: UNAL GOKTAS AND MUHAMMAD USMAN 9:10 – 9:35 Amer Rasheed: Numerical analysis of an anisotropic phase-field model under the action of magnetic-field 9:35 – 10:00 Iftikhar Ahmad, Hira Ilyas: Homotopy Perturbation Method for solution of nonlinear partial differential equation in MHD Jeffery-Hamel flows

10:00-10:30 COFFEE BREAK

10:30 – 12:10 SESSION 15, Mahler: Asymptotics and Applied Analysis-Part VI/VI CHAIRS: ROBERT BUCKINGHAM AND PETER MILLER 10:30 – 10:55 Tom Trogdon: Universality for eigenvalue algorithms

13 10:55 – 11:20 Michael Music: Semiclassical analysis for a 2D completely integrable equation 11:20 – 11:45 Virgil Pierce: Dispersionless limits of DKP equations for continuum limits of the Pfaff lattice equations 11:45 – 12:10 Robert Jenkins: Global Existence and Asymptotics for the Derivative Non-Linear Schrodinger Equation in One Dimension: Part II

10:30 – 12:10 SESSION 14, Room Q: Integrable systems and the geometry of curves and surfaces – Part II/II CHAIRS: ANNALISA CALINI, BAOFENG FENG, KENJI KAJIWARA 10:30 – 10:55 Lynn Heller: Constrained Willmore Minimizers 10:55 – 11:20 Wai Yeung Lam: Minimal surfaces from deformations of circle patterns 11:20 – 11:45 Masashi Yasumoto: Discrete timelike minimal surfaces and discrete wave equations 11:45 – 12:10 Shimpei Kobayashi and Nozomu Matsuura: A construction method for discrete indefinite affine spheres

10:30 – 12:10 SESSION 16, Room R: Nonlocal and full-dispersion model equations in in fluid mechanics– Part IV/IV Chairs: Henrik Kalisch and John Carter 10:30 – 10:55 Chris Curtis and Henrik Kalisch: Surface Waves over Point-Vortices 10:55 – 11:20 Dmitry Pelinovsky: Spectral stability of periodic waves in the generalized reduced Ostrovsky equation 11:20 – 11:45 Jean-Claude Saut: The Cauchy problem for the fractionary kadomtsev-Petvishvili equations 11:45 – 12:10 Mathew Johnson: Oscillation Estimates for Eigenfunctions for Fractional Schrodinger Operators

10:30 – 12:10 SESSION 08, Room J: Analytical and computational methods to study nonlinear partial differential equations – Part I/II CHAIRS: UNAL GOKTAS AND MUHAMMAD USMAN 10:30 – 10:55 Felipe Pereira , Arunasalam Rahunanthan: A GPU Implementation of Central Schemes for Two-Phase Flows 10:55 – 11:20 I. Naeem: Conservation Laws and Exact Solutions of Generalized Kompaneets and Nizhnik-Novikov-Veselov Equations 11:20 – 11:45 Rehana Naz : Nonlocal conservation laws of boundary layer equations on the Surface of a Sphere 11:45 – 12:10 Muhammad Usman, Chi Zhang, Youssef Raffoul, Mudassar Imran: A Study of Bifurcation Parameters in Travelling Wave Solutions of a Damped Forced Korteweg de Vries-Kuramoto Sivashinsky Type Equation

10:30 – 12:10 SESSION 23, Room F/G, Waves and Instabilities in Vlasov plasmas -Part II/IV CHAIR: ANTOINE CERFON 10:30 – 10:55 Zhiwu Lin: The existence of stable BGK waves 10:55 – 11:20 Bedros Afeyan: KEEN and KEEPN waves in Vlasov plasmas 11:20 – 11:45 Bradley Shadwick: Large amplitude plasma waves for particle acceleration 11:45 – 12:10 Carl Schroeder: Properties of nonlinear electron plasma waves driven by intense lasers

10:30 – 12:10 SESSION 13, Room E: Nonlinear wave phenomena in continuum physics: some recent f indings – Part II/II CHAIR: PEDRO M. JORDAN 10:30 – 10:55 J. Alberto Conejero, Carlos Lizama, Marina Murillo-Arcila: On the existence of linear chaos

14 for the viscous van Wijngaarden–Eringen equation 10:55 – 11:20 Sanichiro Yoshida: Deformation wave theory 11:20 – 11:45 Markus Muhr, Vanja Nikolić, Barbara Wohlmuth, Linus Wunderlich: High intensity ultrasound focusing by isogeometric shape optimization 11:45 – 12:10 Pedro M. Jordan: Modeling particle-laden and poroacoustic flow phenomena via the generalized continua approach

10:30 – 12:10 SESSION 12, Room C: Wave Phenomena in Combustion - Part II/II CHAIRS: ANNA GHAZARYAN AND STEPHANE LAFORTUNE 10:30 – 10:55 Jeffrey Humpherys: Viscous detonations in the reactive Navier-Stokes equations 10:55 – 11:20 Yuri Latushkin: Stability of one-dimensional and multi-dimensional fronts in exponentially weighted norms 11:20 – 11:45 Anna Ghazaryan: Stability of wavefronts in a diffusive model for porous media combustion

12:10 – 1:40 Lunch on your own

1:40 – 3:20 SESSION 18, Mahler: Nonlinear waves, dynamics of singularities, and turbulence in hydrodynamics, physical, and biological systems – Part V/V CHAIRS: ALEXANDER KOROTKEVICH AND PAVEL LUSHNIKOV 1:40 – 2:05 Taras Lakoba: Unconditional) numerical instability of the split-step method in simulations of the soliton of the nonlinear Dirac equations 2:05 – 2:30 Katelyn Leisman: The Relatively Small Effective Nonlinearity of the Nonlinear Schrödinger Equation 2:30 – 2:55 Alexander Korotkevich: Simulation of gas transportation networks. Comparison of dynamic and adiabatic approaches 1:40 – 3:20 SESSION 5, Room Q: Solitary and Rogue Waves as Solutions of Generalized Schrodinger Equations. Achievements and Challenges - Part III/III CHAIRS: ANDREI LUDU 1:40 – 2:05 Michail Todorov and Rossen Ivanov: System of Coupled Nonlinear Schrodinger Equations with Different Cross-Modulation Rates 2:05 – 2:30 Gary M. Webb: Magnetohydrodynamic Gauge Field Symmetries and Conservation Laws 2:30 – 2:55 Jose Escorcia and Erwin Suazo: Blow-up results and soliton solutions for a generalized variable coefficient nonlinear Schrodinger equation

1:40 – 3:20 SESSION 22, Room R: Nonlinear evolution equations of quantum physics and their topological solutions, Part II/II CHAIRS: S. GUSTAFSON, I.M. SIGAL, A. SOFFER 1:40 – 2:05 Thomas Chen: Fluctuation dynamics around Bose-Einstein condensates 2:05 – 2:30 Miguel Arturo Ballesteros Montero: Indirect Measurements and Quantum Trajectories 2:30 – 2:55 Gregory Eskin: Gravitational analog of the Aharonov-Bohm effect 2:55 – 3:20 Gennady El, Marco Bertola and Alexander Tovbis: Finite gap (multiphase) solutions of the focusing 1D NLS equation and large amplitude (rogue) wave

1:40 – 3:20 SESSION 23, Room F/G: Waves and Instabilities in Vlasov plasmas - Part III/IV CHAIR: BEDROS AFEYAN 1:40 – 2:05 Antoine Cerfon: Sparse grids for PIC simulations of kinetic plasmas

15 2:05 – 2:30 Jeffrey Hittinger: Simulation of longitudinal and transverse instability of ion acoustic waves using the grid based continuum code LOKI 2:30 – 2:55 Guangye Chen: implicit, charge and energy conserving particle-in-cell multidimensional algorithms for low-frequency plasma kinetic simulations in curvilinear geometries 2:55 – 3:20 Jason Tenbarge: An Eulerian discontinuous Galerkin scheme for the fully kinetic Vlasov-Maxwell system

3:20- 3:50 COFFEE BREAK

3:50 – 5:55 SESSION 23, Room F/G: Waves and Instabilities in Vlasov plasmas – Part IV/IV CHAIR: BEDROS AFEYAN 3:50 – 4:15 Richard Sydora: Fourier-Vlasov simulations in non-inertial reference frames and nonlinear evolution of electromagnetic Cyclotorn waves

END

ABSTRACTS for KEYNOTES

Singular asymptotics for nonlinear waves

Gino Biondini State University of New York at Buﬀalo, Department of Mathematics biondini@buﬀalo.edu

Abstract I will discuss three classical problems in the theory of nonlinear waves, each involving a certain singular asymptotic limit. 1. Modulational instability (MI), namely the instability of a constant background to long- wavelength perturbations, is a ubiquitous nonlinear phenomenon discovered in the mid 1960’s. Until recently, however, a characterization of the nonlinear stage of MI of localized perturbations of the background was still missing. I will first show how MI is manifested in the inverse scattering transform for the focusing nonlinear Schrodinger (NLS) equation. Then I will characterize the nonlinear stage of MI by computing the long-time asymptotics of the NLS equation for localized perturbations of a constant background. For long times, the xt-plane divides into three regions: a left far field and a right far field, in which the solution is approximately constant, and a central region in which the solution is described by a slowly modulated traveling wave. Finally, I will show that this kind of behavior is not limited to the NLS equation, but it is shared among many different models (including PDEs, nonlocal systems and differential-difference equations). 2. As is well known, in 1965 Zabusky and Kruskal (ZK) performed numerical simulations of the Korteweg-deVries (KdV) equation with small dispersion and cosine initial data. The breakup of the initial pulse generated eight solitary waves interacting elastically, which they called solitons. Soon after, Zabusky and others went on to invent the inverse scattering transform, giving birth to the modern theory of integrable systems. Fifty years later, however, a precise analytical description of the ZK simulations was surprisingly still missing. I will show how a careful use of the WKB method in the scattering problem for the KdV equation allows one to completely characterize the problem and obtain explicit expressions for the number, amplitude and speeds of the solitons emerging in any given situation. The theoretical results, which were generalized to other initial conditions and integrable PDEs, are corroborated by shallow water experiments which fully reproduced experimentally for the first time the ZK simulations, including soliton recurrence. 3. In 1965, Whitham formulated his eponymous modulation theory for the KdV equation, which allows one to study the small-dispersion limit by deriving a set of hyperbolic PDEs describing the modulation of the parameters of the traveling-wave solutions of KdV. Whitham modulation theory was subsequently generalized and applied with success in a variety of settings. Most studies, however, have been limited to PDEs in one spatial dimension. I will show how a (2+1)-dimensional generalization of Whitham modulation theory to derive the genus-1 Whitham modulation equations for the Kadomtsev-Petviashvili (KP) equation. I will discuss some basic properties of the resulting KP-Whitham system and I will show how the system can be used to study the stability of the genus-1 solutions of KP. A similar approach was also successfully used for other (2+1)-dimensional nonlinear PDEs. A hierarchy of nonlinear models for tumor growth and clinical applications. Thierry Colin and Olivier Saut Institut de Mathématiques de Bordeaux and INRIA Bordeaux sud-ouest Université de Bordeaux [email protected], [email protected]

A huge number of mathematical/numerical models of tumor growth are available in the literature. Most of them aim at integrating an increasing amount of biological/medical knowledges. These models are able to account at least qualitatively for several complex phenomena (angiogenesis, influence of particular molecular pathway, effects of targeted therapies, ...). They could be useful for clinical applications in order to help to understand the evolution of the disease or the response to the treatment in a personalized clinical context. The challenge is therefore to be able to obtain a parametrization of the models with the available data. If we restrict our self to a clinical context the information is scarce. It consists mainly in the nature of the cancer that is known thanks to biological exams (blood samples, biopsies) and also to imaging data (CT-scans, MRI, PET-scans). The model has therefore to be designed according to the nature of the cancer, its localization but also according to the available imaging data. The images will give information on the volume, but also on the shape and the metabolism of the tumor (thanks to functional imaging technics like perfusion MRI or CT-scans). Moreover, for a particular patient, we often have several successive exams at different times. We therefore have to solve a complex inverse problem in order to be able to give a forecast of the progression of the disease or of the answer to a treatment.

In this talk, we will present several examples of such inverse problems in clinical contexts. These inverse problems will be considered on a hierarchy of models with increasing complexity and with more and more complex data. The models will all have a PDE part that accounts for the spatial variations that are observed on the image. The complexity of the PDE system will correspond to the quantity of information that we are able to extract from the images. The first example will be the analysis of the natural growth of meningioma. Meningioma are intra-cranial tumors that usually grow slowly. The gold standard of care, if there is no symptom, is the follow-up with MRI and then neurosurgery. The forecast of the growth is of course of first interest for the neurosurgeon. We will present a model and a strategy that have been validated on a cohort of 45 patients. Most of cancers will lead to lung metastases. For slowly varying disease with isolated metastases a loco-regional technic may be use (stereotaxic radiotherapy, radiofrequency ablation) and in this case the physician may have to understand what could be the best moment for starting this therapy. Will present an evaluation of such a forecast on a set of 70 patients. Our last example will concern metastatic liver cancer treated with anti-angiogenic drugs for which not only the shape of the tumor is important but also its structure and we will show how a mathematical model can help to anticipate the relapse to the treatment. Our collaborators in this project are: The Institut Bergonié in Bordeaux: Dr. Xavier Buy, Guy Kantor, Michèle Kind, Jean Palussière. CHU of Bordeaux (Bordeaux University Hospital): Pr. Hugues Loiseau, Pr, Alain Ravaud, Dr. François Cornelis This study has been carried out within the frame of the LABEX TRAIL, ANR-10-LABX-0057 with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the “Investments for the future” Programme IdEx (ANR-10-IDEX-03- 02).

Data-driven Discovery of Nonlinear Wave Equations

J. Nathan Kutz Department of Applied Mathematics, University of Washington [email protected]

Abstract

We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity promoting techniques to select the nonlinear and partial derivative terms terms of the governing equations that most accurately represent the data, bypassing a combinatorially large search through all possible candidate models. The method balances model complexity and regression accuracy by selecting a parsimonious model via Pareto analysis. Time series measurements can be made in an Eulerian framework where the sensors are fixed spatially, or in a Lagrangian framework where the sensors move with the dynamics. The method is computationally efficient, robust, and demonstrated to work on a variety of canonical problems of mathematical physics including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation. Moreover, the method is capable of disambiguating between potentially non-unique dynamical terms by using multiple time series taken with different initial data. Thus for a traveling wave, the method can distinguish between a linear wave equation or the Korteweg-deVries equation, for instance. The method provides a promising new technique for discovering governing equations and physical laws in parametrized spatio-temporal systems where first-principles derivations are intractable.

ABSTRACTS for SESSIONS

SESSION 1

Nonlinear Waves

Jerry Bona Department of Mathematics, Statistics and Computer Science The University of Illinois at Chicago Chicago, Illinois 60607-7045 email: [email protected]

Min Chen Department of Mathematics Purdue University West Lafayette, IN. 47907 email: [email protected]

Shenghao Li Department of Mathematics Purdue University West Lafayette, IN. 47907

Dimitrios Mitsotakis School of Mathematics and Statistics Victoria University of Wellington Wellington 6140, NEW ZEALAND

ABSTRACT

This session will be centered around the propagation of waves in water and other media where nonlinearity, dispersion and sometimes dissipation and capillarity are all acting. Fea- tured in the session will be theoretical work, such as existence of multi-humped solutions and existence of two dimensional standing waves, wellposedness of dispersion-managed nonlinear Schoödingersystems, higher order model equations and equations with a nonlocal viscous dispersive term, and numerical investigation on singular solutions of Boussinesq systems and equations with nonlocal terms. The results on shallow water waves on a vertical shear flow and norm-inflation results for the BBM equation will also be presented. A water wave model with a nonlocal viscous dispersive term

Olivier Goubet∗ LAMFA CNRS UMR 7352, Universit´ede Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens Cedex, France [email protected]

Imen Manoubi Unit´ede recherche: Multifractales et Ondelettes, Facult´edes Sciences de Monastir, Av. de l’environnement, 5000 Monastir, Tunisie [email protected]

Abstract

We study the water wave model with a nonlocal viscous term √ ν ∂ Z t u(s) ut + ux + βuxxx + √ √ ds + uux = νuxx, π ∂t 0 t − s

√1 ∂ R t √u(s) where π ∂t 0 t−s ds is the Riemann-Liouville half-order derivative. Here x belongs to IR and ν > 0, β are parameters. We study the initial value problem and the decay rate of solutions to the equilibrium. We follow here [1, 2].

References

[1] I. Manoubi, Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative, DCDS serie B, 19, (2014), n 9, 28372863.

[2] O. Goubet and I. Manoubi, Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diﬀusive approach, to appear in Advances in Nonlinear Analysis. Shallow Water Waves on a Vertical Shear Flow

Maria Bjørnestad and Henrik Kalisch University of Bergen, Department of Mathematics PO Box 7800, 5020 Bergen, Norway [email protected] and [email protected]

Abstract

Water waves in shallow water propagating towards a sloping beach are being investigated. The classical shallow-water equations can be used to obtain an idea of the shoaling processes and runup of long waves approaching a beach. In particular, Carrier and Greenspan obtained explicit solutions to the nonlinear shallow water equations on a linear beach proﬁle, including the development of the waterline [2]. They used a hodograph transform which is a eﬃcient tool for the resolution of systems of conservation laws in the case without forcing. The real novelty of the work of Carrier and Greenspan lay in the fact that they succeeded in applying the hodograph transform in the case of a non-uniform environment, i.e. a bottom forcing.

In coastal areas, the propagation of water waves is often affected by the influence of currents. A first-order approximation of a background current can be obtained by using a linear shear current, such as in [3, 1]. In the current work, we show how the shallow-water equations with a background shear can be solved using a hodograph transformation.

References

[1] A. Ali and H. Kalisch, Reconstruction of the pressure in long-wave models with constant vorticity, Eur. J. Mech. B Fluids 37 (2013), 187–194.

[2] G.F. Carrier and H.P. Greenspan, Water waves of ﬁnite amplitude on a sloping beach, J Fluid Mech 4 (1958), 97–109.

[3] A.F. Teles da Silva and D.H. Peregrine, Steep, steady surface waves on water of ﬁnite depth with constant vorticity, J. Fluid Mech. 195 (1988), 281–302. Standing waves for two-dimensional Boussinesq systems

Shenghao Li and Min Chen Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA [email protected] and [email protected]

Abstract

We consider the abcd-system

( 3 ηt + ∇u + ∇(uη) + a∇ u − b∆ηt = 0, 1 2 3 ut + ∇η + 2 ∇|u| + c∇ η − d∆ut = 0, which was introduced by Bona, Chen and Saut [1] for small-amplitude and long wavelength gravity waves. The parameters a, b, c and d are not independently speciﬁable but satisfy certain physical relevant conditions.

In this paper, we prove existence of a large family of nontrivial bifurcation standing waves for some of the two-dimensional abcd-system. Our proof uses the Lyapunov-Schmidt method to ﬁnd the bifurcation equation. It has been applied to show the one for the one-dimensional model and the traveling waves for two-dimensional model by Chen and Iooss [2, 3].

References

[1] Bona, J. L., Chen, M., and Saut, J. C. (2002). Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory. Journal of Nonlinear Science, 12(4), 283-318.

[2] Chen, M., and Iooss, G. (2005). Standing waves for a two-way model system for water waves. European Journal of Mechanics-B/Fluids, 24(1), 113-124.

[3] Chen, M., and Iooss, G. (2006). Periodic wave patterns of two-dimensional Boussinesq systems. European Journal of Mechanics-B/Fluids, 25(4), 393-405. Existence of multi-hump capillary-gravity waves on water of ﬁnite depth

Shu-Ming Sun Department of Mathematics Virginia Tech Blacksburg, VA 24061 email: [email protected]

Abstract

The talk considers the existence of multi-hump waves with oscillations at infinity on a layer of fluid with finite depth. The fluid is assumed to be incompressible and inviscid with a constant density and the flow is irrotational. The wave is moving with a constant speed on the free surface of the fluid under the influence of gravity and surface tension. If the surface tension is small and the wave speed is near its critical value, it is known that the exact Euler equations have solitary-wave solutions of elevation with small ripples at infinity, called generalized solitary waves. In this talk, it will be shown that under such conditions, the exact Euler equations will have two-hump solutions (i.e., two-solitary-wave solutions) of elevation with small ripples at infinity. The amplitude of the ripples at infinity is algebraically small comparing with the inverse of the wave-length for the part of one solitary wave. The basic idea to prove such existence is to patch two appropriate generalized solitary-wave solutions together using some free parameters. The similar idea works for the existence of multi-hump solutions. (This is a joint work with S. Deng). SINGULAR SOLUTIONS OF A BOUSSINESQ SYSTEM FOR WATER WAVES

JERRY L. BONA1 AND MIN CHEN2 1Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago Chicago, IL 60607, USA 2 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Abstract. Studied here is the Boussinesq system

ηt + ux + (ηu)x + auxxx − bηxxt = 0, 1 u + η + (u2) + cη − du = 0, t x 2 x xxx xxt of partial diﬀerential equations. This system has been used in theory and practice as a model for small-amplitude, long-crested water waves. The issue addressed is whether or not the initial-value problem for this system of equations is globally well posed. The investigation proceeds by way of numerical simulations using a computer code based on a a semi-implicit, pseudo-spectral code. It turns out that larger amplitudes or velocities do seem to lead to singularity formation in ﬁnite time, indicating that the problem is not globally well posed.

1 Higher-order, unidirectional models for surface water waves

Jerry Bona∗ Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago [email protected]

Abstract

Discussed will be a class of higher-order models for the one-way propagation of long crested water waves. Such models have appeared often in the literature, but it has proven diﬃcult to provide global well-posedness results for them in the same way as is done for the lower- order KdV- and BBM-type models. Within this class, one discerns a special subclass of Hamiltonian models. It is shown that these do indeed possess the desired well-posedness theory.

The report is based on joint work with Xavier Carvajal, Mahendra Panthee and Marcia Scialomi. Well-posedness of the dispersion-managed nonlinear Schr¨odinger equation and related equations

John Albert∗ Department of Mathematics, University of Oklahoma, Norman, OK 73019 [email protected]

Abstract

The dispersion-managed nonlinear Schrodinger equation is a model equation for optical pulses in a fiber in which the dispersive properties vary rapidly in the space variable. It provides an interesting illustration of the extent to which even very weak dispersion can significantly affect the propagation of nonlinear waves. We give an overview of what is known about this equation and other related equations, including an account of the well-posedness theory. Norm inflation phenomena for the BBM equation

Jerry Bona and Mimi Dai Department of Mathematics, Stat. and Comp. Sci., University of Illinois Chicago, Chicago, IL 60607,USA [email protected] and [email protected]

Abstract

Considered here is the periodic initial-value problem for the regularized long-wave (BBM) equation ut + ux + uux − uxxt = 0. Adding to previous work in the literature, it is shown here that for any s < 0, there is smooth s initial data that is small in the L2-based Sobolev spaces H , but the solution emanating from it becomes arbitrarily large in arbitrarily small time. This so called norm inﬂation result has as a consequence the previously determined conclusion that this problem is ill-posed in these negative-norm spaces. SESSION 2

Painlev´eEquations, Integrable Systems, and Random Matrices

Anton Dzhamay School of Mathematical Sciences, University of Northern Colorado, Greeley, CO 80639, USA [email protected]

Christopher M. Ormerod Department of Mathematics, University of Maine, Orono, ME 04469, USA [email protected]

Virgil U. Pierce School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, TX 78539, USA [email protected]

Abstract

Modern theory of integrable systems is using tools from diverse areas of mathematics and mathematical physics to perform both qualitative and quantitative analysis of a wide range of important natural phenomena, including, but not limited to, the theory of nonlinear waves. It is also remarkable for uncovering deep and unexpected connections between diﬀerent areas of mathematics.

In this special session we plan to mainly focus on connections and interactions between the theory of Integrable Systems, Random Matrices, Painlev´eEquations, and the theory of Orthogonal Polynomials.

Among important examples described in terms of continous and discrete Painlevétran- scendenta are correlation functions and probability distributions for various random matrix models and determinantal point processes. A large number of such examples belong to an emerging and rapidly developing field of Integrable Probability that studies probabilistic systems, such as random growth, tilings, and percolation, that can be analyzed by analytic methods. One of the most powerful analytic tools in studying such models is the Riemann- Hilbert Problem approach, and solutions of associated Riemann-Hilbert problems can be given in terms of different families of orthogonal polynomial.

We expect that talks in our session will give a broad overview of this research area, highlight important recent developments, and outline possible new research directions. Discontinuity in the asymptotic behavior of planar orthogonal polynomials under a perturbation of the Gaussian weight

Seung-Yeop Lee and Meng Yang 4202 East Fowler Ave, CMC342, Tampa, FL 33620-5700 [email protected] and [email protected]

Abstract

We consider the orthogonal polynomials, {Pn(z)}n=0,1,···, with respect to the measure

|z − a|2ce−N|z|2 dA(z) supported over the whole complex plane, where a > 0, N > 0 and c ∈ (−1/2, ∞) \{0}. We look at the scaling limit where n and N tend to inﬁnity while keeping their ratio, n/N, ﬁxed. The support of the limiting zero distribution is given in terms of certain “limiting potential-theoretic skeleton” of the disk. We show that both the skeleton and the asymptotic distribution of the zeros behave discontinuously at c = 0. The smooth interpolation of the discontinuity is obtained by the further scaling of c = e−ηN in terms of a parameter η ∈ [0, ∞).

References Rational Solutions of Painlev´eEquations

Peter D. Miller Department of Mathematics, University of Michigan East Hall, 530 Church St., Ann Arbor, MI 48109 [email protected]

Abstract

All of the famous six Painlevéequations except the first one admit, for certain integral combinations of parameters in the equation, solutions that are rational functions. The degree of the rational solution relates to the parameter values, and can become large if the parameters are so. The poles and zeros of the rational solutions are distributed in fascinating patterns when the parameters are large. Rational solutions of Painlevéequations of large degree also arise in applications ranging from universality in solutions of integrable wave equations to electrochemistry, eddy dynamics in fluids, and string theory.

In this talk I will describe the large degree asymptotics of rational solutions of the Painlevé-II equation. These results can be obtained in at least three different ways: by an approach based on the Jimbo-Miwa Lax pair (joint work with R. Buckingham [1, 2]), by an approach based on Hankel determinants of complex-orthogonal polynomials (work by M. Bertola and T. Bothner [3]), and by an approach based on the Flaschka-Newell Lax pair (joint work with Y. Sheng [4]). Then I will describe some more recent results on the asymptotic behavior of rational solutions of the Painlevé-III equation (joint with T. Bothner and Y. Sheng [5]).

References

[1] R. Buckingham and P. D. Miller, Large-degree asymptotics of rational Painlev´e-II functions. Noncritical behavior, Nonlinearity, 27 (2014), 2489-2577.

[2] R. Buckingham and P. D. Miller, Large-degree asymptotics of rational Painlev´e-II functions. Critical behavior, Nonlinearity, 28 (2015), 1539-1596.

[3] M. Bertola and T. Bothner, Zeros of large degree Vorob’ev-Yablonski polynomials via a Hankel determinant identity, Int. Math. Research Notices, 19 (2015) 9330-9399.

[4] P. D. Miller and Y. Sheng, Rational Painlev´e-II solutions revisited, (2017). In preparation.

[5] T. Bothner, P. D. Miller, and Y. Sheng, Rational solutions of the Painlev´e-III equation, (2017). In preparation. Eigenvalue densities for the Hermitian two-matrix model

Megan McCormick Stone 617 N Santa Rita Ave Tucson, AZ 85721 [email protected]

Abstract

The Hermitian two-matrix model consists of the space H(N) × H(N) of pairs of Hermi- tian N × N matrices equipped with a prescribed probability distribution. This probability distribution includes an interaction term between the two matrices, making it diﬃcult to directly apply strategies that were successful in characterizing the asymptotic behavior of the Hermitian one-matrix model.

The interaction term in the two-matrix model gives rise to the Harish-Chandra-Itzykson- Zuber (HCIZ) integral. A formula for the HCIZ integral found in 2012 by Goulden, Guay- Paquet, and Novak connects the HCIZ integral to monotone Hurwitz numbers, which count a speciﬁc class of ramiﬁed coverings of the sphere [1]. Using the leading order behavior of this formula, the limiting distribution of eigenvalue pairs in the two-matrix model can be characterized under certain assumptions on the potentials and the coupling constant used for the two-matrix model. In this talk, I will explain these assumptions, and explain why they are reasonable assumptions to make for the two-matrix model.

References

[1] I.P. Goulden, M. Guay-Paquet, and J. Novak, Monotone Hurwitz numbers and the HCIZ integral. (2012) arXiv:1107.1015v3 [math.CO]. Geometry of scattering theory for orthogonal polynomials

Dylan Murphy 617 N. Santa Rita Ave. Tucson, AZ 85721-0089 [email protected]

Abstract

In the late 1970s, Geronimo and Case demonstrated a theory of forward and inverse scattering for orthogonal polynomials on the unit circle and on the real line. This theory includes analogues of the usual objects from scattering theory for Schrödingeroperators on the line, such as Jost functions and Gelfand-Levitan-Marchenko equations for inverse scattering. In this setting, the role of the potential is played by the sequence(s) of coefficients for the recurrence relation which generates the polynomials. Similarly, the machinery of Floquet theory for Schrödingeroperators with periodic potential, which associates to the potential an algebraic curve and some geometric data on that curve, has an analogue in the world of orthogonal polynomials.

In this presentation we describe some current progress in using a long-period limit of Flo- quet theory to construct versions of these algebrogeometric data which can be associated to scattering quantities. The result is a “curve” of continuum genus with continuum versions of the divisor and theta function which moderate the inverse spectral problem in the periodic case.

References

[1] N. Ercolani and H.P. McKean. Geometry of KdV(4): Abel sums, Jacobi variety, and theta function in the scattering case. Inventiones mathematicae 99 (Springer-Verlag, 1990).

[2] D. Mumford and P. van Moerbeke. The spectrum of diﬀerence operators and algebraic curves. Acta Mathematica 143(1) (Springer-Verlag, 1979).

[3] J. S. Geronimo and K. M. Case. Scattering theory and polynomials orthogonal on the unit circle. Journal of Mathematical Physics 20(2) (American Institute of Physics, 1979).

[4] B. Simon. Orthogonal polynomials on the unit circle. American Mathematical Society Colloquium Publications, 2004. Semiclassical dynamics of the three-wave resonant interaction equations

Robert Buckingham∗ University of Cincinnati [email protected]

Robert Jenkins University of Arizona [email protected]

Peter Miller University of Michigan, Ann Arbor [email protected]

Abstract

The three-wave resonant interaction equations describe the time evolution of the complex amplitudes of three resonant wave modes. We analyze the collision of two or three packets in the semiclassical limit by applying the inverse-scattering transform. Using WKB analysis, we construct an associated semiclassical soliton ensemble, a family of reflectionless solutions intended to accurately approximate the initial data in the semiclassical limit. Plots of the soliton ensembles indicate the space-time plane is partitioned into regions containing either quiescent, slowly varying, or rapidly oscillating waves. This behavior resembles the well- known generation of dispersive shock waves in equations such as the Korteweg-de Vries and nonlinear Schrödingerequations, although the physical mechanism must be different as the system is non-dispersive.

References

[1] R. Buckingham, R. Jenkins, and P. Miller, Semiclassical soliton ensembles for the three- wave resonant interaction equations, arXiv:1609.05416 (2016). The birth and death of a random matrix

Sevak Mkrtchyan∗ Department of Mathematics, University of Rochester, 500 Joseph C. Wilson Blvd., Rochester, NY 14627 [email protected]

Abstract

In the thermodynamic limit of the lozenge tiling model the frozen boundary develops special points where the liquid region meets with two different frozen regions. These are called turning points. In a paper titled ”The birth of a random matrix” it was conjectured by Okounkov and Reshetikhin [1] that in the scaling limit of the model the local point process near turning points should converge to the GUE corner process. We will discuss a result establishing the GUE corner process when the underlying measure is the homogeneous q to the volume” measure. We’ll also see how this process is modified when weights are not homogeneous anymore. The modified process does not correspond to a random matrix model anymore. A portion of the results presented is based on joint work with L.Petrov.

References

[1] A. Okounkov and N. Reshetikhin, The birth of a random matrix, Mosc. Math. J., 6(3), (2006), 553 – 566. Three facets of the theta divisor associated with the autonomous Garnier system of type 9/2

Akane Nakamura Department of Mathematics, Faculty of Science, Josai University 1-1 Keyakidai, Sakado, Saitama 350-0295, Japan [email protected]

Abstract

Our starting point is a classification of 4-dimensional Painlevé-type equations from isomonodromic point of view [6, 3, 2]. In a previous work [4], we considered the 40 types of autonomous 4-dimensional Painlevé-type equations and studied the Namikawa-Ueno-type degenerations [5] of their genus two spectral curves. In this talk, we treat the autonomous 9 Garnier system of type 2 , one of the most degenerated system, as an example to illustrate two other important curves associated with the system: the Painlevédivisor[1] and the separation curve[7]. All three curves can be considered as the theta divisor of the Liouville tori, which in turn can be considered as the Jacobian of these curves. It might be helpful to consider the Namikawa-Ueno-type degenerations of these curves, for identifying or distinguishing integrable systems that do not initially come along with a Lax pair. Following Vanhaecke [7], we can also give a Lax pair using its separation curve.

References [1] M. Adler, P. van Moerbeke, P. Vanhaecke, Algebraic integrability, Painlevégeometry and Lie algebras, A Series of Modern Surveys in Mathematics, 47, 2004. [2] H. Kawakami, Four-dimensional Painlevé-type equations associated with ramified linear equations I: Matrix Painlevé systems, arXiv:1608.03927, II: Sasano systems, arXiv:1609.05263, III (in preparation). [3] H. Kawakami, A. Nakamura, H. Sakai, Degeneration scheme of 4-dimensional Painlevé- type equations, arXiv:1209.3836, 2012. [4] A. Nakamura, Autonomous limit of 4-dimensional Painlevé-type equations and degeneration of curves of genus two, arXiv:1505.00885, 2015. [5] Y. Namikawa, K. Ueno, The complete classification of fibres in pencils of curves of genus two, Manuscripta Math., 9, 1973, 143-186. [6] H. Sakai, Isomonodromic deformation and 4-dimensional Painlevé type equations, preprint, University of Tokyo, Graduate School of Mathematical Sciences, 2010. [7] P. Vanhaecke, Integrable systems in the realm of algebraic geometry, LNM, 1638, 2001. The generic quantum superintegrable system on the sphere and Racah operators

Plamen Iliev School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA [email protected]

Abstract

I will discuss the generic quantum superintegrable system on the d-sphere with potential Pd+1 bk V (y) = k=1 2 , where bk are parameters. This system has been extensively studied in yk the literature as an important example of a second-order superintegrable system, possessing (2d − 1) second-order algebraically independent symmetries.

Appropriately normalized, the symmetry operators for the Hamiltonian deﬁne a representation of the Kohno-Drinfeld Lie algebra on the space of polynomials orthogonal with respect to the multivariate beta distribution. The Gaudin subalgebras generated by Jucys-Murphy elements are diagonalized by families of Jacobi polynomials in d variables on the simplex.

I will define a set of generators for the symmetry algebra and prove that their action on the Jacobi polynomials is represented by the multivariable Racah operators introduced in a joint paper with Geronimo. The constructions also yield a new Lie-theoretic interpretation of the bispectral property for Tratnik’s multivariable Racah polynomials. Complex reflection groups, bi-flat F-manifolds and Painlevéequations

Alessandro Arsie∗ and Paolo Lorenzoni The University of Toledo, Toledo, Ohio, USA University of Milano-Bicocca, Milan, Italy [email protected] and [email protected]

Abstract

We show that bi-flat F-manifolds can be interpreted as natural geometrical structures en- coding the almost duality for Frobenius manifolds without metric. Using this, we extend Dubrovins duality between orbit spaces of Coxeter groups and Veselovs v-systems, to the orbit spaces of exceptional well-generated complex reflection groups of rank 2 and 3. We will also show how non-semisimple bi-flat F-manifolds in dimension 3 are parametrized by the full family of PainlevéV and PainlevéIV equations.

References

[1] A. Arsie and P. Lorenzoni, F-manifolds, multi-ﬂat structures and Painlev´etranscen- dents, submitted, https://arxiv.org/abs/1501.06435.

[2] A. Arsie ans P. Lorenzoni, Complex reﬂection groups, logarithmic connections and bi-ﬂat F-manifolds, submitted, https://arxiv.org/abs/1604.04446. SESSION 3

Evolution Equations and Integrable Systems

Alex A. Himonas Department of Mathematics, University of Notre Dame Notre Dame, IN 46556 email: [email protected]

Curtis Holliman Department of Mathematics, The Catholic University of America Aquinas Hall 116, Washington, DC 20064 email: [email protected]

Dionyssis Mantzavinos Department of Mathematics and Statistics, University of Massachusetts Amherst Amherst, MA 01003-9305 email: [email protected]

ABSTRACT

Linear and nonlinear evolution equations have been at the forefront of advances in partial diﬀerential equations for a long time. They are involved in beautiful, yet extremely challenging problems, with a strong physical background, for which progress is achieved through a mixture of techniques lying at the interface between analysis and integrable systems. Topics studied for these equations include, among others, local and global well-posedness, inverse scattering, stability, integrability and travelling waves. Linearization and exact solvability of the Burgers equation with time-dependent coeﬃcients and nonlinear forcing term

Axel Schulze-Halberg Department of Mathematics and Actuarial Science and Department of Physics Indiana University Northwest, 3400 Broadway, Gary IN 46408, USA [email protected]

Abstract

We construct and discuss a new linearization method [3] for solving the inhomogeneous Burgers equation with time-dependent coeﬃcients and a nonlinear forcing term. Our results are shown to contain and generalize former ﬁndings [1] [2]. We apply our method for the construction of solutions to several initial- and boundary-value problems involving the time-dependent Burgers equation with forcing terms of sinusoidal, polynomial, as well as exceptional orthogonal polynomial (X1-Laguerre) type.

References

[1] S. Buyukasik, O. Pashaev, Exact solutions of forced Burgers equations with time variable coeﬃcients, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 1635-1651

[2] P. Miskinis, New exact solutions of one-dimensional inhomogeneous Burgers equation, Rep. Math. Phys., 48 (2001), 175-181

[3] A. Schulze-Halberg, Burgers equation with time-dependent coeﬃcients and nonlinear forcing term: Linearization and exact solvability, Commun. Nonlinear Sci. Numer. Sim- ulat., 22 (2015), 1068-1083. Multi-component nonlinear waves in nonlinear Schr¨odinger (NLS) systems

E. G. Charalampidis∗ and P. G. Kevrekidis Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003-4515, USA [email protected] and [email protected]

D. J. Frantzeskakis Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece [email protected]

B. A. Malomed Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel [email protected]

P. E. Farrell Mathematical Institute, University of Oxford, Oxford, UK [email protected]

Abstract

In this talk, we will present a two-component NLS system in one and two spatial dimensions. The formation of bright solitonic bound states in the second component will be discussed and their bifurcation points will be identified by the underlying linear limit. This way, nonlinear states can be identified and their stability will be studied. Finally, we will discuss a deflated continuation approach for the numerical computation of states in NLS systems.

References

[1] E. G. Charalampidis, P. G. Kevrekidis, D. J. Frantzeskakis, and B. A. Malomed. Dark- bright solitons in coupled nonlinear Schr¨odinger equations with unequal dispersion coef- ﬁcients. Phys. Rev. E, 91, 012924, 2015.

[2] E. G. Charalampidis, P. G. Kevrekidis, D. J. Frantzeskakis, and B. A. Malomed. Vortex- soliton complexes in coupled nonlinear Schrödinger equations with unequal dispersion coefficients. Phys. Rev. E, 94, 022207, 2016. [3] E. G. Charalampidis, P. G. Kevrekidis, and P. E. Farrell. Computing stationary solutions of the two-dimensional Gross-Pitaevskii equation with Deflated continuation. arXiv:1612.08145 From Discrete Solitons to Discrete Breathers and to Lattice Traveling Waves: A Discussion of Spectral Stability and Associated Criteria

P.G. Kevrekidis Department of Mathematics and Statistics University of Massachusetts, Amherst, MA 01003 [email protected]

Keywords: Stability, DNLS, FPU, Klein-Gordon, Solitons, Breathers, Traveling Waves

Abstract

The aim of this talk is to give an overview of stability criteria as they apply to a variety of coherent structures on infinite dimensional lattice dynamical systems. We will start with solitary waves of the discrete nonlinear Schrodinger equation (DNLS), discussing both a stability classification from the anti-continuum (uncoupled site) lattice limit and the famous Vakhitov-Kolokolov (VK) criterion. We will then extend considerations to discrete breathers primarily in nonlinear Klein-Gordon lattices, and will show how a direct analogy to the stability of their periodic orbits exists in connection to DNLS. Moreover, we will discuss a recently put forth criterion for their spectral stability which is analogous to the VK criterion and “falls back” on it upon reduction to the DNLS case. Lastly, we will discuss some intriguing connections of the discrete breather problem with that of traveling waves in (chiefly Fermi-Pasta-Ulam type) lattices and will devise yet another spectral stability criterion in that case too which will once again be the proper analogue of the VK one for the lattice traveling waves. Decay Properties of Solutions to a 4-parameter Family of Wave Equations

Ryan C. Thompson Department of Mathematics University of North Georgia Dahlonega, GA [email protected]

Abstract

In this presentation, persistence properties of solutions are investigated for a 4-parameter family (k − abc equation) of evolution equations having (k + 1)-degree nonlinearities and containing as its integrable members the Camassa-Holm, the Degasperis-Procesi, Novikov and Fokas-Olver-Rosenau-Qiao equations. These properties will imply that strong solutions of the k − abc equation will decay at inﬁnity in the spatial variable provided that the initial data does. Furthermore, it is shown that the equation exhibits unique continuation for appropriate values of the parameters k, a, b, and c.

References

[1] A. Himonas, D. Mantzavinos, The Cauchy problem for a 4-paramter family of equations with peakon traveling waves, Nonlinear Analysis, 133 (2016), 161-199.

[2] A. Himonas, R. Thompson, Persistence properties and unique continuation for a generalized Camassa-Holm equation, Journal of Math. Phys., 55, 091503 (2014).

[3] A. Himonas, G. Misio lek,G. Ponce, Y. Zhou, Persistence Properties and Unique Contin- uation of Solutions of the Camassa-Holm Equation, Commun. Math. Phys., 271 (2007), 511-522.

[4] V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002, 14 pp. Lax integrability and the peakon problem for the modiﬁed Camassa-Holm equation: Part I and Part II

Xiangke Chang and Jacek Szmigielski LSEC, Institute of Computational Mathematics and Scientiﬁc Engineering Computing, AMSS, Chinese Academy of Sciences, P.O.Box 2719, Beijing 100190, PR China, Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, S7N 5E6, Canada [email protected] and [email protected]

Abstract

Peakons are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. We show that the natural concept of weak solutions in the case of the modified Camassa-Holm equation studied in this paper is dictated by the distributional compatibility of its Lax pair and, as a result, it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the Burgers type. In the follow-up talk (Part II), we give a complete construction of peakon solutions satisfying the modified Camassa-Holm equation in the sense of distributions; our approach is based on solving certain inverse boundary value problem, the solution of which hinges on a combination of classical techniques of analysis involving Stieltjes’ continued fractions and multi-point Padéapproximations. We propose sufficient conditions needed to ensure the global existence of peakon solutions and analyze the large time asymptotic behaviour whose special features include a formation of pairs of peakons which share asymptotic speeds, as well as Toda-like sorting property. Nonlinear Profile Decomposition and the Concentration Phenomenon for the supercritical gKdV Equation

Luiz Gustavo Farah and Brian Pigott Universidade Federal de Minas Gerais - Brazil. Woﬀord College - US, [email protected] and pigottbj@woﬀord.edu

Abstract

A nonlinear profile decomposition is established for solutions of the supercritical generalized Korteweg-de Vries equation. As a consequence, we obtain a concentration result for finite time blow-up solutions that are of Type II. This is a joint work with Brian Pigott (Wofford College). Particle Trajectories and Wave Breaking in the KdV Approximation

Henrik Kalisch University of Bergen, Department of Mathematics PO Box 7800, 5020 Bergen, Norway [email protected]

Abstract

The KdV equation 3 1 ηt + ηx + 2 ηηx + 6 ηxxx = 0 (1) is a completely integrable diﬀerential equation which arises as a simple model for the evolution of long short-crested waves of small amplitude in a shallow ﬂuid.

In this talk we will investigate to what extent the KdV equation can be used to describe particle paths in the ﬂuid below the wave, and whether the KdV equation can predict breaking of surface waves. Special attention is paid to the eﬀect of a background shear current with constant vorticity.

This is joint work with Handan Borluk, Mats Brun, Chistopher Curtis and Amutha Senthilku- mar.

References

[1] A. Ali and H. Kalisch, On the formulation of mass, momentum and energy conservation in the KdV equation, Acta Appl. Math. 133 (2014), 113–131.

[2] M. Bjørkav˚agand H. Kalisch, Wave breaking in Boussinesq models for undular bores, Phys. Lett. A 375 (2011), 1570–1578.

[3] H. Borluk and H. Kalisch, Particle dynamics in the KdV approximation, Wave Motion 49 (2012), 691–709. Integral transform approach to evolution equations in the curved spacetime

Karen Yagdjian School of Mathematical and Statistical Sciences, University of Texas RGV, Edinburg, TX 78539, U.S.A. [email protected]

Abstract

In this talk we will present the integral transform that allows to construct solutions of the hyperbolic partial differential equation with variable coefficients via solution of a simpler equation. This transform was suggested by the author and it was used to investigate several well-known equations such as generalized Tricomi equation [1], the Klein–Gordon equation in the de Sitter and Einstein-de Sitter space-times [2, 3]. In the talk a special attention will be given to the global in time existence of self-interacting scalar field in the de Sitter universe [4].

References

[1] K. Yagdjian, Integral transform approach to generalized Tricomi equations. J. Diﬀeren- tial Equations 259 (2015), 5927–5981.

[2] K. Yagdjian, Integral transform approach to solving Klein-Gordon equation with variable coeﬃcients. Math. Nachr. 288 (2015), no. 17-18, 2129–2152.

[3] K. Yagdjian, Integral Transform Approach to Time-Dependent Partial Diﬀerential Equa- tions, in Mathematical Analysis, Probability and Applications – Plenary Lectures (2016) Springer Proceedings in Mathematics & Statistics, 177, 281–336

[4] A. Galstian and K. Yagdjian, Global in time existence of self-interacting scalar ﬁeld in de Sitter spacetimes, Nonlinear Analysis: Real World Applications, 34, April 2017, 110–139 On the Whitham and related equations

Jean-Claude Saut Laboratoire de Math´ematiques Universit´eParis-Sud [email protected]

Abstract

We will present various results on the Whitham equation, a full dispersion surface wave asymptotic model in the long wave, weak amplitude regime. In particular we will prove rigorous error estimates between solutions of the Whitham and the KdV equations for well- prepared initial data. Qualitative features of the Whitham solutions will be illustrated by numerical simulations. We will also consider the system version (`ala Boussinesq) of the Whitham equation which can be viewed as a regularization of the (ill-posed) Kaup- Kupperschmidt system.

This is based on work with Christian Klein, Felipe Linares and Didier Pilod. Well-posedness of evolutions equations via the uniﬁed transform method

Athanassios S. Fokas Dept. Appl. Math. Theor. Phys. University of Cambridge, Cambridge, CB3 0WA email: [email protected]

Alex A. Himonas* Department of Mathematics, University of Notre Dame Notre Dame, IN 46556 email: [email protected]

Dionyssis Mantzavinos Department of Mathematics and Statistics University of Massachusetts, Amherst, MA 01003 email: [email protected]

ABSTRACT

The uniﬁed transform method was introduced in late nineties as the analogue of the inverse scattering transform machinery for integrable nonlinear equations on the half-line. It was later understood that it also has signiﬁcant implications for linear initial-boundary value problems. In this talk, this method is used for showing well-posedness of nonlinear dispersive equations on the half-line with data in Sobolev spaces. A note on the non-periodic compressible Euler equations

John Holmes*, Barbara Keyﬁtz, Feride Tiglay 100 Math Tower, 231 W 18th Ave, Columbus OH, 43210 [email protected]

Abstract

We consider the Cauchy problem correspoding to the compressible Euler equations with data in the Sobolev space Hs(R2). This system of equations can be written in the form

ρt + ρ0ux + (ρu)x + ρ0vy + (ρv)y = 0 (1) h0 + h ut + uux + vuy + hx + ρx = 0 (2) ρ0 + ρ h0 + h vt + uvx + vvy + hy + ρy = 0 (3) ρ0 + ρ ht + uhx + vhy + (γ − 1)(h0 + h)(ux + vy) = 0, (4) where γ > 1, ρ0 > 0 and h0 > 0 are constant. In order to arrive at this from the standard form of the equations for ideal compressible gas dynamics we have written the density as ρ0+ρ and have replaced the pressure p by a multiple of the internal energy, h0+h = p/(ρ0+ρ). The velocity components are u and v. We have also written the system in nonconservative form, as we are considering only classical solutions. Pointwise restrictions on the initial data allow us to stay a positive distance from a vacuum state.

Local in time well-posedness in the sense of Hadamard for the system (in d space dimensions) is well known when s > 1 + d/2. The idea of the proof goes back to G˚arding,Leray, Lax and Kato. In particular, if the initial data is in the Sobolev space Hs, for any s > 1 + d/2, then there exists a unique solution for some time interval which depends upon the Hs norm of the initial data, and the solution depends continuously on the initial condition. Classical solutions to the compressible Euler equations do not exist globally in time. Indeed, it has been shown that even for almost constant initial data, there is generally a critical time, T C , at which the classical (Hs) solution breaks down.

We improve upon the well-posedness results by showing that the continuity of the data- to-solution map is sharp. In particular, the data-to-solution map for this system is not uniformly continuous from any bounded subset of Hs to the solution space C([−T,T ]; Hs). The uniﬁed transform method and well-posedness of the general NLS on the half line

Fangchi Yan Department of Mathematics University of Notre Dame [email protected]

Abstract

We shall discuss the initial-boundary value problem for the general nonlinear Schrödinger equation on the half-line. First, applying the unified transform method (UTM), which is also known as the Fokas transform method [1], we shall solve the initial-boundary value problem with forcing to obtain a formula that defines the iteration map for the nonlinear equation. Then, following the methodology developed for the cubic NLS in [2], we shall prove well-posedness in Sobolev spaces.

References

[1] A.S. Fokas, A uniﬁed approach to boundary value problems, SIAM, 2008.

[2] A.S. Fokas, A.A. Himonas and D. Mantzavinos, The nonlinear Schr¨odingerequation on the half-line. Trans. AMS (2017), Vol. 369, No.1, January 2017, Pages 681-709 Well-Posedness for a Modiﬁed NLS equation

Curtis Holliman and Ethan Robinett The Catholic University of America, Washington DC 20064 [email protected] and [email protected]

Abstract

Rogue waves are large spontaneous surface waves that can unpredictably appear, and thus present a great deal of danger to ocean going vessels. Recent work [1] has used a probabilistic algorithm applied to the modiﬁed nonlinear Schr¨odingerequation (MNLS) to predict where these waves will form.

The MNLS equation, in non-dimensional form, is

1 i 1 i 2 3 2 1 2 ut + ux + uxx − uxxx + |u| u + |u| ux + u ux + iuϕx = 0. (1) 2 8 16 2 2 4 z=0 where function ϕ in the above equation is the nonlinear Fourier multiplier (F being the Fourier transform) −1 2 ϕx|z=0 = −F [|k|F[|u| ]]/2. (2) The MNLS has been experimentally tested, [2] and [3], and performs as a good model for unidirectional wave envelopes. Our work develops the basic well-posedness properties of this equation.

References

[1] Cousins, Will, and Themistoklis P. Sapsis. Reduced-Order Precursors of Rare Events in Unidirectional Nonlinear Water Waves. Journal of Fluid Mechanics 790 (February 11, 2016): 368–388.

[2] Lo, E. and Mei, C. C. 1985 A numerical study of water wave Modulation based on a higher-order nonlinear Schrodinger equation. J. Fluid Mech. 150, 395 – 416.

[3] Goullet, A. and Choi, W. 2011 A numerical and experimental study on the nonlinear evolution of long-crested irregular waves. Physics of Fluids (1994-present) 23 (1), 16601.

[4] J. Gorsky, A. Himonas, C. Holliman & G. Petronilho, The Cauchy problem of a periodic higher order KdV equation in analytic Gevrey spaces, J. Math. Anal. Appl. 405 (2013), no. 2, 349–361. Well-posedness theory for dispersive equations on the half line

M. B. Erdogan and N. Tzirakis University of Illinois at Urbana–Champaign [email protected] and [email protected]

Abstract

In this talk we present the local and global regularity properties of certain dispersive partial diﬀerential equations, on the half line, with rough initial data. We focus, in particular, on two well-known models: the cubic nonlinear Schrodinger equation and the Zakharov system. Semilinear Hyperbolic Equation in the de Sitter Spacetime with Hyperbolic Spatial Part

Anahit Galstyan School of Mathematical and Statistical Sciences, University of Texas RGV, 1201 W. University Drive, Edinburg, TX 78539, USA [email protected]

Abstract

We present new results on the semilinear massless waves propagating in the de Sitter spacetime. The global in time existence of the solutions for the Klein-Gordon equation in the de Sitter spacetime is known (see e.g. [2]-[5]) with a weak restriction on the order of nonlinearity. However, for the Cauchy problem for the semilinear wave equation in the de Sitter spacetime the global in time existence of the solutions is still an open problem. We give estimates for the lifespan of the solutions of semilinear wave equation in the de Sitter spacetime with ﬂat and hyperbolic spatial parts under some conditions on the order of the nonlinearity [1]. In the case of hyperbolic spatial part the order of nonlinearity is less than the critical value given by Strauss conjecture for the semilinear wave equation in the Minkowski spacetime.

References

[1] A. Galstian, Semilinear wave equation in the de Sitter spacetime with hyperbolic spatial part, Birkhauser series Trends in Mathematics/Research Prospectives, Springer Interna- tional Publishing (2017) doi: 10.1007/978-3-319-48812-7-62.

[2] A. Galstian and K. Yagdjian, Global in time existence of self-interacting scalar ﬁeld in de Sitter spacetimes, Nonlinear Analysis: Real World Applications, 34 (2017), 110-139.

[3] A. Galstian and K. Yagdjian, Global solutions for semilinear Klein-Gordon equation in FLRW spacetimes, Nonlinear Analysis: Theory, Methods & Applications, 113 (2015), 339-356.

[4] M. Nakamura, The Cauchy problem for semi-linear Klein-Gordon equations in de Sitter spacetime, J. Math. Anal. Appl., 410 (2014), 445-454.

[5] K. Yagdjian, Global existence of the scalar ﬁeld in de Sitter spacetime, J. Math. Anal. Appl., 396 (2012), 323–344. On rigorous aspects of the uniﬁed transform method: linear and nonlinear evolution equations on the half-line

Dionyssios Mantzavinos Department of Mathematics and Statistics University of Massachusetts Amherst [email protected]

Alex Himonas Department of Mathematics University of Notre Dame [email protected]

Athanassios S. Fokas Department of Applied Mathematics and Theoretical Physics University of Cambridge [email protected]

Abstract

This talk is devoted to a new approach for proving well-posedness of nonlinear evolution equations on the half-line supplemented with initial and boundary data in appropriate Sobolev spaces. Utilizing the uniﬁed transform method of Fokas for the associated linear equations, one obtains directly a solution formula for the forced linear problem involving integrals along contours in the complex spectral (Fourier) plane. This formula is in turn used to deﬁne an iteration map for the nonlinear problem, which is subsequently analyzed by means of suitably adapted harmonic analysis techniques. Non-uniform dependence on initial data for equations of Whitham type

Mathias Nikolai Arnesen Department of Mathematical Sciences, Norwegian University of Science and Technology 7491 Trondheim, Norway [email protected]

Abstract

We consider the Cauchy problem

∂tu + u∂xu + L(∂xu) = 0, u(0, x) = u0(x) for a class of Fourier multiplier operators L and prove that the solution map u0 7→ u(t) is s 3 not uniformly continuous in H on the real line or the torus for s > 2 . The class of equations considered includes in particular the Whitham equation and fractional Korteweg-de Vries. The result is proved by constructing two sequences of solutions converging to the same limit at the initial time, while the distance at a later time is bounded below by a positive constant. Stability of solitary wave solutions to a coupled system

Hongqiu CHEN University of Memphis

[email protected]

Abstract

Considered here is a system

Ut + Ux − Uxxt + (∇H(U))x = 0 (1) of nonlinear dispersive equations, where U = U(x, t) is an R2-valued function, and ∇H is the gradient of a homogeneous polynomial function H : R2 → R. We present existence and stability criteria for explicit solitary wave solutions. Using the idea by Bona, Chen and Karakashian [1] and exploiting the accurate point spectrum information of the associated Schr¨odinger operator, with Xiaojun Wang, we improve the stability results previously obtained by Pereira [2] and also observe the criteria for instability of solitary wave solutions.

References

[1] J. L. Bona, H. Chen and O. A. Karakashian, Stability of solitary-wave solutions of systems of dispersive equations, Appl Math Optim on line, Nov. 17 (2015) DOI 10.1007/s00245-015-9322-4.

[2] J.M. Pereira, Stability and instability of solitary waves for a system of coupled BBM equations, Appl. Anal. 84 (2005), no. 8, 807–819. Short pulse systems produced through the negative WKI hierarchy

Qilao Zha, Qiaoyi Hu, and Zhijun Qiao School of Mathematical and Statistical Sciences, University of Texas - Rio Grande Valley [email protected]

Abstract

In this paper, we study a two-component short pulse system, which was produced through a negative integrable ﬂow associated with the WKI hierarchy. The multi-soliton solutions for the two short pulse system investigated, in particular, one-, two-, three-loop soliton, and breather soliton solutions are discussed in details with interesting dynamical interactions and shown through ﬁgures. BLOW-UP RESULTS AND SOLITON SOLUTIONS FOR A GENERALIZED VARIABLE COEFFICIENT NONLINEAR SCHR’ODINGER EQUATION

Jose Escorcia and Erwin Suazo Department of Mathematics, University of Puerto Rico, Arecibo, P.O. Box 4010, Puerto Rico 00614-4010. School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 W. University Drive, Edinburg, Texas, 78539-2999. [email protected] and [email protected]

Abstract

In this paper, by means of similarity transformations we study exact analytical solutions for a generalized nonlinear Schrödingerequation with variable coefficients. This equation appears in literature describing the evolution of coherent light in a nonlinear Kerr medium, Bose-Einstein condensates phenomena and high intensity pulse propagation in optical fibers. By restricting the coefficients to satisfy Ermakov-Riccati systems with multiparameter solutions, we present conditions for existence of explicit solutions with singularities and a family of oscillating periodic soliton-type solutions. Also, we show the existence of bright-, dark- and Peregrine-type soliton solutions, and by means of a computer algebra system we exem- plify the nontrivial dynamics of the solitary wave center of these solutions produced by our multiparameter approach.

Keywords. Soliton-like equations, Nonlinear Schr¨odingerlike equations, Fiber optics, Gross-Pitaevskii equation, Similarity transformations and Riccati-Ermakov systems.

References

[1] J. Escorcia and E. Suazo, Blow-up results and soliton solutions for a generalized variable coeﬃcient nonlinear Schr’odinger equation, to appear in Applied Mathematics and Computation. Peakons: weak solutions or distributional solutions?

Stephen Anco Department of Mathematics and Statistics Brock University, Canada [email protected]

Abstract

Recent work [1] has shown that a wide family of peakon equations mt + f(u,ux)m + (g(u,ux)m)x = 0 admits multi-peakon solutions in the setting of a weak formulation. Typ- ically, for the Camassa-Holm, Degasperis-Procesi, Novikov equations (and their b-family extensions), multi-peakon solutions can formulated in an equivalent way as distributional solutions. In this talk, it will be shown that, in contrast, a diﬀerence arises between the weak setting and the distributional setting for peakon equations with stronger nonlinearities such as occur in the mCH/FORQ equation. In particular, for the mCH/FORQ equation, single and multi peakons solutions in the weak setting are shown not to be solutions in the distributional setting, and vice versa. Moreover, this diﬀerence is directly related to the open question of existence of a Hamiltonian structure for the peakon sector of the mCH/FORQ equation in the weak setting: it will be shown that no Hamiltonian structure appears to exist for that sector, while a Hamiltonian structure does exist for peak sector in the distributional setting.

References

[1] S.C. Anco and E. Recio, A general family of multi-peakon equations, arXiv: 1609.04354 math-ph (2016).

SESSION 4

Applied Nonlinear Waves

David Kaup, Constance Schober, Thomas Vogel Department of Mathematics University of Central Florida

Abstract

The theoretical aspects of solitons and nonlinear waves are relevant to a broad spectrum of fields. This Session will bring together researchers actively working on the theoretical, experimental and computational aspects of nonlinear wave phenomena in novel physical settings. Topics include applications in hydrodynamics and nonlinear optics as well as structure preserving algorithms for solving nonlinear wave equations.

Structure-Preserving Exponential Integrators for Simulating Damped-Driven Nonlinear Waves

Ashish Bhatt and Brian E. Moore∗ Department of Mathematics University of Central Florida [email protected] and [email protected]

Abstract

Many PDE models for nonlinear waves have conservative properties (such as energy, momentum, mass, etc.) which are desirable to preserve in numerical simulations. In the presence of a driving force or damping terms those conservative properties inevitably break down. Yet, in cases where the forcing and/or damping is linear, with coeﬃcients that depend on time, those properties may sometimes be reformulated as conservation laws, which can be preserved through discretization using exponential integrators. In particular, exponential Runge-Kutta methods preserve these properties under certain restrictions on their coeﬃ- cient functions [2]. As time-stepping schemes these methods have certain advantages for damped-driven nonlinear wave equations [1], because they can preserve dissipation rates up to machine precision. In some applications it may also be useful to apply such methods as spacial discretizations for the sake of structure preservation [3].

References

[1] A. Bhatt, D. Floyd, and B.E. Moore, Second order conformal symplectic schemes for damped Hamiltonian systems, Journal of Scientiﬁc Computing 66 (2016) 1234-1259.

[2] A. Bhatt and B.E. Moore, Structure-preserving exponential Runge-Kutta methods, SIAM Journal on Scientiﬁc Computing, to appear, 2017.

[3] B.E. Moore, Multi-conformal-symplectic PDEs and discretizations, submitted for publication, 2016. Breathers and rogue waves on a vortex ﬁlament with nontrivial axial ﬂow

Matthew Russo Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364 USA [email protected]

Abstract

The motion of an isolated vortex filament with nontrivial axial flow within the vortex core is approximately described by a modified localized induction equation (mLIA)[1, 2]. Through Hasimoto’s transformation, the mLIA has been shown to be equivalent to Hirota’s equation. In this talk we make use of standard and generalized Darboux transformations[3] to derive spatially periodic breather solutions and rogue wave solutions, respectively, to Hirota’s equation. For each case, we then recover the corresponding dynamics on the vortex filament.

References

[1] D. W. Moore, P. G. Saﬀman (1972) ‘The Motion of a Vortex Filament with Axial Flow’, Phil. Trans. R. Soc. Lond. A, 272, pp. 403-429.

[2] Fukumoto, Y. and Miyazaki, T. (1991) ‘Three-dimensional distortions of a vortex ﬁla- ment with axial velocity’, Journal of Fluid Mechanics, 222, pp. 369416.

[3] B. Guo, L. Ling, and Q.P. Liu, (2012) ‘Nonlinear Schr¨odinger equation: Generalized Darboux transformation and rogue wave solutions’, Phys. Rev. E, 85, pp. Rogue waves over non-constant backgrounds

C. M. Schober∗ Department of Mathematics, University of Central Florida Orlando, FL, USA [email protected]

A. Calini Department of Mathematics, College of Charleston Charleston, SC, USA [email protected]

Abstract

Rogue waves in random sea states modeled by the JONSWAP power spectrum are high amplitude waves arising over non-uniform backgrounds that cannot be viewed as small amplitude modulations of Stokes waves. In the context of Nonlinear Schrödinger(NLS) models for waves in deep water, this poses the challenge of identifying appropriate analytical solutions for JONSWAP rogue waves, investigating possible mechanisms for their formation, and examining the validity of the NLS models in these more realistic settings. In this talk we investigate JONSWAP rogue waves using the inverse spectral theory of the periodic NLS equation for moderate values of the period. For typical JONSWAP initial data, numerical experiments show that the developing sea state is well approximated by the first few dom- inant modes of the nonlinear spectrum and can be described in terms of a 2- or 3-phase periodic NLS solution. As for the case of uniform backgrounds, proximity to instabilities of the underlying 2-phase solution appears to be the main predictor of rogue wave occurrence, suggesting that the modulational instability of 2-phase solutions of the NLS is a main mechanism for rogue wave formation and that heteroclinic orbits of unstable 2-phase solutions are plausible models of JONSWAP rogue waves. To support this claim, we correlate the maximum wave strength as well as the higher statistical moments with elements of the nonlinear spectrum. Finally, we examine the validity of NLS models for JONSWAP data, and show that NLS solutions with JONSWAP initial data are described by non-Gaussian statistics, in agreement with the TOPLEX field studies of sea surface height variability.

References

[1] A. Calini and C. M. Schober, Characterizing JONSWAP rogue waves and their statistics via spectral data, Wave Motion, (2016) DOI:10.1016/j.wavemoti.2016.06.007. Modeling cholera spread in a stream environment

Zhisheng Shuai Department of Mathematics, University of Central Florida [email protected]

Abstract

Cholera is a waterborne disease caused by the aquatic bacterium Vibrio cholerae that can persist for extended time outside of the human host. The recent cholera outbreak in Haiti, which was initiated along Artibonite river in October 2010, has highlighted the need in understanding the impact of pathogen movement in a stream environment on disease dynamics. Mathematical models can be used to investigate this impact and guide the design of cholera control and intervention strategies. Internally Driven Oceanic Surface Waves

Tom Vogel and David Kaup Stetson University 421 N Woodland Blvd DeLand, FL 32723 University of Central Florida 4000 Central Florida Blvd. Orlando, FL 32816 [email protected] and [email protected]

Over the last several decades many advances have been made in understanding the nonlinear dynamics of oceanic surface waves. The present work considers a two-layer incompressible stratified fluid 2D+1 model in order to investigate the connection between the geometry of the thermocline and its corresponding effect on the evolution of the free surface. The evolution in the geometry of the thermocline-surface coupled system is considered in this model to be induced by horizontally propagating internal gravity waves in the lower layer. This analysis leads to a generalization of the Benjamin-Ono equation and recovery of the BO equation under the appropriate limits will be discussed. SESSION 5

Solitary and Rogue Waves as Solutions of Generalized Schr¨odingerEquations. Achievements and Challenges

Michail Todorov Dept. of Applied Mathematics and Informatics, Technical University of Soﬁa, Bulgaria mtod@tu-soﬁa.bg

Andrei Ludu Dept. of Mathematics, Embrie-Riddle Aeronautical University, Daytona, FL, USA [email protected]

Abstract

This session covers various types of nonlinear waves as solutions of the scalar, vector and multidimensional Schr¨odingerequations, including high order derivatives versions. The papers presented will employ a diverse array of methods and techniques towards the analysis of nonlinear wave propagation and interaction in various contexts. Based on the Boussinesq Paradigm about the stability of solitary waves and concept for the interacting solitons as quasi-particles the session is extended towards Boussinesq-like equations in biological context. The session exempliﬁes how the mathematical modeling leads to challenging problems giving rise to nonlinear wave dynamics. The focus will be on the breaking waves criteria and on approaches on switching between solitary and rogue waves. Rotating Hollow Patterns in Fluids

Andrei Ludu Embry-Riddle Aeronautical University, Dept. Mathematics Daytona Beach, FL 32114 USA [email protected]

Abstract

Rotating hollow polygonal patterns were obtained for liquid nitrogen Leidenfrost drops con- fined in circular boundaries [1, 2]. The patterns range from convex or concave regular polygons (triangles to heptagons) to higher frequency periodic waves. During the rapid evaporation of the liquid various stable modes (number of edges of polygons) take over in the process. This incompressible, inviscid flow is basically 2-dimensional dominated by gravity, tension surface and evaporation. When placed between convex shapes, the liquid shows stable large amplitude bridge oscillations controlled by curvatures combinations. The model is based on shallow water theory and predicts the existence of sharp rotational polygonal waves with peakon solutions. PIV measurements showed that the inner rotating polygons are generated by energy transfer waves (apparent shape rotation), and do not involve matter transport. It is proved that other models of nonlinear waves (cnoidal, solitons, etc.) cannot insure the high stability of such sharp corner polygons [1]. Similar hollow polygonal patterns are noticed in eye of hurricanes, in fast rotational bodies of water, in Saturn’s hexagon, and in some plasma systems. Consequently, a tentative universality model for these rotational effects is presented.

References

[1] Andrei Ludu, Boundaries of a Complex World (Springer-Verlag, Heidelberg 2016).

[2] Andrei Ludu, Nonlinear Waves and Solitons on Contours and Closed Surfaces (Springer- Verlag, Heidelberg 2012). BLOW-UP RESULTS AND SOLITON SOLUTIONS FOR A GENERALIZED VARIABLE COEFFICIENT NONLINEAR SCHR’ODINGER EQUATION

Abstract

In this paper, by means of similarity transformations we study exact analytical solutions for a generalized nonlinear Schr¨odinger equation with variable coecients. This equation appears in literature describing the evolution of coherent light in a nonlinear Kerr medium, Bose-Einstein condensates phenomena and high intensity pulse propagation in optical ﬁbers. By restricting the coecients to satisfy Ermakov-Riccati systems with multiparameter solutions, we present conditions for existence of explicit solutions with singularities and a family of oscillating periodic soliton-type solutions. Also, we show the existence of bright-, dark- and Peregrine-type soliton solutions, and by means of a computer algebra system we exem- plify the nontrivial dynamics of the solitary wave center of these solutions produced by our multiparameter approach.

Keywords. Soliton-like equations, Nonlinear Schr¨odinger like equations, Fiber optics, Gross-Pitaevskii equation, Similarity transformations and Riccati-Ermakov systems.

References

[1] J. Escorcia and E. Suazo, Blow-up results and soliton solutions for a generalized variable coecient nonlinear Schr’odinger equation, to appear in Applied Mathematics and Computation. Formation of rogue waves in NLS systems: Theory and Computation

E. G. Charalampidis∗ and P. G. Kevrekidis Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003-4515, USA [email protected] and [email protected]

J. S. He Department of Mathematics, Ningbo University, Ningbo , Zhejiang 315211, P. R. China [email protected]

R. Babu Mareeswaran and T. Kanna Post Graduate and Research Department of Physics, Bishop Heber College, Tiruchirapalli-620 017, Tamil Nadu, India [email protected] and [email protected]

D. J. Frantzeskakis Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece [email protected]

J. Cuevas-Maraver Grupo de F´ısica No Lineal, Departamento de F´ısica Aplicada I, Universidad de Sevilla. Escuela Politécnica Superior, C/ Virgen de Africa,´ 7, 41011-Sevilla, Spain Instituto de Matemáticas de la Universidad de Sevilla (IMUS). Edificio Celestino Mutis. Avda. Reina Mercedes s/n, 41012-Sevilla, Spain [email protected]

Abstract

In this talk, we will discuss the dynamics of rogue waves in one- and two-component nonlinear Schrödinger (NLS) equations with variable coefficients. Upon utilizing suitable transformations for the wavefunctions, the formation and existence of such soliton solutions will be presented. Finally, the initial value problem (IVP) with Gaussian wavepacket initial data for the scalar (NLS) will be discussed where some novel features will be presented. Hopefully, such findings might be of particular importance towards realizing experimentally extreme events in BECs. Self-focusing and Spatio-Temporal Dynamics of Nonresonant co-Filaments in air.

Alexey Sukhinin∗, Alejandro Aceves Department of Mathematics and Statistics, University of Vermont, Burlington, VT, 05405, USA Department of Mathematics, Southern Methodist University, Dallas, TX, 75275, USA [email protected], [email protected]

Abstract

Self-focusing of a gaussian beam is a well known optical effect. It is the main factor that responsible for the beam collapse in air[1]. Mathematically, this phenomenon is well modeled by the (2+1)D Nonlinear Schrodinger Equation (NLSE). In reality collapse of the beam produces new optical effects that arrest it. One such effect is the ionization of air that may lead to filamentation. Laser filamentation is a process that can be described by the balance between optical self-focusing and multiphoton ionization. In this talk I will describe the self- focusing and collapse events of various configurations of two nonresonant beams. Collapse is linked to the unstable fundamental and vortex soliton solutions that was obtained[2]. I will describe the role of the total beam power and individual powers as the conditions for the collapse. Then the spatio-temporal dynamics of non-resonant filamentation will be considered with the simplified model[3, 4].

References

[1] Moll, K. D., Alexander L. Gaeta, and Gadi Fibich. ”Self-similar optical wave collapse: observation of the Townes proﬁle.” Physical review letters 90, no. 20 (2003): 203902.

[2] Sukhinin Alexey, Aceves Alejandro, Diels Jean-Claude, Arissian Ladan. ”Collapse events of two-color optical beams.” (preprint)

[3] Sukhinin, Alexey, Alejandro Aceves, Jean-Claude Diels, and Ladan Arissian. ”On the co-existence of IR and UV optical ﬁlaments.” Journal of Physics B: Atomic, Molecular and Optical Physics 48, no. 9 (2015): 094021.

[4] Sukhinin Alexey, Downes Edward, Aceves Alejandro, Diels Jean-Claude, Arissian Ladan. ”Temporal dynamics of the co-propagating two-color pulses.” (preprint) System of Coupled Nonlinear Schr¨odingerEquations with Diﬀerent Cross-Modulation Rates

Michail Todorov∗ Technical University of Sofia, 8 Kl.Ohridski Blvd., 1000 Sofia, Bulgaria mtod@tu-sofia.bg

Rossen Ivanov School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland [email protected]

Abstract

In oceanography, freak waves (or rogue waves, extreme waves) are usually defined as waves whose height is more than twice the “significant” wave height Hs.(Hs is the mean of the largest third of waves in a wave record). The nonlinear Schrödinger(NLS) equation is a fundamental model for the slowly varying complex group envelope of surface waves over a deep water. The Peregrine’s soliton of the NLS equation [1] is often considered as a prototype model of the deterministic freak waves’ generation. The solution is with constant boundary conditions at x → ∞ and has the property that at x = 0 the amplitude of the hump is 3 times higher than the uniform constant background. This suggests that an initially small “hump” in a nearly monochromatic wave train may indeed evolve into a wave with a much larger amplitude. Thus the nonlinear wave interaction is a mechanism of creation of waves with significantly larger amplitude. We investigate in silico wave interactions modeled by two coupled NLS equations, where each component describes an envelope of a separate wave packet. We show that such nonlinear interaction causing polarization shock [2] can lead to an amplification of the amplitude of one of the packets and hence providing a possible mechanism for the formation of freak waves. We use constructed earlier by us conservative fully implicit scheme for investigation of the head-on collisions of solitary waves of coupled NSE for superposition of linearly polarized soliton envelopes in the initial configuration. We elucidate numerically the role of nonlinear coupling on their quasi-particle (QP) dynamics. We have uncovered many other different scenarios of the QP behavior upon collision including multiplying the soliton envelopes after the collision, dramatic change of amplitudes, and velocity shift.

References [1] V. I. Shrira, V. V. Geogjaev, What makes the Peregrine soliton so special as a prototype of freak waves? J Eng Math 67 (2010), 11–22. [2] M. D. Todorov, C. I. Christov, Collision Dynamics of elliptically polarized solitons in Coupled Nonlinear Schr¨odingerEquations, Math Comput Simul 82 (2012), 1221–1232. On numerical modelling of solitary waves in lipid bilayers and complexity

Kert Tamm, Tanel Peets and J¨uriEngelbrecht Tallinn University of Technology, Department of Cybernetics Akadeemia Tee 21, 12618, Tallinn, Estonia [email protected], [email protected], [email protected]

Abstract

The nerve pulse propagation is a well investigated problem with several popular models for modelling such a phenomenon. Probably the most common family of models is Hodgkin- Huxley [1] and its modifications. However, while the electrophysiology family of models give excellent results they do not describe some of the effects that are associated with the nerve pulse propagation, like, for example, the mechanical wave travelling along the axon with the action potential which has been experimentally observed [2]. There exist separate models for describing such a mechanical wave – see [3, 4] and references therein. The next logical step would be coupling the existing models into a system that takes into account several aspects of the nerve pulse propagation [5]. For a start one could use, for example, a FitzHugh Nagumo model for the action potential which is a simplification of the Hodgin-Huxley model, Navier Stokes equations for an elastic tube for taking into account the axoplasm inside the axon and the improved Heimburg-Jackson model for describing the mechanical wave along the axon with additional force terms for taking into account the coupling effects as proposed in [5]. The focus of the present study is a preliminary numerical investigation of such a coupled system of models with a particular attention directed at the influence of the coupling terms on the behaviour of the solutions of the models for individual components.

References [1] A. L. Hodgkin and A. F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 117 (1952) 500-544. [2] K. Iwasa, I. Tasaki and R. Gibbons, Swelling of nerve fibers associated with action potentials. Science, 210, (1980) 338-339. [3] T. Heimburg and A. D. Jackson. On soliton propagation in biomembranes and nerves. Proc. Natl. Acad. Sci. U.S.A., 102 (2005) 9790-9795. [4] J. Engelbrecht, K. Tamm and T. Peets, On mathematical modelling of solitary pulses in cylindrical biomembranes. Biomech. Model. Mechanobiol., 14 (2015) 159-167. [5] J. Engelbrecht, T. Peets, K. Tamm, M. Laasmaa and M. Vendelin, On modelling of physical effects accompanying the propagation of action potentials in nerve fibres. arXiv:1601.01867 [physics.bio-ph] On solitonic solutions of a Boussinesq-type equation modelling mechanical waves in biomembranes

Tanel Peets, Kert Tamm and J¨uriEngelbrecht Department of Cybernetics, Tallinn University of Technology Akadeemia tee 21, 12618 Tallinn, Estonia [email protected], [email protected], [email protected]

Abstract

The original Boussinesq equation was derived for waves in ﬂuid but nowadays such equations are also used in solid mechanics [1]. In this contribution an analysis of a Boussinesq-type equation with displacement-dependent nonlinearity is presented. Such a model was proposed by Heimburg and Jackson for describing longitudinal waves in biomembranes [2] and later improved by Engelbrecht et al. [3] taking into account the microinertia of a biomembrane:

h 2 2 i utt = (c + pu + qu )ux − h1uxxxx + h2uxxtt, (1) 0 x where u = ∆ρA is a density change, c0 is the velocity of low amplitude sound, p, q are coefficients determined from experiments and h1, h2 are dispersion coefficients. Equation (1) has an analytical solution in the form of a soliton and/or periodic waves [4]. The influence of nonlinear and dispersive terms over the wide range of possible sets of coefficients is analysed over the phase space. It is also demonstrated by the numerical analysis how the solutions arise from an arbitrary input. The numerical analysis of interaction of solitons demonstrates that the interaction is accompanied by radiation effects and solutions arising from Eq. (1) are not strictly speaking solitons.

References

[1] C. I. Christov, G. A. Maugin, and A. V. Porubov, On Boussinesq’s paradigm in nonlinear wave propagation, Comptes Rendus M´ecanique, 335 (2007), 521–535.

[2] T. Heimburg and A. D. Jackson, On soliton propagation in biomembranes and nerves, Proc. Natl. Acad. Sci. USA, 102 (2005), 9790–5.

[3] J. Engelbrecht, K. Tamm, and T. Peets, On mathematical modelling of solitary pulses in cylindrical biomembranes, Biomech. Model. Mechanobiol., 14 (2015), 159–167.

[4] J. Engelbrecht, T. Peets, and K. Tamm, On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes, Philos. Mag., in press. Design of supercontinuum optical sources aided by high performance computing

Andreas Falkenstrøm Mieritz∗, Mads Peter Sørensen and Allan Peter Engsig-Karup Department of Applied Mathematics and Computer Science Technical University of Denmark, Kongens Lyngby, Denmark [email protected], [email protected] and [email protected]

Rasmus Dybbro Engelsholm, Ivan Bravo Gonzalo and Ole Bang Department of Photonics Engineering Technical University of Denmark, Kongens Lyngby, Denmark [email protected], [email protected] and [email protected]

Abstract

Spectral white light is used in Optical Coherence Tomography (OCT) systems for e.g. medical diagnostics purposes. A powerful way of generating a broad band white light source is to use supercontinuum generation in nonlinear optical crystal ﬁbers illuminated by a monochromatic laser. In this presentation we shall discuss the development of a new generation of supercontinuum light sources with unprecedented low noise and shaped power spectra that are optimal for use in the next generation ultra-high resolution Optical Coherence Tomog- raphy (UHROCT) systems.

In this presentation the main focus is on invoking high performance computing of the generalized nonlinear Schr¨odingerequation for aiding the optimal design of supercontinuum generation. Our model includes higher order dispersion, delayed Raman response and ta- pering in order to construct ﬁber design features for reducing the noise in a supercontinuum light source and shape its spectrum.

The overall goal is to use UHROCT for cost eﬀective diagnose of glaucoma, the second leading cause of blindness worldwide, and to develop equipment easy to use for a local clinic contrary to current practice. The project is conducted in collaboration with NKT Photonics, designing supercontinuum and OCT systems, and Bispebjerg Hospital, Denmark. On the application of 2D spectal analysis in case of the KP equation

Andrus Salupere and Mart Ratas Department of Cybernetics, School of Science, Tallinn University of Technology Akadeemia tee 21, 12618, Tallinn, Estonia [email protected] and [email protected]

Abstract

Authors of the present paper have demonstrated that in case of 1D wave propagation the discrete spectral analysis is very helpful tool in order to analyze the space-time behavior of diﬀerent wave structures [1, 2, 3]. Here we generalize the method proposed in [1] to 2D case. The KPI equation is applied as a model equation. For numerical integration the pseudospectral method is applied. We demonstrate how 2D spectral characteristics can be applied for analysis of complicated wave structures that can be formed from diﬀerent initial pulses in case of the KPI equation, see [4] for details. Recurrence phenomenon, temporal periodicity and temporal symmetry of the solution and number of emerging solitary waves (which can behave like solitons in some cases) will be discussed in the presentation.

References

[1] A. Salupere, The pseudospectral method and discrete spectral analysis. In E. Quak, T. Soomere (Eds.), Applied Wave Mathematics: Selected Topics in Solids, Fluids, and Math- ematical Methods, Heidelberg, Springer, 2009, pp. 301–333.

[2] A. Salupere, M. Lints, J. Engelbrecht, On solitons in media modelled by the hierarchical KdV equation. Archive of Applied Mechanics, 84 (2014), 1583–1593.

[3] A. Salupere, M. Ratas, On solitonic structures and discrete spectral analysis. In G. Biondini, T. Taha (Eds.), Book of Abstracts: The Ninth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory, Athens, Georgia, April 1-4, 2015. Athens, University of Georgia, 2015, p. 30.

[4] M. Ratas, A. Salupere, 2D spectral analysis of the KPI equation, Research Report, Insti- tute of Cybernetics at Tallinn University of Technology, Tallinn, 2017. Magnetohydrodynamic Gauge Field Symmetries and Conservation Laws

G. M. Webb∗ Center for Space Plasma and Aeronomic Research, The University of Alabama in Huntsville, Huntsville AL 35805 [email protected]

S. C. Anco Department of Mathematics, Brock University, St. Catharines, ON L2S3A1, Canada [email protected]

Abstract

A Clebsch potential formulation of magnetohydrodynamics is developed based in part on the work of [1] (see [2] for a more complete account). Gauss’s equation, (divergence of B is zero), Faraday’s equation, the mass continuity equation, the gas entropy advection equation, and Lin constraints are enforced by means of Lagrange multipliers in the action. We consider gauge symmetries of the action, in which the density ρ, fluid velocity u, magnetic field induction B and entropy S do not change, but the Lagrange multipliers change. Noether’s theorem, and the gauge symmetries are used to derive conservation laws for (a) magnetic helicity (b) cross helicity, (c) fluid helicity for non-magnetized fluids, and (d) a class of conservation laws related to curl and divergence equations, which applies to Faraday’s equation and Gauss’s equation. The magnetic helicity conservation law is due to a gauge symmetry, which is not a fluid relabelling symmetry. The analysis is carried out for the case of a non- barotropic gas, in which the gas pressure p = p(ρ, S). The cross helicity and fluid helicity conservation laws in the non-barotropic case, are nonlocal conservation laws, that reduce to local conservation laws in the barotropic case. The connections between gauge symmetries, Clebsch potentials and Casimirs are developed (see e.g. [3]).

References

[1] M.G. Calkin, An action principle for magnetohydrodynamics, Canad. J. Physics, 41 (1963), 2241-2251.

[2] G. M. Webb and S.C. Anco, On Magnetohydrodynamic Gauge Field Theory, J. of Physics A, Math. and Theoret., submitted December 30, 2016, arxiv:1701.0052v1.

[3] F. S. Henyey, Canonical construction of a Hamiltonian for dissipation-free magnetohydrodynamics, Phys. Rev. A, 26, no. 1., (1982), 480-483. SESSION 6

Discrete Integrable Systems

Nalini Joshi School of Mathematics and Statistics F07, The University of Sydney, NSW 2006, Australia. [email protected]

Christopher Lustri and Nobutaka Nakazono Numeracy Centre Faculty of Science, Macquarie University, NSW 2109, Australia Department of Physics and Mathematics, Aoyama Gakuin University, Fuchinobe, Kanagawa 252-5258, Japan. [email protected] and [email protected]

Milena Radnovic and Yang Shi School of Mathematics and Statistics F07, The University of Sydney, NSW 2006, Australia. [email protected] and [email protected]

Abstract

The study of integrability and integrable systems addresses important questions from mathematics and physics. Many of these questions arise from the study of models involving finite operations, and require the analysis of discrete integrable systems in order to be answered. Discrete integrable systems are difference equations whose integrability are guaranteed in various ways, such as singularity confinement, algebraic entropy, or the solutions to associated linear problems. The study of these discrete systems reveals a range of fascinating mathematical features, including the symmetry group structure of solutions, and the existence of solutions that may be expressed in terms of known special functions. Consequently, these systems are of great interest due to both their rich mathematical structure, as well as their wide applicability. This session will bring together researchers with expertise in a range of fields related to discrete integrable systems, with purpose of facilitating the exchange of methods and ideas.

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Nobutaka Nakazono∗ and Nalini Joshi Department of Physics and Mathematics, Aoyama Gakuin University, Kanagawa, Japan. School of Mathematics and Statistics, The University of Sydney, New South Wales, Australia. [email protected], [email protected]

Abstract

Discrete Painleve´ equations are nonlinear ordinary diﬀerence equations of second order, which include discrete analogues of the six Painleve´ equations: PI, ... ,PVI. It is well known that discrete Painleve´ equations are classiﬁed into 19 types by the rational surface called space of initial values [3]. Discrete Painleve´ equations on the highest surface are called elliptic Painleve´ equations since their time evolutions are described by the elliptic functions. In [3], Sakai derived an elliptic Painleve´ equation. We now regard it as the standard elliptic Painleve´ equation, and its properties have been well investigated.

On the other hand, in [2] the following elliptic Painleve´ equation is obtained from the reduction of lattice Krichever-Novikov (KN) system: (1 − k2sz4)cg dg xy − (cg2 − cz2)cz dz − (1 − k2sg2sz2)cz dz x2 y˜ = e e e e , (1a) 2 2 − 2 2 − − 2 4 + − 2 2 2 k (cge cz )cz dz x y (1 k sz )cgedge x (1 k sgesz )cz dz y (1 − k2szb4)cg dg yx˜ − (cg2 − czb 2)czb dzb − (1 − k2sg2szb2)czb dzb y ˜2 x˜ = o o o o , (1b) 2 2 − b 2 b b 2 − − 2 b4 + − 2 2 b2 b b k (cgo cz )cz dzy ˜ x (1 k sz )cgodgo y˜ (1 k sgosz )cz dz x where k is the modulus of the elliptic sine, x = x(z0), y = y(z0), sz = sn (z0) , cz = cn (z0) , dz = dn (z0) , b = + δ , b = + δ , b = + δ , = γ , = γ , sz sn (z0 ) cz cn (z0 ) dz dn (z0 ) sge sn ( e) sgo sn ( o) = γ , = γ , = γ , = γ , δ = γ + γ , cge cn ( e) cgo cn ( o) dge dn ( e) dgo dn ( o) e o and ˜ : (γe, γo, z0) 7→ (γe, γo, z0 + 2δ). The geometry of Equation (1): its space of initial values and corresponding Cremona isometries is investigated in [1]. Although the geometry of Equation (1) has been clariﬁed, the realization of Equation (1) from the birational action of Cremona isometries was missing. The present study ﬁlls this gap, that is, our main result provides the realization of the Equation (1) from the birational action of Cremona isometries. Using the birational action, we derive not only Equation (1) but also other two new elliptic Painleve´ equations.

References [1] J. Atkinson, P. Howes, N. Joshi, and N. Nakazono. J. Lond. Math. Soc. (2), 93(3):763–784, 2016. [2] A. Ramani, A. S. Carstea, and B. Grammaticos. elliptic Painleve´ equations. J. Phys. A, 42(32):322003, 8, 2009. [3] H. Sakai. Comm. Math. Phys., 220(1):165–229, 2001. Regular ﬂat structures and generalized Okubo systems

Toshiyuki Mano Department of Mathematical Sciences, Faculty of Science University of the Ryukyus Nishihara-cho, Okinawa 903-0213 JAPAN [email protected]

Abstract

B. Dubrovin [2] introduced the notion of Frobenius manifold in order to interpret geomet- rically the WDVV equation appeared in 2D topological field theory, and showed that there is a correspondence between three dimensional regular semisimple Frobenius manifolds and solutions to a one-parameter family of the sixth Painlevéequation. This correspondence was generalized to that between three dimensional regular semisimple flat structures (without potentials) and solutions to the (full-parameter) sixth Painlevéequation in [5, 3]. Recently, A. Arsie and P. Lorenzoni [1] showed a correspondence between three dimensional regular (non-semisimple) bi-flat F-manifolds and solutions to the PainlevéV and IV equations.

In this talk, we study a correspondence between regular flat structures and isomonodromic deformations of generalized Okubo systems (which were introduced by H. Kawakami [4]). As an application, we can define flat structures on the spaces of independent variables of the (classical) Painlevéequations (except for PI). This talk is based on a joint work with H. Kawakami.

References

[1] A. Arsie and P. Lorenzoni: F-manifolds, multi-ﬂat structures and Painlev´etranscen- dents, arXiv:1501.06435

[2] B. Dubrovin: Geometry of 2D topological ﬁeld theories. In:Integrable systems and quantum groups. Montecatini, Terme 1993 (M. Francoviglia, S. Greco, eds.) Lecture Notes in Math. 1620, Springer-Verlag 1996, 120-348.

[3] M. Kato, T. Mano and J. Sekiguchi: Flat structure on the space of isomonodromic deformations. arXiv:1511.01608

[4] H. Kawakami: Generalized Okubo Systems and the Middle Convolution. Int. Math. Res. Not. 2010, no.17 (2010), 3394-3421.

[5] P. Lorenzoni: Darboux-Egorov system, bi-flat F-manifolds and PainlevéVI, IMRN (2014), Vol. 12, 3279-3302. Regular and finite time blowup solutions for discrete integrable equations

Yasuhiro Ohta Department of Mathematics, Kobe University Rokko, Kobe 657-8501, Japan [email protected]

Abstract

Rogue waves are nonlinear waves localized in time which describe spontaneous freak waves in ocean. A class of algebraic solutions models the rogue waves in some soliton equations such as nonlinear Schr¨odingerequation, Davey-Stewartson equation and Yajima-Oikawa equation. It has been revealed that the algebraic solutions have rich algebraic structures.

It is known that these solutions are even richer for discrete systems. The Ablowitz-Ladik equation admits rogue wave solutions (algebraic solutions) of both regular type and ﬁnite time blowup type [1]. In this paper we consider some algebraic solutions with free parameters for discrete integrable equations and discuss the properties of regularity and blowing up in ﬁnite time for the solutions.

References

[1] Y. Ohta and J. Yang, General rogue waves in the focusing and defocusing Ablowitz-Ladik equations, J. Phys. A: Math. Theor., 47 (2014), 255201. Darboux integrability of the trapezoidal H4 and H6 equations

G. Gubbiotti1,∗ and R.I. Yamilov2 1 Dipartimento di Matematica e Fisica, Universit`adegli Studi Roma Tre and Sezione INFN di Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy 2 Institute of Mathematics, Ufa Scientiﬁc Center, Russian Academy of Sciences, 112 Chernyshevsky Street, Ufa 450008, Russian Federation [email protected] and [email protected]

Abstract

In this talk we review the basic concepts of Darboux integrability for quad equations [2]. We then prove that the trapezoidal H4 and the H6 families of quad equations, introduced in [1, 3, 4, 5], are Darboux integrable. This results sheds light on the fact that such equations are linearizable as it was suggested using the Algebraic Entropy test [6]. We discuss with an example some consequences of the Darboux integrability.

References

[1] V. E. Adler, A. I. Bobenko, and Y. B. Suris. Discrete nonlinear hyperbolic equations. classiﬁcation of integrable cases. Funct. Anal. Appl., 43 (2009), 3-17,.

[2] V. E. Adler and S.Ya. Startsev. Discrete analogues of the Liouville equation. Theor. Math. Phys., 121 (1999), 1484-1495.

[3] R. Boll. Classiﬁcation of 3D consistent quad-equations. J. Nonlinear Math. Phys., 18 (2011), 337-365.

[4] R. Boll. Classiﬁcation and Lagrangian structure of 3D consistent quad-equations. PhD thesis, TU Berlin, (2012).

[5] R. Boll. Corrigendum classiﬁcation of 3D consistent quad-equations. J. Nonlinear Math. Phys., 19 (2012), 1292001.

[6] G. Gubbiotti, C. Scimiterna, and D. Levi. Algebraic entropy, symmetries and linearization of quad equations consistent on the cube. J. Nonlinear Math. Phys., 23 (2016), 507-543.

[7] G. Gubbiotti, C. Scimiterna, and D. Levi. On partial diﬀerential and diﬀerence equations with symmetries depending on arbitrary functions. Acta Polytechnica, 56 (2016), 193- 201. Spaces of initial conditions for nonautonomous mappings of the plane

Takafumi Mase Graduate School of Mathematical Sciences, the University of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan [email protected]

Abstract

Spaces of initial conditions are one of the most important and powerful tools to analyze mappings of the plane. In this talk, we study the basic properties of general nonautonomous equations that have spaces of initial conditions. We will consider the minimization of spaces of initial conditions for nonautonomous systems and we shall discuss a classiﬁcation of nonautonomous integrable mappings of the plane with a space of initial conditions. Discriminant Separability and Integrability: from the Kowalevski Top to the Quad-graphs

Vladimir Dragovi´c The University of Texas at Dallas 800 West Campbell Road, 75080 Richardson, TX [email protected]

Abstract

We will present the classiﬁcation of discriminantly separable polynomials in three variables and of degree two in each variable. We will discuss connections between such polynomials and some well-known continuous and discrete integrable systems, including the Kowalevski top and the ABS quad-graphs. Some of the results are obtained jointly with Katarina Kuki´c.

References

[1] Dragovi´c, Vladimir, Geometrization and generalization of the Kowalevski top. Comm. Math. Phys. 298 (2010), no. 1, 3764

[2] Dragovi´c, Vladimir, Kuki´c, Katarina, Discriminantly separable polynomials and quad- equations. J. Geom. Mech. 6 (2014), no. 3, 319333

[3] Dragovi´c, Vladimir, Kuki´c, Katarina, Systems of Kowalevski type and discriminantly separable polynomials. Regul. Chaotic Dyn. 19 (2014), no. 2, 162184

[4] Dragović, Vladimir, Kukić, Katarina, The Sokolov case, integrable Kirchhoff elasticae, and genus 2 theta functions via discriminantly separable polynomials. Proc. Steklov Inst. Math. 286 (2014), no. 1, 224239 Irregular conformal blocks and Painlevétau functions

Hajime Nagoya Kanazawa, Japan [email protected]

Abstract

Painlev´etau functions may be expressed as Fourier transforms of some functions: ∑ τ(t) = snG(θ,⃗ ν + n; t). (1) n∈Z

Here, the function G(θ,⃗ ν + n; t) can be computed using the third-order differential equation satisfied by the tau function. In several cases, these functions G(θ,⃗ ν + n; t) are identified with conformal blocks in the two dimensional conformal field theory: series expansions of PVI tau function at regular singular points in [1], series expansions of PV and PIII at the regular singular point 0 in [2] and series expansions of PV and PIV at the irregular singular points ∞ in [3].

We present irregular vertex operators of rank n/2 (n ∈ N) and deﬁne irregular conformal blocks of ramiﬁed type. We identify them with series expansions of the functions G(θ,⃗ ν+n; t) for PIII and PII at the irregular singular point ∞.

References

[1] O. Gamayun, N. Iorgov and O. Lisovyy, Conformal ﬁeld theory of Painlev´e VI, [arXiv:1207.0787]

[2] O. Gamayun, N. Iorgov and O. Lisovyy, How instanton combinatorics solves Painlev´e VI, V and III ’s, [arXiv: 1302.1832]

[3] H. Nagoya, Irregular conformal blocks, with an application to the ﬁfth and fourth Painlev´eequations, J. Math. Phys. 56, 123505 (2015) Mutations of cluster algebras and discrete integrable systems

Atsushi Nobe Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan [email protected]

Abstract

In [1], Okubo presented a direct connection between time evolutions of the discrete Toda lattice and quiver mutations of some cluster algebra. Since the discrete Toda lattice of type (1) AN is arising from the point addition on the hyperelliptic curve as its spectral curve, this fact leads us to a geometric interpretation of the quiver mutation. In particular, for the (1) discrete Toda lattice of type A1 , which is the non-trivial lowest dimensional Toda lattice with periodic boundary, a small quiver consisting of four vertices emulates the time evolution. Thus the quiver mutation is regarded as the point addition P 7→ P + T on the elliptic curve, where T is a fixed point on the curve. Moreover, if we restrict the time evolution alternately, which corresponds to the point addition P 7→ P + 2T on the curve, then we find that all (1) sub-quivers of rank two appearing in the evolutions are of type A1 . (The type of a quiver is defined to be the one of the Cartan matrix of the corresponding anti-symmetric integral (1) matrix. It is known that the cluster algebra associated with the quiver of type A1 is of infinite type, that is, the set of all seeds obtained by applying its mutations to the initial seed is an infinite set.) This observation suggests a direct link between the quiver mutation (1) and the time evolution of the Toda lattice both of which are of type A1 , which enables us (1) to give a geometric interpretation of the quiver mutation of type A1 .

(1) We show that we can realize the mutation of the quiver of type A1 with tropical coeﬃcients as the QRT map by using a variable transformation. We also realize the time evolution (1) of the Toda lattice of type A1 as the QRT map by applying an appropriate birational transformation which simultaneously sends the unit of addition and the adding point on the spectral curve to the ones on the invariant curve of the QRT map, respectively. We then construct a direct connection between the alternate time evolution of the Toda lattice and the (1) quiver mutation both of which are of type A1 by appropriate choices of the parameters and the initial points. This connection gives a geometric interpretation of the quiver mutation (1) of type A1 as a degenerate limit of the point addition on the elliptic curve [2].

References [1] N. Okubo, Discrete Integrable Systems and Cluster Algebras, RIMS Kˆokyˆuroku Bessatsu B41 (2013) 25-42.

(1) [2] A. Nobe, Mutations of the cluster algebra of type A1 and the periodic discrete Toda lattice, J. Phys. A: Math. Theor., 49 (2016), 285201. Non-autonomous discrete hungry integrable systems and asymptotic expansions of their determinant solutions

Masato Shinjo and Yoshimasa Nakamura Graduate School of Informatics, Kyoto University, Yoshida-Hommachi, Sakyo-ku, Kyoto 606-8501, Japan [email protected] and [email protected]

Masashi Iwasaki Faculty of Life and Environmental Science, Kyoto Prefectural University, 1-5, Nakaragi-cho, Shimogamo, Sakyo-ku, Kyoto, 606-8522, Japan [email protected]

Koichi Kondo Graduate School of Science and Engineering, Doshisha University, 1-3 Tatara miyakodani, Kyotanabe, Kyoto 610-0394, Japan [email protected]

Abstract

The discrete hungry Toda equation and discrete hungry Lotka-Volterra system often appear in the studies of discrete hungry integrable systems, and contribute to computing eigenvalues of Hessenberg-type totally nonnegative matrices [1, 2].

In this talk, we comprehensively clarify determinant solutions and Lax pairs associated with non-autonomous discrete hungry integrable systems, and then show asymptotic convergence of the determinant solutions to constants concerning matrix eigenvalues as discrete-time goes to inﬁnity through expanding the determinants.

References

[1] A. Fukuda et al.,Annali di Matematica Pura ed Applicata, 192 (2013), 423–445.

[2] R. Sumikura et al.,AIP Conf. Proc., 1648 (2015), 690006. On a bilateral series solution of the Hahn-Exton q-Bessel type equation

Takeshi MORITA Center for Japanese Language and Culture, Osaka University, Minoh, Osaka 562-0022, Japan [email protected]

Abstract

In this talk, we show the connection formula for the bilateral basic hypergeometric series ν −ν 2ψ1(p , p ; 0; p, x) as follows:

( ) L+ ◦ B+ ψ (pν, p−ν; 0; p, x) (x) q,λ q 2 1 ( ) ν 2ν+1 ν−1 ν+1 ν+1 (p , p , p; p)∞ θp(p λ) θp(p x/λ) 2ν+1 p = 1−ν 2ν 1−2ν 1ϕ1 0; p ; p, (p , p , p ; p)∞ θp(λ/p) θp(px/λ) ( x ) −ν −2ν+1 −ν−1 −ν+1 −ν+1 (p , p , p; p)∞ θp(p λ) θp(p x/λ) −2ν+1 p + 1+ν −2ν 1+2ν 1ϕ1 0; p ; p, . (p , p , p ; p)∞ θp(λ/p) θp(px/λ) x

B+ L+ Here, the notations q and q,λ are the q-Borel-Laplace transformations [1]. The bilateral basic hypergeometric series 2ψ1 is essentially given by the (second) solution of the Hahn- Exton q-Bessel type q-diﬀerence equation [2] [ { } ] 2 − ν −ν − 2 2−ν σp (p + p ) x p σp + 1 y(x) = 0 where σpy(x) = y(px). Since the series 2ψ1 is a divergent series around the origin, we apply the q-Borel-Laplace resummation method to obtain the formula above.

References

[1] T. Morita, The Stokes phenomenon for the q-diﬀerence equation satisﬁed by the Ra- manujan entire function, The Ramanujan Journal, 34 (2014), 329-346, ISSN: 1382-4090 (Print) 1572-9303 (Online)

[2] T. Morita, A connection formula of the Hahn-Exton q-Bessel Function, SIGMA,7 (2011), 115, 11pp. Asymptotic analysis of stochastic cellular automata

Kazushige Endo∗ Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan [email protected]

Abstract

We consider the exact analysis of asymptotic behavior for some stochastic cellular automata (CA) with conserved quantities as nonlinear transport systems. In this talk, we introduce a method to derive fundamental diagrams (FD) of the stochastic CA in order to understand the relation between density and mean ﬂow of the conserved quantities for the asymptotic solution of the stochastic CA. In particular, we explain max-plus algebra, reduction relation, equilibrium equations to derive the FD[1].

References

[1] K. Endo, D. Takahashi, J. Matsukidaira, On fundamental diagram of stochastic cellular automata with a quadratic conserved quantity, NOLTA, Vol.7 No.3 (2016), 313-323. Plane partitions and the discrete two-dimensional Toda molecule

Shuhei Kamioka Graduate School of Informatics, Kyoto University Yoshida-hommachi, Sakyo-ku, Kyoto 606-8501, Japan [email protected]

Abstract

A plane partition is a two-dimensional array of nonnegative integers such that its support is ﬁnite and each of its rows and columns are weakly decreasing. Plane partitions are combinatorial objects of great importance for which product formulae exist for generating functions such as the triple product formula by MacMahon [4] and the trace generating functions by Stanley [5] and Gansner [1].

In this talk a close connection between plane partitions and an integrable dynamical system, the discrete two-dimensional Toda molecule [2], is discussed. In particular it is shown that several well-known product formulae including MacMahon’s formula and the trace generating functions and unknown product formulae which generalize the known ones can be derived from solutions to the discrete 2D Toda molecule [3].

References

[1] E. R. Gansner, The Hillman-Grassl correspondence and the enumeration of reverse plane partitions, J. Combin. Theory Ser. A 30 (1981), 71–89.

[2] R. Hirota, S. Tsujimoto, and T. Imai, Diﬀerence scheme of soliton equations, State of the art and perspectives of studies on nonlinear integrable systems (Japanese) (Kyoto, 1992), S¯urikaisekikenky¯ushoK¯oky¯uroku,822, 1993, pp. 144–152.

[3] S. Kamioka, Multiplicative partition functions for reverse plane partitions derived from an integrable dynamical system, arXiv:1701.06762 [math.CO].

[4] P. A. MacMahon, Combinatory analysis, volume 2, Cambridge University Press, Cam- bridge, 1916.

[5] R. P. Stanley, The conjugate trace and trace of a plane partition, J. Combinatorial Theory Ser. A 14 (1973), 53–65. SESSION 8

Analytical and computational methods to study nonlinear partial diﬀerential equations

Unal Goktas Turgut Ozal University, Ankara, Turkey [email protected]

Muhammad Usman Department of Mathematics, University of Dayton, Dayton OH 45469-2316, USA [email protected]

Abstract

In general, Nonlinear Partial Differential Equations (NPDEs) cannot be solved analytically, in order to predict the behavior of the system these models must be solved numerically. There has been significant progress in the development of numerical methods, including algorithms implementation on GPUs. Most well-known methods include finite difference methods, col- location methods and finite element methods. While testing numerical techniques, when exact solutions of initial and boundary value problem of NPDEs are not available, conservation laws play an important role. This session is dedicated to study the dynamical system obtained by the steady state solutions of NPDEs, Homotopy Perturbation Method (HPM), conservation laws and numerical techniques for NPDEs. A GPU Implementation of Central Schemes for Two-Phase Flows

Felipe Pereira Mathematical Sciences Department University of Texas at Dallas, Richardson, Texas [email protected]

Arunasalam Rahunanthan∗ Department of Mathematics and Computer Science Central State University, Wilberforce. Ohio [email protected]

Abstract

We consider a two-phase (water and oil) model in heterogeneous porous media. In solving the hyperbolic problem associated with such a model, one could use a high-resolution, non-oscillatory central scheme [2]. In this talk we describe a parallelization on a Graphics Processing Unit (GPU) for the central scheme in the simulation of two-phase ﬂows in three space dimensions [1]. The speedup using a GPU (with respect to the time needed to solve the same problem on a CPU) can be up to 60, indicating that our procedure makes an eﬀective use of the hardware.

References

[1] Pereira, F., Rahunanthan, A.: Numerical simulation of two-phase ﬂows on a GPU. In: 9th International meeting on High Performance Computing for Computational Science (VECPAR ’10). Berkeley, CA (2010)

[2] F. Pereira and A. Rahunanthan, A semi-discrete central scheme for the approximation of two-phase ﬂows in three space dimensions, Mathematics and Computers in Simulation, 81(10) (2011), 2296-2306. A Study of Bifurcation Parameters in Travelling Wave Solutions of a Damped Forced Korteweg de Vries-Kuramoto Sivashinsky Type Equation

Muhammad Usman∗, Chi Zhang, Youssef Raﬀoul University of Dayton, 300 College Park, Dayton OH 45469-2316, USA [email protected]

Mudassar Imran Department of Mathematics and Natural Sciences Gulf University of Science and Technology, Mishref, Kuwait

Abstract

We consider the damped externally excited Korteweg de Vries-Kuramoto Sivashinsky (KdV- KS) type equation and use an asymptotic perturbation method to analyze the stability of the traveling wave solutions. We consider the primary resonance defined by the detuning parameter. External-excitation and frequency-response curves are shown to exhibit jump and hysteresis phenomena (discontinuous transitions between two stable solutions) for the KdV-KS type equation. Homotopy Perturbation Method for solution of nonlinear partial differential equation in MHD Jeffery-Hamel flows

Iftikhar Ahmad∗, Hira Ilyas Department of Mathematics, University of Gujrat, Gujrat, Pakistan. [email protected]∗, [email protected]

Abstract

In this study, Homotopy Perturbation Method (HPM) is employed to find solution of nonlinear MHD Jeffery-Hamel arterial blood flow problem. Primarily, two-dimensional nonlinear Navier-Stoke equations are transformed into one-dimensional third order equation, later the governing equation is applied to construct Homotopy Perturbation Method. The proposed numerical results show a good agreement with reference solution for finite interval and em- phasizes to understand the human arterial blood flow rate and find out the effects of changing of Reynolds Numbers and angles between plates on blood flow rate in human artery. Further, we increased the iteration process up to 3rd order to ensure the accuracy and reliability of proposed method. Finally, it describes the effect of product of and Re on flow rate that is, as product increases the flow rate of MHD Jeffery-Hamel also increases. Conservation Laws and Exact Solutions of Generalized Kompaneets and Nizhnik-Novikov-Veselov Equations

I. Naeem Department of Mathematics, School of Science and Engineering, LUMS, Lahore Cantt 54792, Pakistan [email protected]

Abstract

The conservation laws for the generalized Kompaneets equations with one functional parameter are computed via partial Noether’s approach. A complete classification for functional parameter is presented and for each case the conservation laws are derived. Then the Lie point symmetries are calculated and the association between symmetries and conserved vectors are established using symmetries conservation laws relationship. A number of exact solutions for the nonlinear Generalized Kompaneets equations are obtained in each case. A similar analysis is performed for the generalized Nizhnik-Novikov-Veselov equation. Numerical analysis of an anisotropic phase-field model in the presence of magnetic-field

Amer Rasheed Department of Mathematics, Lahore University of Management Sciences, Opposite Sector U, DHA,Lahore Cantt 54792, Pakistan [email protected]

Abstract

A numerical scheme is proposed and numerical analysis of anisotropic phase-field model of binary mixtures under the influence of magnetic field in two-dimensional geometry is performed. Precisely, the numerical stability and error analysis of this approximation scheme which is based on mixed finite-element method are performed. An application of a nickel- copper binary alloy is considered. The study substantiates a good agreement between the numerical and theoretical results, and demonstrates the efficiency of the presented method. Nonlocal conservation laws of boundary layer equations on the Surface of a Sphere

Rehana Naz Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, 53200, Pakistan [email protected]

Abstract

The system of partial diﬀerential equations (PDEs) can be analyzed by its nonlocally related systems. It plays essential role in constructing solutions of given system of PDEs as the nonlocally related systems and subsystems yield same set of solutions as the given system [1, 2]. The potential systems and nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere have been investigated. The multiplier approach yields two local conservation laws for the Prandtl boundary layer equations on the surface of a sphere. Two potential variables ψ and φ are introduced corresponding to ﬁrst and second conservation law. Moreover, another potential variable p is introduced by considering the linear combination of both conservation laws. Two level one potential systems involving a single nonlocal variable ψ or φ are constructed. One level two potential system involving both nonlocal variables ψ and φ is established. The nonlocal variable p is utilized to derive a spectral potential system. The nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere are derived by computing the local conservation laws of its potential systems. The nonlocal conservation laws are utilized to derive the further nonlocally related systems.

References

[1] Bluman, G., Cheviakov, A. F. (2005). Framework for potential systems and nonlocal symmetries: Algorithmic approach. Journal of mathematical physics, 46(12), 123506.

[2] Bluman, G., Cheviakov, A. F., Ivanova, N. M. (2006). Framework for nonlocally related partial diﬀerential equation systems and nonlocal symmetries: Extension, simpliﬁcation, and examples. Journal of mathematical physics, 47(11), 113505. SESSION 9

Nonlinear Schr¨odinger and Related Models: a Session in Honor of Boris Malomed

R. Carretero-Gonz´alez Nonlinear Dynamical Systems Group, San Diego State University [email protected]

P.G. Kevrekidis Department of Mathematics and Statistics, University of Massachusetts [email protected]

Abstract

This session is intended to celebrate Boris Malomed’s contribution to the nonlinear community. His drive, motivation and throughput have driven and inspired a broad and diverse array of research areas and their applications. The session will focus on nonlinear and related models.

The nonlinear Schr¨odinger (NLS) equation describes a very large variety of physical systems since it is the lowest order nonlinear (cubic) partial diﬀerential equation that describes the propagation of modulated waves. For example, two of the most salient applications of NLS equations, that will be the main theme of this session, stem from the realm on nonlinear optics and Bose-Einstein condensates (BECs). In addition, in recent years, numerous variants of these themes both Hamiltonian ones such as spin-orbit coupling, Bose-Fermi mixtures, dipolar systems with long-range interactions, as well as ones bearing gain-loss such as PT-symmetric media, and exciton-polariton condensates have opened novel avenues of exploration of NLS themes.

The aim of this session is to bring together experts, as well as young researchers working in the theory, the numerical simulation and the experimental study of nonlinear Schr¨odinger and related equations and its applications. The focus is to establish a fruitful discussion of the current state-of-the-art and an examination of future challenges and directions of interest. This should be a session appealing to theoretical physicists, experimental physicists and applied mathematicians alike and will be a vehicle for the exchange of ideas that could cross- fertilize diﬀerent disciplines and spurt the initiation of new collaborations that could address some of the pertinent open problems. Along this path, we will also have ample opportunity to remember Boris’ numerous substantial contributions and celebrate his career. Vortex solitons in spin-orbit coupled Bose Einstein condensates Hidetsugu Sakaguchi Department of Applied Science for Electronics and Materials, Kyushu University, Fukuoka, Japan

The Gross-Pitaevskii (GP) equations have been intensively studied for Bose-Einsten condensates (BECs) with weak inter-atomic interaction. Recently, the spin-orbit (SO) coupling was experimentally realized in Bose-Einstein condensates. The coupled GP equations with spin-orbit coupling terms are used for the SO-BECs. Various states such as a semi-vortex state and one-dimensional solitons have been studied in the coupled GP equations. We found a semi-vortex state and a mixed-mode state in the coupled GP equations with attractive interaction even if the confinement harmonic potential is absent. They are two-dimensionally localized state with vorticity and are called vortex solitons. The vortex solitons can move although there is a critical velocity. We furthermore investigated the coupled GP equations including several other terms. For example, we studied the influence of the Zeeman effect to the vortex solitons. There is a transition from the mixed mode state to the semi-vortex state when the Zeeman-splitting strength is increased. Under a rotating harmonic potential, a vortex state with multi quantum vorticity has a lower energy than the semi-vortex state, if the inter-atomic interaction is absent. If the interaction is repulsive, a vortex lattice around the multi quantum vortex appears. If the interaction is attractive, a crescent state appears at a position distant from the origin. Enhanced fractal dynamics of a BEC induced by dipolar interactions

Jessica R. Taylor∗ and Boaz Ilan Univerisity of California, Merced 5200 N. Lake Road, Merced, CA 95340 [email protected] & [email protected]

Kevin A. Mitchell Univerisity of California, Merced 5200 N. Lake Road, Merced, CA 95340 [email protected]

Abstract

Bose-Einstein condensation (BEC) is arguably one of the most fundamental quantum mechanical structures in condensed matter physics. Of current interest is the ability to stabilize a BEC. Earlier studies show that various configurations of the Nonlinear Schrödingerequa- tion effect the flux / escape rate of a BEC. We extend these studies to the dipolar NLS equation. Here, we consider the effects of a defocusing dipolar potential on the fractal escape-rates of a BEC relative to a dipolar ground state initial condition. We conclude that the addition of dipolar effects reduces the dispersion of the system and arrests collapse of representative wavefunction. Intermittent Many-Body Dynamics at Equilibrium

Sergej Flach Center for Theoretical Physics of Complex Systems Institute for Basic Science, Daejeon, South Korea sﬂ[email protected]

Abstract

The equilibrium value of an observable defines a manifold in the phase space of an ergodic and equipartitioned many-body system. A typical trajectory pierces that manifold infinitely often as time goes to infinity. We use these piercings to measure both the relaxation time of the lowest frequency eigenmode of the Fermi-Pasta-Ulam chain, as well as the fluctuations of the subsequent dynamics in equilibrium. We show that previously obtained scaling laws for equipartition times are modified at low energy density due to an unexpected slowing down of the relaxation. The dynamics in equilibrium is characterized by a power-law distribution of excursion times far off equilibrium, with diverging variance. The long excursions arise from sticky dynamics close to regular orbits in the phase space. Our method is generalizable to large classes of many-body systems.

References

[1] C. Danieli, D. K. Campbell and S. Flach, Intermittent FPU dynamics at equilibrium, arXiv:1611.00434 (2016). Two-dimensional dipolar gap solitons in free space with spin-orbit coupling

Yongyao Li, Yan Liu, and Zhiwei Fan College of Electronic Engineering, South China Agricultural University Guangzhou 510642, China

Wei Pang Department of Experiment Teaching, Guangdong University of Technology Guangzhou 510006, China

Shenhe Fu Department of Optoelectronic Engineering, Jinan University Guangzhou 510632, China

Boris A. Malomed Department of Physical Electronics, School of Engineering, Faculty of Engineering Tel Aviv University, Tel Aviv 69978, Israel email: malomedpost.tau.ac.il

ABSTRACT

We present gap solitons (GSs) that can be created in free two-dimensional (2D) space in two-component dipolar Bose-Einstein condensates with the spin-orbit coupling (SOC). In the limit of strong SOC, the kinetic-energy terms in the respective coupled Gross-Pitaevskii equations may be omitted, which gives rise to a bandgap in the system’s spectrum, in the presence of the Zeeman splitting between the components [1]. Stable isotropic and anisotropic 2D solitons of the semi-vortex type (with vorticities 0 and 1 in the two components) are found, by means of analytical and numerical methods, when the dipoles are polarized perpendicular and parallel to the 2D plane, respectively. The GS families extend, in the form of embedded solitons (ESs), into spectral bands, a part of the ES branch being stable for isotropic solitons. Mobility and collision of the solitons are studied too, revealing negative and positive eﬀective masses of the isotropic and anisotropic GSs, respectively.

References

[1] H. Sakaguchi and B. A. Malomed, One- and two-dimensional solitons in PT-symmetric systems emulating spin–orbit coupling, New J. Phys., 18 (2016), 105005. A PT -symmetric necklace of optical waveguides with a gain and loss ombr´e

Igor Barashenkov and Gwyneth Allwright Department of Mathematics, University of Cape Town, Rondebosch 7701, South Africa [email protected] and [email protected]

We consider a ring-like PT -symmetric necklace of 2N coupled optical waveguides with the clockwise gain-loss variation. As the necklace is traced quarter-way around, the waveguide’s loss coefficient decreases from its maximum value γ to zero and switches to gain. The gain coefficient grows to its maximum value of γ at the half-way point around the necklace and then starts decreasing to become loss again, with its maximum value reached at the starting waveguide. We show that unlike necklaces with alternating or clustered fixed-value gain and loss [1], the symmetry-breaking threshold γc in this system does not tend to zero as N → ∞. Despite the complex structure of the ombrénecklace, the system is found to be exactly solvable and the value of γc(N) is available in closed form:

1/N γc(N) = 2 .

References

[1] I. V. Barashenkov, L. Baker, and N. V. Alexeeva PRA 82 (2010) 043803 Nonlinear Excitations in Lattices with Long Range Interactions

Christopher Chong Bowdoin College [email protected]

Abstract

Spatially localized solutions of lattices with interactions beyond nearest neighbor are considered. Two prototypical cases are examined: (i) In the context of a discrete nonlinear Schrodinger equation with next-nearest-neighbors, discrete solitary waves are studied via numerical computations and a variational approximation, revealing that solutions with a non-trivial phase are possible (ii) Breathers are studied in a lattice of repelling magnets. The long range interaction of this system allows the possibility of breathers with spatial decay of tails that transition from exponential to algebraic. The breathers are studied both numerically and experimentally, yielding qualitative agreement. Surface Breathers in Graphene

G. T. Adamashvili Technical University of Georgia, Kostava str.77, Tbilisi, 0179, Georgia email: guram−[email protected]

D. J. Kaup∗ Department of Mathematics & Institute for Simulation and Training University of Central Florida, Orlando, Florida, 32816-1364, USA email: [email protected]

Abstract

Graphene is a single atomic layer of carbon, molded into a two-dimensional (2D) honeycomb lattice. It is the first experimental attempt to construct 2D atomic crystals, whose features are significantly different from 3D crystals. Graphene and other similar materials have some very unique optical properties and are being extensively investigated as the next generation material in nano-optics and nano-electronics applications. SPP (surface plasmon polariton) is a surface optical wave which undergoes a strong enhancement and spatial confinement of its wave power near an interface of 2D layered structures. Its amplitude has a maximum at the interface and decays exponentially away from the interface. Graphene and other two-dimensional graphene-like materials are recognized as promising materials for potential future applications of SPPs.

We have developed a theory of optical breathers of the self-induced transparency (SIT) for SPP waves. Starting from the optical nonlinear wave equation for surface TM-modes interacting with a two-dimensional layer of atomic systems (or semiconductor quantum dots), under a graphene monolayer (or two-dimensional graphene-like material), we have obtained the evolution equations for the electric field. Due to the presence of the doped atoms (or semiconductor quantum dots), we obtain the conditions for SIT solitons (McCall Hahn soliton and breather)to occur, given sufficient intensity. Proceeding forward from this, we consider the evolution when the area of a pulse is O(), with its scale-length being of order O(−1). In this case, one finds that the evolution of these pulses become described by the damped nonlinear Schrödinger equation instead. For small intensity SPP fields, breathers are found to occur, although they will undergo damping. Explicit relations of the dependence of breathers on the local media, graphene conductivity, transition layer properties and transverse structures of the SPP, are obtained and will be given. Nonlinear quantum optics via highly nonlocal interactions

Gershon Kurizki Weizmann Institute of Science [email protected]

Abstract

Nonlinear optical phenomena are typically local. We predict the possibility of highly nonlocal optical nonlinearities for light propagating in atomic media trapped near a nano-waveguide, where long-range interactions between the atoms can be tailored. When the atoms are in an electromagnetically-induced transparency configuration, the atomic interactions are translated to long-range interactions between photons and thus to highly nonlocal optical nonlinearities. We derive and analyze the governing nonlinear propagation equation, finding a roton-like excitation spectrum for light and the emergence of order in its output intensity. For atoms coupled to a waveguide with a bandgap spectrum illuminated by an off-resonant laser, the resulting dynamics of the atoms is predominantly affected by an extremely long- range conservative force that can enhance their interaction. Even more dramatic, giant, enhancement of the interaction is achievable via the control of the geometry, for dipolar forces induced by the electromagnetic vacuum, namely, the Casimir and van der Waals (vdW) forces. The idea is to consider atoms coupled to an electric transmission line (TL), such as a coaxial cable or coplanar waveguide, which support the propagation of quasi- 1d transverse electromagnetic (TEM) modes. Then, virtual excitations (photons) of these extended modes can mediate much stronger and longer-range Casimir and vdW forces than in free-space. These predictions open the door to studies of unexplored wave dynamics and many-body physics with highly-nonlocal interactions of optical fields in one dimension.

References

[1] E. Shahmoon et al., Optica 3, 725 (2016); Phys. Rev. A 89, 043419 (2014); Phys. Rev. A 87, 03383 (2013); PNAS 111, 10485 (2014). Vortex Rings in Bose-Einstein Condensates

Ricardo Carretero-Gonz´alez San Diego State University [email protected]

Abstract

We review recent results for the emergence, existence, dynamics and interactions of vortex rings in Bose-Einstein condensates (BECs) modelled by the Gross-Pitaevskii equation (GPE). We focus our attention on the two opposite regimes of low and high atomic density limits in the BEC as well as in the intermediate transition between these two limits.

In the low density limit, corresponding to the linear limit, we study the emergence of single and multiple vortex rings emanating from planar 3D dark solitons through bifurcations. We characterize such bifurcations quantitatively using a Galerkin-type approach, and ﬁnd good qualitative and quantitative agreement with our Bogoliubov-de Gennes (BdG) numerical analysis. Under appropriate conditions for the trapping strengths, we ﬁnd that vortex rings might be stabilized for large enough atomic densities (large chemical potentials).

On the other hand, in the large density limit, the vortex rings acquire stability and are effectively robust coherent structures. We study different single and multi-vortex-ring configurations together with their (normal) modes of vibration. Exotic structures such as Hop- fions, the one-component counterpart to Skyrmions, are also constructed and tested for stability. Finally, we discuss some interactions dynamics between vortex rings such as periodic leapfrogging of co-axial vortex rings and the scattering behavior for co-planar collisions between vortex rings. SESSION 11

Traveling waves and spectral theory

Yuri Latushkin Department of Mathematics, University of Missouri Columbia, MO 65211 [email protected]

Sam Walsh Department of Mathematics, University of Missouri Columbia, MO 65211 [email protected]

Abstract

The purpose of this special session is to bring together researchers working on various stability issues for such special solutions of partial diﬀerential equations as periodic and solitary waves and water waves. All aspects of stability/instability will be discussed, from spectral to nonlinear, with special emphasis on methods of spectral theory. A special attention will be given to geometric methods, in particular, involving the Maslov index, a topological invariant that recently made its way in stability theory for traveling waves.

It is expected that the speakers will spend some time of their talks to address possible perspectives in the field of their work as we believe that such a perspective would be not only interesting for the audience but it can also stimulate further discussion and further research in the field. Nondegeneracy of Antiperiodic Standing Waves for Fractional Nonlinear SchrödingerEquations

Kyle M. Claassen and Mathew A. Johnson Department of Mathematics, University of Kansas, 1460 Jayhawk Boulevard, Lawrence, KS 66045 [email protected]

ABSTRACT

In the stability and blowup analyses for traveling and standing waves in nonlinear Hamil- tonian dispersive equations, the nondegeneracy of the linearization about such waves is of paramount importance. That is, one must verify that the kernel of the second variation of the Hamiltonian is generated by the continuous symmetries of the PDE. The proof of this property can be far from trivial, especially when the dispersion admits a nonlocal description where shooting arguments, Sturm-Liouville theories, and other ODE methods may not be applicable. In this talk, we discuss the nondegeneracy of the linearization associated with antiperiodic constrained energy minimizers in a class of defocusing NLS equations having fractional dispersion. Key to our analysis is the development of ground state and oscillation theories for linear periodic Schrödingeroperators with antiperiodic boundary conditions. The antiperiodic nature of the problem greatly complicates the analysis, as linear Schrödinger operators with periodic potentials need not have simple antiperiodic ground states even in the classical (local) case. As an application, we obtain the nonlinear (orbital) stability of antiperiodic standing waves with respect to antiperiodic perturbations. The Maslov and Morse indices for Schrödingeroperators on R

Peter Howard∗ Mathematics Department,Texas A&M University, College Station, TX 77843, USA [email protected]

Yuri Latushkin Mathematics Department, University of Missouri, Columbia, MO 65211, USA [email protected]

Alim Sukhtayev Mathematics Department, Indiana University, Bloomington, IN 47405, USA [email protected]

Abstract

Assuming a symmetric matrix-valued potential that approaches constant endstates with a suﬃcient asymptotic rate, we relate the Maslov and Morse indices for Schr¨odinger operators on R. In particular, we show that with our choice of convention, the Morse index is precisely the negative of the Maslov index. The Maslov index and the stability of traveling waves

Paul Cornwell∗ and Christopher K.R.T. Jones University of North Carolina at Chapel Hill [email protected] and [email protected]

Abstract

The Maslov index is a topological invariant assigned to curves of Lagrangian subspaces of a symplectic vector space. It has been used to locate positive eigenvalues for self-adjoint operators deﬁned on various domains. In particular, it is a valuable tool in the stability analysis of standing wave solutions for reaction-diﬀusion equations with gradient nonlinearity.

Although many results exist relating the Maslov index to the Morse index of linear operators, two major obstacles hamper the application of the index to stability analysis. First, the class of problems amenable to this analysis–namely, reaction diﬀusion equations with gradient nonlinearity–is quite restrictive. Second, actually calculating the Maslov index using geometric features of the wave itself has remained a signiﬁcant challenge.

Using a doubly-diﬀusive FitzHugh-Nagumo model as an example, we aim to address both of these issues. As the quintessential activator-inhibitor system, this equation is certainly not of the form typically analyzed using the Maslov index. In particular, the Maslov index has not been used to study traveling waves of this system. Furthermore, there is a timescale separation in the equations that allows us to compute the Maslov index. Using this calculation, we show that fast traveling waves are stable. Absolute Instability for Travelling Waves in a Chemotaxis Model

Paige Davis and Peter van Heijster Queensland University of Technology [email protected] and [email protected]

Robby Marangell∗ University of Sydney [email protected]

Abstract

In the 1970’s Keller and Segel introduced a class of models for bacterial chemotactic motion through a consumable substrate. In general these models exhibit travelling wave solutions. I will discuss the spectral stability of such travelling wave solutions with a logarithmic chemosensitivity function and a constant consumption rate. Linearising around the travelling wave solutions, one locates the essential and absolute spectrum of the associated linear operators and ﬁnds that all travelling wave solutions have essential spectrum in the right half plane. However, there exists a range of parameters such that the absolute spectrum is contained in the open left half plane and the essential spectrum can thus be weighted into the open left half plane. A critical parameter value is determinede for which the absolute spectrum crosses into the right half plane, indicating the onset of an absolute instability of the travelling wave solution. This crossing always occurs oﬀ of the real axis. If there is time, I will discuss the point spectrum for such a model. Traveling Waves in Diatomic Fermi-Pasta-Ulam-Tsingou lattices.

J. Douglas Wright∗ Drexel University, Philadelphia PA [email protected]

Abstract

Consider an infinite chain of masses, each connected to its nearest neighbors by a nonlinear spring. This is an FPUT lattice. In the instance where the masses are identical, there is a well-developed theory on the existence, dynamics and stability of solitary waves and the system has come to be one of the paradigmatic examples of a dispersive nonlinear equation. In this talk, I will discuss recent rigorous results of mine (together with T. Faver, A. Hoffman, R. Perline, A. Vainchstein and Y. Starosvetsky) on the existence of traveling waves in the setting where the masses alternate in size. In particular I will address in the limit where the mass ratio tends to zero. The problem is inherently singular and as such the existence theory becomes rather complicated. In particular, we find that the traveling waves are not true solitary waves but rather “nanopterons”, which is to say, waves which asymptotic at spatial infinity to very small amplitude periodic waves. Moreover, we can only find solutions when the mass ratio lies in a certain open set. The difficulties in the problem all revolve around understanding Jost solutions of a nonlocal Schrodinger operator in its semi-classical limit. Break waves and solitary waves to the rotation-two-component Camassa-Holm system

Ming Chen∗ Department of Mathematics, University of Pittsburgh, PA 15260, USA [email protected]

Lili Fan College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China [email protected]

Hongjun Gao School of Mathematical Sciences, Institute of Mathematics, Nanjing Normal University, Nanjing 210023, China [email protected]

Yue Liu Department of Mathematics, University of Texas, Arlington, TX 76019, USA [email protected]

Abstract

The rotation-two-component Camassa-Holm system models the equatorial water waves with the eﬀect of the Coriolis force [2]. The quasi-nonlinear nature of the system suggests that smooth initial data may develop ﬁnite time singularity, giving rise to the so-call wavebreaking phenomenon. On the other hand, the balance between the nonlinearity and dispersion makes it possible to support solitary waves, sometimes even with singularities. In this talk I will discuss these two types of solutions.

References

[1] L. L. Fan, H. J. Gao and Y. Liu, On the rotation-two-component Camassa-Holm system modelling the equatorial water waves, Adv. Math., 291 (2016), 59-89. Diﬀusive stability of spatially periodic patterns

Alim Sukhtayev and Kevin Zumbrun Indiana University [email protected] and [email protected]

Soyeun Jung Kongju National University [email protected]

Raghavendra Venkatraman Indiana University [email protected]

Abstract

Applying the Lyapunov–Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth–order scalar Swift–Hohenberg equation, we carry out a rigorous small–amplitude stability analysis of Turing patterns for the canonical second–order system of reaction diffusion equations given by the Brusselator model, and for the model for pattern formation with a conservation law. Our results confirm that stability is accurately predicted in the small–amplitude limit by the formal Ginzburg–Landau amplitude equations for the Brusselator model, and by the formal modified Ginzburg-Landau system (mGL) consisting of a coupled Ginzburg–Landau equation and mean mode equation for the model with a conservation law, rigorously validating the standard weakly unstable approximation. Spectral stability of solutions to the Vortex Filament Hierarchy

St´ephaneLafortune and Thomas Ivey Department of Mathematics College of Charleston 66 George Street, Charleston South Carolina, 29424, U.S. [email protected] and [email protected]

Abstract

The Vortex Filament Equation (VFE) is part of an integrable hierarchy of filament equations. Several equations in this hierarchy have been derived to describe vortex filaments in various situations. Inspired by these results, we develop a general framework for studying the existence and the linear stability of closed solutions of the VFE hierarchy. The framework is based on the correspondence between the VFE and the nonlinear Schrödinger(NLS) hierarchies. Our results establish a connection between the AKNS Floquet spectrum and the stability properties of the solutions of the filament equations. We apply our machinery to solutions of the filament equation associated to the Hirota equation. We also discuss how our framework applies to soliton solutions. The Maslov index and the spectra of second order elliptic operators

Yuri Latushkin and Selim Sukhtaiev Department of Mathematics University of Missouri Columbia, MO 65211, USA [email protected] and [email protected]

Abstract

In this talk I will discuss a formula relating the spectral flow of the one-parameter families of second order elliptic operators to the Maslov index, the topological invariant counting the signed number of conjugate points of paths of Lagrangian planes in H1/2(∂Ω) × H−1/2(Ω). In addition, I will present a formula relating the Morse index, the number of negative eigenvalues, to the Maslov index for several classes of the second order operators: the θ~−periodic Schrödingeroperators on a period cell Q ⊂ Rn, the elliptic operators with Robin-type boundary conditions, and the abstract self-adjoint extensions of the Schrödingeroperators on star-shaped domains. Global bifurcation of rotating vortex patches

Zineb Hassainia, Nader Masmoudi, and Miles H. Wheeler∗ Courant Institute of Mathematical Sciences New York University [email protected], [email protected], [email protected]

Abstract

Rotating vortex patch solutions of the two-dimensional Euler equations, also known as V - states, have been rigorously constructed [1] as perturbations of circular patches. For every m ≥ 3, there is a curve of nearly-circular patches with the symmetry of a regular m-gon. In this talk we prove that these local bifurcation curves can be extended to global ones. At the end of each global curve, the minimum value of the angular ﬂuid velocity along the boundary of the patch becomes arbitrarily small. This is consistent with the formation of sharp corners which is observed numerically [2].

References

[1] T. Hmidi, J. Mateu, and J. Verdera, Boundary regularity of rotating vortex patches. Arch. Ration. Mech. Anal., 209 (2013), 171–208.

[2] H. M. Wu, E. A. Overman II, and N. J. Zabusky, Steady-state solutions of the Euler equations in two dimensions: rotating and translating V -states with limiting cases. I. Numerical algorithms and results. J. Comput. Phys., 53 (1984), 42–71. Traveling waves for the mass-in-mass model of granular chains

Atanas Stefanov, Panos Kevrekidis University of Kansas UMASS-Amherst [email protected] and [email protected]

Haitao Xu University of Minnesota [email protected]

Abstract

We consider the mass in mass (or mass with mass) system of granular chains, namely a granular chain involving additionally an internal resonator, the prototypical model being

∂ X = [(X − X )p − (X − X )p ] + k˜(x − X ) tt i i−1 i + i i+1 + i i ˜ ν∂ttxi = −k(xi − Xi)

For these chains, we rigorously establish that under suitable “anti-resonance” conditions connecting the mass of the resonator and the speed of the wave, bell-shaped traveling wave solutions continue to exist in the system, in a way reminiscent of the results proven for the standard granular chain of elastic Hertzian contacts.

References

[1] P.G. Kevrekidis, A. Stefanov, H. Xu, Traveling waves for the Mass in Mass model of granular chains, Lett. Math. Phys. 106 (2016), no. 8, p. 1067–1088. Global bifurcation of gravity water waves with multiple critical layers

Kristoﬀer Varholm Department of Mathematical Sciences Norwegian University of Science and Technology kristoﬀ[email protected]

Abstract

We establish the existence of global curves of steady periodic gravity water waves with analytic vorticity distributions, extending curves from previous results for small-amplitude waves [1, 3, 4]. For this, we employ the analytic global bifurcation theory due to Dancer, Buﬀoni and Toland [2]. The formulation used allows for waves with interior stagnation and an arbitrary number of critical layers, at least suﬃciently close to the bifurcation point. This is a work in progress.

References

[1] A. Aasen and K. Varholm, Traveling gravity water waves with critical layers, to appear in J. Math. Fluid Mech.

[2] B. Buﬀoni and J. F. Toland, Analytic theory of global bifurcation, Princeton University Press, Princeton, 2003.

[3] M. Ehrnstr¨om,J. Escher and E. Wahl´en,Steady water waves with multiple critical layers, SIAM J. Math. Anal., 43 (2011), no. 3, 1436–1456.

[4] V. Kozlov and N. Kuznetsov, Dispersion equation for water waves with vorticity and Stokes waves on ﬂows with counter-currents, Arch. Ration. Mech. Anal., 214 (2014), no. 3, 971–1018. Modulational Stability of Quasiperiodic Traveling Waves

Jared C. Bronski University of Illinois Second line of the address, if necessary [email protected]

Mat Johnson University of Kansas [email protected]

Robert Marangell University of Sydney [email protected]

Abstract

In this paper we compute a rigorous normal form for the linearization of the nonlinear Schr¨odingerequation 2 iψt = ψxxV (|ψ| )ψ about a quasi-periodic traveling wave in a neighborhood of the origin in the spectral λ plane. The quasi-periodic solutions are critical points of the Hamiltonian subject to the constraints of constant mass, momentum and quasi-momentum, and the stability of these solutions can be expressed in terms of the derivatives of these quatitities with respect to the associated Lagrange multipliers. A dynamical approach to semilinear elliptic equations

Graham Cox∗ Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, NL, Canada [email protected]

Abstract

It is well known that the ODE uxx = F (u) can be reduced to a first-order system in the variables (u, ux). The same reduction has been carried out for PDE when the spatial domain is a cylinder or is radially symmetric. We describe a similar reduction for the semilinear elliptic equation ∆u + g(u) = 0 on a domain Ω ⊂ Rn with arbitrary geometry. It is shown that as Ω is deformed through a smooth family {Ωt}, the boundary data of u on ∂Ωt satisfies an ill-posed first-order system of equations. This system possesses Hamiltonian structure when the elliptic equation is selfadjoint.

The linearized system admits an exponential dichotomy if Ω is deformed to a point in a suﬃ- ciently regular manner, with the unstable subspace at time t corresponding to the boundary data of weak solutions to the PDE on Ωt. Whenu ¯ is a steady state, with ∆¯u + g(¯u) = 0, this construction provides a geometric connection between the stateu ¯ and the unstable subspaces for the ﬁrst-order system corresponding to the linearized operator L = ∆ + g0(¯u). This family of unstable subspaces is precisely what one needs to evaluate the Maslov index ofu ¯ (and thus evaluate its stability).

This is joint work with Margaret Beck (Boston University), Christopher Jones (UNC Chapel Hill), Yuri Latushkin (University of Missouri) and Alim Sukhtayev (Indiana University). SESSION 12

Wave phenomena in combustion

Anna Ghazaryan Department of Mathematics Miami University, 301 S. Patterson Ave Oxford, OH 45056, USA, Ph. 1-513-529-0582 [email protected]

St´ephaneLafortune Department of Mathematics College of Charleston Charleston, SC 29424 [email protected]

Abstract

This mini-symposium will bring together researchers who study fronts, pulses, wave trains and patterns of more complex structure which occur in combustion phenomena. In mathematics, these objects are realized as solutions of nonlinear partial differential equations. Specifically, the mini-symposium will focus on problems of existence, stability, dynamic properties, and bifurcations of those solutions. The techniques used will be both analytical and numerical in nature. Gelfand type problem for co-flow laminar jets

Peter V. Gordon Department of Mathematics The University of Akron Akron, OH 44325 [email protected]

Abstract

In this talk I will discuss a model for auto-ignition of laminar co-flow jets. Such jets consist of two parts: an inner part with oxidizer that is surrounded by an outer part with fuel or reverse. The derivation of the model is based on combination of Burke-Schumann theory of diffusion flames and Semenov-Frank-Kamenerskii theory of thermal explosion. The main advantage of this model is that it gives a sharp characterization for auto-ignition of a jet as blow up of solution of underlying PDE. This model falls into a general class of Gelfand type problems which were studied in mathematical literature since early 1960’s. I will also discuss analysis of the model that reveals dependency of the auto-ignition position on principal physical and geometric parameters involved. Moreover, explicit expressions for auto-ignition position in asymptotic regimes relevant to applications will be given. The detailed account of results presented in this talk can be found in [1] This a joint work with U.G. Hegde, M.C. Hicks and M.J. Kulis of NASA Glenn Research Center.

References

[1] P.V. Gordon, U.G. Hegde, M.C. Hicks and M.J. Kulis, On autoignition of co-ﬂow laminar jets, SIAM J. Appl. Math., 47(4) (2016), 976-994. Stability of one-dimensional and multi-dimensional fronts in exponentially weighted norms

Anna Ghazaryan Miami University [email protected]

Yuri Latushkin and Xinyao Yang University of Missouri and Xian Jiaotong-Liverpool University, China [email protected] and [email protected]

Abstract

We consider a class of systems of reaction diﬀusion equations that frequently appears in combustion theory and chemical modeling. We study stability of traveling fronts in both one- dimensional and multi-dimensional cases. The essential spectrum of the operator obtained by linearizing the system about the front touches the imaginary axis, and thus we have to work in the intersection of the spaces of functions with and without exponential weights. For the one-dimensional case we prove the existence of stable foliation in vicinity of the front (these results are obtained jointly with R. Schnaubelt). For the multi-dimensional case we extend the stability theorems to the case of exponentially weighted spaces, and prove algebraic decay of perturbations of the front. Stability of Wavefronts in a Diﬀusive Model for Porous Media Combustion

Anna Ghazaryan Department of Mathematics Miami University Oxford, OH 45056 [email protected]

St´ephaneLafortune Department of Mathematics College of Charleston Charleston, SC 29424 [email protected]

Peter McLarnan formerly: Miami University [email protected]

Abstract

We study the stability of fronts in a reduction of a model of combustion in hydraulically resistant porous media. We first consider the model with the Lewis number chosen in a specific way and with initial conditions of a specific form. We then show that the stability results for that system extend to the fronts in the full system with the same Lewis number. The fronts are either absolutely unstable or convectively unstable. Evans-function techniques for the stability of viscous detonation waves

Gregory Lyng Department of Mathematics University of Wyoming [email protected]

Abstract

We survey results pertaining to the stability of detonation waves. Mathematically, the centerpiece is the Evans function, a spectral determinant whose zeros agree in location and multiplicity with the eigenvalues of the linearized operator about the wave; it enters the analysis at both the nonlinear and linear/spectral levels. We discuss both theoretical aspects of the Evans function and also issues related to its practical computation. Physically, the novelty of this work stems from the inclusion of oft-neglected diﬀusive eﬀects.

References

[1] B. Barker, J. Humpherys, G. Lyng, and K. Zumbrun, Viscous hyperstabilization of detonation waves in one space dimension, SIAM J. Appl. Math., 75 (2015), 885–906.

[2] J. Hendricks, J. Humpherys, G. Lyng, and K. Zumbrun, Stability of viscous weak detonation waves for Majda’s model. J. Dynam. Diﬀerential Equations, 27 (2015), 237–260.

[3] J. Humpherys, G. Lyng, and K. Zumbrun, Kevin, Stability of viscous detonations for Majda’s model. Phys. D, 259 (2013), 63–80.

[4] G. Lyng, M. Raooﬁ, B. Texier, and K. Zumbrun, Pointwise Green function bounds and stability of combustion waves. J. Diﬀerential Equations 233 (2007), 654–698. Viscous detonations in the reactive Navier-Stokes equations

Jeﬀrey Humpherys Brigham Young University, Provo, Utah USA jeﬀ[email protected]

Gregory Lyng University of Wyoming, Laramie, Wyoming USA [email protected]

Joshua Lytle Brigham Young University, Provo, Utah USA [email protected]

Abstract

A detonation is a hypersonic traveling wave arising in combustion problems. The standard practice is to study detonations with the reactive Euler (or ZND) equations, which do not account for viscosity. Barker et. al. [1] has revealed new behavior in viscous detonations of the reactive Navier-Stokes equations in one spatial dimension. Evans function computation shows that unstable eigenvalues re-stabilize as activation energy increases. Our research extends this Evans function study into the multi-dimensional setting.

References

[1] Blake Barker, Jeﬀrey Humpherys, Gregory Lyng and Kevin Zumbrun, Viscous hyperstabilization of detonation waves in one space dimension., SIAM Journal on Applied Mathematics, 75(3) 885-906, 2015. SESSION 13

Nonlinear Wave Phenomena in Continuum Physics: Some Recent Findings

P. M. Jordan Acoustics Division, U.S. Naval Research Laboratory, Stennis Space Ctr., Mississippi, 39529, USA [email protected]

Abstract

The talks presented in this session relate to recent advances in our understanding of the propagation, evolution, etc., of nonlinear waves in physical systems that may be regarded as a continuous medium; in other words, to wave phenomena described by mathematical models involving nonlinear partial diﬀerential equations (PDEs).

On the topic of solid mechanics, the talks include a study of acceleration waves in random media, which is modeled by a class of random ﬁelds that allows for the decoupling of the fractal dimension and Hurst parameter, and another wherein a discussion of recent developments in the study of deformation waves and fractures in solids is presented.

In the second group of talks, questions relating to the physical basis and current/future use of artiﬁcial viscosity in numerical simulations are examined and a traveling wave analysis of a modiﬁed form of Fisher’s equation, wherein a square-root nonlinearity appears in place of the usual quadratic one, is performed via a phase space approach.

In the third group of talks, the findings of a study in which the concept of Bäcklund charts is employed to study Korteweg–de Vries-type evolution equations are detailed and the occurrence of Devaney and distributional chaos in the solutions of the telegraph equation and PDEs describing traffic flow and sound waves in dispersive media is explored.

The last group of talks falls under the topic of nonlinear acoustics and consist of the following: an analysis of oscillating shock waves in a tube based on a higher-order, driven, weakly- nonlinear acoustic model; the determination, via a combined analytic-numerical methodology, of the optimal shape of an acoustic lens used to focus high-intensity ultrasound; and a discussion of the application of generalized continua methods to modeling particle-laden and poroacoustic ﬂows. Modeling particle-laden and poroacoustic ﬂow phenomena via the generalized continua approach

P. M. Jordan Acoustics Division, U.S. Naval Research Laboratory, Stennis Space Ctr., Mississippi, 39529, USA [email protected]

Abstract

With a focus on acoustic phenomena, we investigate several dual-phase (i.e., fluid-solid) flows using the generalized continua (GC) modeling approach, where by “GC”we mean modern generalizations of the constitutive relations of classical continuum mechanics that seek to capture the impact of sub-scale structure/dynamics on the (macroscopic) field variables. Working under the finite-amplitude framework, we derive and analyze generalizations of the weakly nonlinear versions of the Euler and Navier–Stokes equations for the case of particle-laden and poroacoustic flows. Using both analytical and numerical methods, we examine the impact of the solid phase on the propagation and evolution of (1D) shock and acceleration waves in perfect gases and common liquids. Along the way, the advantages and disadvantages of this modeling approach will be discussed and applications to other fields noted. The presentation concludes with a mention of possible follow-on studies. (Work supported by ONR funding.) Oscillating Shock Waves in Nonlinear Acoustics

Yuri Gaididei Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine [email protected]

Anders Rønne Rasmussen GreenHydrogen, Kolding, Denmark [email protected]

Peter Leth Christiansen Department of Physics and Department of Applied Mathematics and Computer Science Technical University of Denmark, Kongens Lyngby, Denmark [email protected]

Mads Peter Sørensen∗ Department of Applied Mathematics and Computer Science Technical University of Denmark, Kongens Lyngby, Denmark [email protected]

Abstract

We investigate oscillating shock waves in a tube using a higher order weakly nonlinear acoustic model[1]. The model includes thermoviscous effects and is non isentropic. The oscillating shock waves are generated at one end of the tube by a sinusoidal driver. Numerical simulations show that at resonance a stationary state arise consisting of multiple oscillating shock waves. Off resonance driving leads to a nearly linear oscillating ground state but superim- posed by bursts of a fast oscillating shock wave. Based on a travelling wave ansatz for the fluid velocity potential with an added 2nd order polynomial in the space and time variables, we find analytical approximations to the observed single shock waves in an infinitely long tube. Using perturbation theory for the driven acoustic system approximative analytical solutions for the off resonant case are determined.

References

[1] Yu.B. Gaididei, A.R. Rasmussen, P.L. Christiansen and M.P. Sørensen, Oscillating Non- linear Acoustic Shock Waves, Evolution Equations and Control Theory, 5 (2016), 367- 381. Deformation wave theory

Sanichiro Yoshida Department of Chemistry and Physics, Southeastern Louisiana University Hammond, LA 70402 [email protected]

Abstract

Deformation and fracture of solids are formulated as a comprehensive dynamic theory [1]. The present formalism is based on two postulates and a physical principle known as local symmetry. The two postulates are (a) a solid of any stage in deformation locally obeys the law of linear elasticity (call these local regions the deformation structural elements) and (b) as far as the solid remains as a continuum the deformation structural elements are logically connected. Requesting that the linear transformation representing elastic deformation be locally symmetric, the formalism introduces a compensation ﬁeld to connect the deformation structural elements. Applying the least action principle to the associated Lagrangian, the formalism derives ﬁeld equations.

The field equations describe dynamics that governs deformation of all stages including fracture comprehensively. The general solution to the field equations represents wave characteristics of the displacement field. Each stage of deformation is characterized by the restoring mechanism that generates the oscillatory dynamics propagated as a wave. Elastic deformation is characterized by restoring longitudinal force generating longitudinal waves. Plastic deformation is characterized by longitudinal velocity damping force and transverse restoring force generating decaying transverse waves. Deformation in the pre-fracturing stage is characterized by localized restoring force generating soliton-like waves.

In this paper, recent development of the above formalism will be presented. Supporting experimental data and numerical analysis of the formalism will be reported as well.

References

[1] S. Yoshida, Deformation and Fracture of Solid-State Materials, Springer, (2015). Waves in Random Media with Fractal and Hurst Characteristics

Vinesh Nishawala and Martin Ostoja-Starzewski Department of Mechanical Science & Engineering

University of Illinois at Urbana-Champaign

[email protected] and [email protected]

Abstract

We discuss recent results on waves and wavefronts in random media with fractal and Hurst characteristics. Such media are modeled by wide-sense statistically homogeneous and isotropic, multiscale random fields (RF). Two particular problems are investigated: (i) Lamb’s problem (a linear elastic, infinite half-space with a mass density RF), subject to a normal line load, and (ii) acceleration wave’s growth and decay in a 1D domain with a vector RF of nonlinear elasticanddampingproperties.TheRFsaremodeledbycorrelationfunctionsofCauchyor Dagum type; these fields are unique in that they are able to model and decouple the field’s fractal dimension and Hurst parameter.

In the case of Lamb’s problem, we employ the cellular automata (CA) method to simulate thewavepropagation[1].Wefirst evaluate the response of CA to an uncorrelated (white- noise) mass-density field of varying coarseness as compared to CA’s node density. Then we evaluate the response of CA to multiscale mass-density RFs of Cauchy and Dagum type. We determine to what extent the fractal or the Hurst parameter is a significant factor in altering the solution to the planar stochastic Lamb’s problem by evaluating the coefficient of variation (CV) of the response as compared to the CV of the RF [2].

In the second problem we study the eﬀect of the vector RF on the probability of shock formation, the critical amplitude, and the distance to form a shock. The focus is on determining the driving parameter, either fractal or Hurst, which is signiﬁcant in altering the system’s response, the latter being governed by a stochastic Bernoulli equation [3].

References [1] V. Nishawala, M. Ostoja-Starzewski, M. Leamy and P.N. Demmie, Simulation of elastic wave propagation using cellular automata and peridynamics, and comparison with experiments, Wave Motion 60 (2016), 73-83. [2] V. Nishawala, M. Ostoja-Starzewski, M. Leamy and E. Porcu, Lamb’s problem on random mass density fields with fractal and Hurst effects, Proc.Roy.Soc.A(2016), online. [3] V. Nishawala and M. Ostoja-Starzewski, Acceleration waves in random media with fractal and Hurst effects, (2017), unpublished. High intensity ultrasound focusing by isogeometric shape optimization

Markus Muhr, Vanja Nikoli´c∗, Barbara Wohlmuth and Linus Wunderlich M2 - Department of Mathematics, Technical University of Munich Boltzmannstr. 3, 85748 Garching, Germany [email protected], [email protected], [email protected], [email protected]

Abstract

We study the problem of ﬁnding the optimal shape of an acoustic lens that focuses high- intensity ultrasound. This problem arises, for instance, in lithotripsy where ultrasound should be focused at kidney stones without damage to the surrounding tissue.

To model propagation of ultrasound, we use a quasilinear strongly damped wave equation that involves quadratic nonlinearities. Since the acoustic lens in lithotripsy is typically immersed in a nonlinear acoustic fluid with higher speed of sound, our shape optimization problem is governed by an acoustic-acoustic interface coupling (cf. [3]). We use a pressure- tracking cost functional, whose shape derivative can be obtained by a variational approach (introduced in [2]) that replaces shape differentiability of the state variable by Hölderregu- larity with respect to the geometry perturbations.

In the algorithmic realization, we employ a gradient-descent type algorithm, combining shape optimization with the concept of isogeometric analysis (cf. [1]). In this way our lens and ﬂuid subdomains can be represented exactly by using Non-Uniform Rational B-Spline (NURBS) base, whereby the same basis representation is employed for both the geometry and the numerical simulations. Update of the NURBS geometry in every gradient step is performed by moving control points that belong to its NURBS description in accordance with the computed derivative. Only the interface control points act as optimization variables, whereas the internal control points are updated in order to prevent mesh tangling. Numerical tests in a two-dimensional setting will illustrate the performance of the algorithm.

References [1] C. J. Austin, T. Jr Hughes and Y. Bazilevs, Isogeometric analysis: toward integration of CAD and FEA, John Wiley & Sons, (2009). [2] K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivatives, ESAIM: COCV, 14, (2008), 517-539. [3] S. Veljovi´c,Shape Optimization and Optimal Boundary Control for High Intensity Fo- cused Ultrasound (HIFU), PhD thesis, University of Erlangen-Nuremberg, (2009). Chaotic semigroups from second order partial diﬀerential equations

J. Alberto Conejero∗ IUMPA - Universitat Polit`ecnicade Val`encia [email protected]

Carlos Lizama Universidad de Santiago de Chile [email protected]

Marina Murillo-Arcila IMAC - Universitat Jaume I [email protected]

Abstract

We give general conditions on given parameters to ensure Devaney and distributional chaos for the solution C0-semigroup corresponding to a class of partial diﬀerential equations of order two on certain spaces of analytic functions of slow growth. Moreover, we provide a critical parameter that lead us to distinguish between stability and chaos for these semigroups. In the case of chaos, we prove that the C0-semigroup admits a strongly mixing measure with full support. We also give concrete examples of partial diﬀerential equations, such as the telegraph equation, whose solutions satisfy these properties.

Some applications to particular equations such as telegraph, Lighthill-Whitham-Richards, Moore-Gibson-Thompson or van Wijngaarden-Eringen. equations were considered.

References

[1] J. A. Conejero, C. Lizama, and M. Murillo-Arcila. Chaotic semigroups from second order partial differential equations. Preprint. (2017). [2] J. A. Conejero, F. Mart´ınez-Giménez,A. Peris, and F. Ródenas.Chaotic asymptotic behaviour of the solutions of the Lighthill-Whitham-Richards equation. Nonlinear Dynam., 84 (2016), 127–133. [3] J. A. Conejero, C. Lizama, and M. Murillo-Arcila. Chaotic behaviour of the solutions of the Moore-Gibson-Thompson equation. Appl. Math. Inf. Sci., 9(5) (2015), 2233–2238. [4] J. A. Conejero, C. Lizama, and M. Murillo-Arcila. On the existence of chaos for the viscous van Wijngaarden–Eringen equation. Chaos, 89 (2016), 100–104. The reality of artificial viscosity

Len G. Margolin Computational Physics Division, Los Alamos National Laboratory Los Alamos, NM, 87545 [email protected]

Abstract

The use of artificial viscosity to regularize numerical simulations of shock wave propagation dates back to the dawn of computational fluid dynamics [1]. Over the past 60 years, there has been considerable effort and success in improving the computational aspects of artificial viscosity, but it is only recently that a deeper understanding has begun to clarify its physical basis. In this talk, I will begin with a brief account the origins of artificial viscosity. I will then describe the relation of artificial viscosity to physical concepts such as enslavement, coarse–graining, and nonequilibrium thermodynamics [2]. I will end with some thoughts on the role of artificial viscosity in the future when computers begin to resolve the transport scales of the Navier–Stokes equations in practical problems.

References

[1] J. von Neumann, & R.D. Richtmyer, A method for the numerical calculation of hydro- dynamic shocks, J. Appl. Phys., 21 (1950), 232-237.

[2] L.G. Margolin, Finite scale theory: the role of the observer in classical fluid flow, Mech. Res. Comm. 57 (2014), 10–17. KdV-type Nonlinear Evolution Equations: non-Abelian versus Abelian Bäcklund Charts

Sandra Carillo Dip. “Scienze di Base e Applicate per l’Ingegneria”, Sapienza - University of Rome, 16, Via A. Scarpa, 00161 Rome, Italy & I.N.F.N. - Sez. Roma1, Gr. IV - Mathematical Methods in NonLinear Physics, Rome, Italy [email protected] and [email protected]

Abstract

Bäcklund charts which, in turn, connect third order evolution equations of Korteweg–de Vries-type in the non-Abelian and in the Abelian cases are compared. Novel results are presented in both cases: they are based on a reasearch program which takes its origin in [2, 1] and, now, is mostly devoted to study operator nonlinear evolution equations [3, 4, 5, 6, 7]. Specifically, analogies and dicrepancies between the Bäcklund chart in [1], recently extended, is compared with the analog non-Abelian one [5, 7]. Recursion operators are explicitely given and compared.

References

[1] B. Fuchssteiner and S. Carillo, Soliton structure versus singularity analysis: Third order completely integrable nonlinear equations in 1+1-dimensions, Phys. A 154 (1989), 467- 510. [2] C. Rogers and S. Carillo, On reciprocal properties of the Caudrey-Dodd-Gibbon and Kaup-Kupershmidt hierarchies, Physica Scripta 36 (1987), 865-869. [3] S. Carillo and C. Schiebold, Non-commutative KdV and mKdV hierarchies via recursion methods, J. Math. Phys. 50 (2009), 073510. [4] S. Carillo and C. Schiebold, Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies: Non-commutative soliton solutions, J. Math. Phys. 52 (2011), 053507. [5] S. Carillo, M. Lo Schiavo and C. Schiebold, Bäcklund Transformations and Non Abelian Nonlinear Evolution Equations: a novel Bäcklund chart, SIGMA, 12 (2016), 087, 17 pages, doi:10.3842/SIGMA.2016.087. [6] S. Carillo, M. Lo Schiavo and C. Schiebold, Recursion Operators admitted by non-Abelian Burgers equations: Some Remarks, (2016), 12 pages, arXiv:1606.07270v1. [7] S. Carillo, M. Lo Schiavo and C. Schiebold, (2017), submitted. Analysis of the Traveling Wave Solutions to a Modified Diffusionless Fisher Equation

Ronald E. Mickens∗ Clark Atlanta University Atlanta, GA 30314 [email protected]

’Kale Oyedeji Morehouse College Atlanta, GA 30314-3773 [email protected]

Abstract

We investigate traveling wave solutions to a modiﬁed Fisher PDE

√ Ut + V0Ux = DUxx + λ1 U − λ2U, (1)

where V0 is the constant advection velocity, D is the diffusion coefficient, and (λ1, λ2) are positive parameters. Our analysis shows that for the case where D = 0, three possibilities exist for the traveling wave solutions. We determine the general qualitative features of this class of solutions and give finite difference discretizations based on the nonstandard finite difference methodology of Mickens. SESSION 14

Integrable Systems and the Geometry of Curves and Surfaces

Annalisa Calini Department of Mathematics College of Charleston Charleston, SC 29424, U.S.A [email protected]

Baofeng Feng School of Math & Statistical Sciences University of Texas Rio Grande Valley Edinburg, TX 78539, U.S.A. [email protected]

Keniji Kajiwara Institute of Mathematics for Industry Kyushu University Fukuoka 819-0395, Japan [email protected]

Abstract

The aim of this special session is to provide researchers in diﬀerential geometry and integrable systems with an opportunity for interaction, focusing on curves and surfaces. The timeliness and relevance of this topic is based on interesting recent developments, including for example discretizations and discrete maps, B¨acklund and Darboux transformations, integrability in various geometries, and geometric visualization. Some Examples of Integrable Geometric Curve Flows

Hsiao-Fan Liu Institute of Mathematics, Academia Sinica 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan [email protected]

Abstract

In this talk, we study some examples of geometric curves flows whose invariants flow according to some soliton equations. One classical example is the vortex filament equation(VFE), that is related to the nonlinear Schrödingerequation(NLS). We also discuss the correspon- dences between the Schödingerflows on Hermitian symmetric spaces and equations of NLS type. The existence of solutions to the Cauchy problems of VFE in R3 and the Schödinger flow on a 2-sphere for periodic boundary conditions follows from this correspondence. Geo- metric algorithms are then obtained to solve periodic Cauchy problems numerically.

References

[1] C. Terng and G. Thorbergsson, Completely integrable curve ﬂows on adjoint orbits. In Dedicated to Shiing-Shen Chern on his 90th birthday, Results Math., 40 (2001), 286-309.

[2] C. Terng and K. Uhlenbeck, Schrödingerflows on grassmannians. In Integrable systems, Geometry, and Topology, AMS/IP Stud. Adv. Math. 36 (2006), 235-256. dNLS flow on discrete space curves

Sampei Hirose Shibaura Institute of Technology, Saitama, Japan [email protected]

Jun-ichi Inoguchi Institute of Mathematics, University of Tsukuba, Japan [email protected]

Kenji Kajiwara∗ Institute of Mathematics for Industry, Kyushu University, Japan [email protected]

Nozomu Matsuura Department of Applied Mathematics, Fukuoka University, Japan [email protected]

Yasuhiro Ohta Department of Mathematics, Kobe University, Japan [email protected]

Abstract

The binormal flow on the space curves γ(x, t) ∈ R3 given byγ ˙ = κB, x is the arclength, κ(x, t) is the curvature and B(x, t) is the binormal vector, is one of the most important deformations of the space curves, since√ it describes the dynamics of vortex filaments. The −1 R λ(x,t) dx complex curvature u(x, t) = κ√(x, t)e , where λ is the torsion, satisfies the nonlinear 00 1 2 Schrödingerequation (NLS) −1u ˙ + u + 2 |u| u = 0, as shown by Hasimoto (1972). In this paper, we consider the discrete NLS (dNLS) proposed by Ablowitz-Ladik (1976)

√ 2 m+1 √ 2 m m m+1 2 m 2 m −1 −1 δ − 1 un − δ + 1 un + (un+1 + un−1 )(1 + |un | )Γn = 0,

m 2 m 2 Γn+1 1 + |un | m m m = 2 m+1 2 , n, m ∈ Z, un ∈ C, Γn , , δ ∈ R, Γn 1 + |un | and construct the discrete deformation of discrete space curves whose discrete curvature is described by dNLS. We formulate the deformation in terms of the discrete Frenet frame, which is regarded as a discrete model of vortex filaments. We also construct the explicit formulas of both smooth and discrete curves in terms of the τ functions of two-component KP hierarchy. Numerical results of both open and closed discrete curves will be presented. A construction method for discrete indefinite affine spheres

Shimpei KOBAYASHI∗ and Nozomu MATSUURA Department of Mathematics, Hokkaido University Department of Applied Mathematics, Fukuoka University [email protected] and [email protected]

Abstract

It is known that special classes of smooth surfaces, such as constant Gaussian curvature surfaces, constant mean curvature surfaces or aﬃne spheres, are deﬁned by integrable systems, such as sine-Gordon equation, sinh-Gordon equation or Tzitzeica equation, respectively. More precisely, the moving frame of the surface gives the Lax pair of an integrable system and a natural parameter can be introduced into the Lax pair, and thus the surface has a loop group structure.

We can discretize those surfaces by keeping the loop group structure, and the discrete structure equation of the discrete surface often becomes a well-known discrete integrable system, [1, 2]. Recently in [3], the ﬁrst named author gave a construction method for discrete constant negative Gaussian curvature surfaces using a loop group method, the so-called Birkhoﬀ decomposition of the loop group. This is a discrete analogue of separation of variables for sine-Gordon equation by Krichever, [4].

In this presentation, we give a construction method for discrete indefinite affine spheres using a loop group method. In particular, we give an explicit formula for discrete indefinite improper affine spheres.

References [1] A. I. Bobenko and U. Pinkall, Discretization of surfaces and integrable systems. Discrete integrable geometry and physics (Vienna, 1996), 3–58, Oxford Lecture Ser. Math. Appl., 16, Oxford Univ. Press, New York, 1999. [2] A. I. Bobenko and W. K. Schief, Affine spheres: discretization via duality relations, Exp. Math., 8 (1999), 261–280. [3] S.-P. Kobayashi, Nonlinear d’Alembert formula for discrete pseudospherical surfaces, arXiv:1505.07189, 2015. [4] I. M. Krichever, An analogue of the d’Alembert formula for the equations of a principal chiral field and the sine-Gordon equation. Soviet Math. Dokl., 22 (1980), no. 1, 79-84. [5] S.-P. Kobayashi and N. Matsuura, A loop group method for discrete indefinite affine spheres, in preparation, 2017. Discrete timelike minimal surfaces and discrete wave equations

Masashi Yasumoto Osaka City University Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan [email protected], [email protected]

Abstract

In the continuous case, a timelike immersion with vanishing mean curvature in 3-dimensional Minkowski space is called a timelike minimal surface. Timelike minimal surfaces are highly related to linear and nonlinear wave equations. In fact, their metric function satisﬁes a nonlinear integrable wave equation called a hyperbolic Liouville equation ([1], [2]), and each coordinate function is a solution of a 1D wave equation.

In this talk we describe a theory of discrete timelike minimal surfaces in 3-dimensional Minkowski space ([3]). First we introduce a Weierstrass-type representation for them. Fur- thermore, by a reparametrizion for discrete surfaces, we show that each coordinate function of a discrete timelike minimal surface satisﬁes a discrete version of the 1D wave equation.

This result provides not only the geometric meaning of special solutions for a discrete 1D wave equation but also a new representation formula for discrete timelike minimal surfaces.

References

[1] J. Dorfmeister, J. Inoguchi and M. Toda, Weierstraß type representation of timelike surfaces with constant mean curvature, Contemp. Math., 308 (2002), 77-99.

[2] H.J. de Vega and N. Sanchez, Exact integrability of strings in D-dimensional de Sitter spacetime, Phys. Rev. D, 47 (1993), 3394.

[3] M. Yasumoto, Discrete timelike isothermic surfaces and discrete wave equations, in preparation. Minimal surfaces from deformations of circle patterns

Wai Yeung Lam Brown University [email protected]

Abstract

William Thurston introduced circle packings to approximate holomorphic functions. Burt Rodin and Dennis Sullivan proved the convergence of the analogue of Riemann maps for circle packings. Oded Schramm further extended the idea by considering circle patterns, where circles are allowed to intercept with each other.

We present a discrete analogue of the Weierstrass representation for minimal surfaces in terms of discrete holomorphic quadratic differentials, which are induced from infinitesimal deformations of circle patterns. In this way, we unify the earlier notions of discrete minimal surfaces: circular minimal surfaces via the integrable systems approach and conical minimal surfaces via the curvature approach. Furthermore, each discrete holomorphic quadratic differential obtained from discrete integrable systems yields a family of discrete minimal surfaces which are critical points of the total area. Constrained Willmore Minimizers

Lynn Heller Leibniz Universit¨atHannover, Welfengarten 1, 30167 Hannover (Germany) [email protected]

Abstract

I consider compact immersed surfaces minimizing the Willmore energy under the constraint of prescribed conformal class. For spheres, where there exist only one conformal structure, the constrained Willmore minimizer is the round sphere. For topological tori the Willmore conjecture, solved by Marquez and Neves, shows that the Cliﬀord torus minimizes the Will- more energy in the class of all immersions, and thus it clearly also minimizes the energy in its conformal class - the square class. The only other case where the constrained Willmore minimizers are determined (by Ndiaye and Schtzle) is for rectangular conformal classes in a small neighborhood of the square class, where the homogenous tori minimizes. In my talk I want t o show how to generalize this result to non-rectangular conformal classes close to the square class with emphasis on the construction of the appropriate candidate surfaces. This is joint work with Cheikh Brahim Ndiaye. On the stability of the Hasimoto Filament

Annalisa Calini, St´ephaneLafortune, and Brenton Lemesurier Department of Mathematics, College of Charleston Charleston, SC 29424, USA [email protected], [email protected], [email protected]

Abstract

We adapt the classical approach by Grillakis, Shatah, and Strauss to study the orbital stability of the one-soliton solution of the Vortex Filament Equation (VFE), without making recourse to its well-known correspondence with the Nonlinear Schr¨odinger equation. This relies on formulating the VFE as a Hamiltonian system that is invariant under a group of symmetries on a suitable space of curves. Hamiltonian structures for lattice W n algebras and centro-aﬃne geometry

Gloria Mar´ı-Beﬀa∗ Mathematics Department, University of Wisconsin-Madison Madison, WI 53707, USA maribeﬀ@math.wisc.edu

Annalisa Calini Department of Mathematics, College of Charleston Charleston, SC 29424, USA [email protected]

Abstract

n The authors of “Hamiltonian structures and integrable evolutions of twisted gons in RP ”, G. Mari-Beﬀa, J.P. Wang, Nonlinearity 26 (2013) deﬁned two Hamiltonian structures naturally associated to geometric evolutions of projective polygons. They proved that certain integrable discretizations of Wn algebras were Hamiltonian with respect to both Hamiltonian structures, and they showed that in the cases of dimension 2 and 3 the structures formed a biHamiltonian pencil.

In this talk we will describe how these structures can be lifted to the space of discrete vectors fields in centro-affine geometry, to produce two pre-symplectic forms. These two forms are much simpler than the original Hamiltonian structures, and the connection opens the possibility of not only proving that the pair forms a pencil for all dimensions, but also of finding further integrable hierarchies generated by the kernel of one of the forms. The geometric interpretation of the complex short pulse equation

Baofeng Feng School of Mathematical and Statistical Sciences University of Texas Rio Grande Valley Edinburg,TX, 78539-2999 U.S.A.

ABSTRACT

In this talk, we are concerned with the geometric meaning of the complex short pulse equation. First, we will show that the complex short pulse equation of focusing and defocusing type can be derived from the motion of three-dimensional curves in Euclidean space and Minkowski space, respectively. Furthermore, we show that the fundamental forms of surfaces can also give rise to the focusing and defocusing complex short pulse equation and their curve ﬂows. SESSION 15 Asymptotics and Applied Analysis

Robert Buckingham Department of Mathematical Sciences University of Cincinnati [email protected]

Peter Miller Department of Mathematics University of Michigan, Ann Arbor [email protected]

Abstract

Recently, analytic and asymptotic methods originally developed to study nonlinear wave equations have led to significant advances in fields ranging from spectral theory to probability. This session will bring together researchers in dispersive equations, stability theory, integrable systems, random matrix theory, and related areas to present new results and foster further connections between fields. On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo-Miwa-Ueno differential

Thomas Bothner∗ University of Michigan 530 Church Street, Ann Arbor, MI 48109 [email protected]

Alexander Its and Andrei Prokhorov Indiana University-Purdue University Indianapolis 402 N. Blackford St., Indianapolis, IN 46202 [email protected] and [email protected]

Abstract

Recently A. Its, O. Lisovyy and A. Prokhorov [1] have found a way to extend the classical Jimbo-Miwa-Ueno isomonodromic tau function [2] to the full space of extended monodromy data of systems of linear ordinary diﬀerential equations with rational coeﬃcients. We shall use this method to analyze incomplete spectra in unitary random matrix models.

References

[1] A. Its, O. Lisovyy, A. Prokhorov, Monodromy dependence and connection formulae for isomonodromic tau functions. preprint arXiv:1604.03082

[2] M.Jimbo, T.Miwa, K.Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coeﬃcients. I, Physica D2, (1981), 306-352. Dispersionless limits of the DKP equations for continuum limits of the Pfaﬀ lattice equations

Virgil U Pierce University of Texas Rio Grande Valley 1201 W University Drive Edinburg, Texas 78539 email: [email protected]

ABSTRACT

The Pfaff lattice equations are generalization of the Toda lattice equations obtained by replacing the QR or LU-factorization by SR-factorization. The resulting equations are integrable with similar expressions for their constants of motion as the Toda lattice equations. As with the Toda lattice equations the solutions can be expressed in terms of a tau-function, which is a partition function of the Gaussian orthogonal random matrix ensemble. This implies that the log-tau functions should possess nice (combinatorically significant) asymptotic expansions, and to that end one wishes to search for continuum limits of the Pfaff lattice equations. Unlike in the case of the Toda lattice equations, the naive limits are ill- conditioned. The tau-functions, as generators of solutions to the lattice equations, satisfy their own hierarchy of equations, the D-type Kadomtsev-Petviashvili equations (DKP), and we will use this to show that one may find governing equations for the asymptotic expansion of the log-tau functions. Spectral and nonlinear stability of traveling fronts for a hyperbolic Allen-Cahn model with relaxation

Corrado Lattanzio and Corrado Mascia Universit´adegli Studi dell’Aquila, and Universit´adi Roma ‘La Sapienza’ [email protected] and [email protected]

RamónG. Plaza∗ Universidad Nacional Autónomade México [email protected]

Chiara Simeoni Universit´ede Nice Sophia-Antipolis [email protected]

Abstract

In this talk, I discuss existence and stability of traveling wave solutions to hyperbolic systems of equations which result from a modiﬁcation of the standard parabolic Allen–Cahn equation, and determined by the substitution of Fick’s diﬀusion law with a relaxation relation of Ca- ttaneo-Maxwell type [1, 3]. We show (cf. [2]) that traveling wave solutions exist, and that they are nonlinearly stable. The stability analysis involves Evans function techniques to handle the spectral problem for the perturbations. The spectral information is used in a key way to prove that the waves are nonlinearly stable under small perturbations. In addition, numerical studies are performed in order to determine the propagation speed, to compare it to the speed for the parabolic case, and to explore the dynamics of large perturbations of the front.

References

[1] C. Cattaneo, Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena 3 (1949) 83–101.

[2] C. Lattanzio, C. Mascia, R. G. Plaza and C. Simeoni, Analytical and numerical investigation of traveling waves for the Allen–Cahn model with relaxation, Math. Models Methods Appl. Sci., 26 (2016), no. 5, 931-985.

[3] J. C. Maxwell, On the dynamical theory of gases, Trans. Soc. London 157 (1867), no. 9, 49–88. Asymptotics for the normal matrix model and the mother body problem

Guilherme Silva University of Michigan 530 Church Street, 48108 Ann Arbor, Michigan, USA [email protected]

Abstract

We consider the normal matrix model with cubic plus linear potential. We introduce multiple orthogonal polynomials that should describe the model in the thermodynamic limit. Developing the Deift-Zhou nonlinear steepest descent method to the associated Riemann- Hilbert problem we obtain asymptotics for these polynomials, showing a connection with the so-called mother body problem. The associated g-functions are constructed with the help of quadratic diﬀerentials as we also plan to explain.

This is a joint work with Pavel Bleher (IUPUI). The semi-classical sine-Gordon equation, universality at phase transition and the gradient catastrophe

Bingying Lu∗ and Peter Miller University of Michigan, Ann Arbor Department of Mathemtatics [email protected] and [email protected]

Abstract

In this talk I am going to discuss the universal behaviours of the semi-classical limit of the sine-Gordon equation. We consider a class of solutions with pure impulse initial data below critical value such that within small time only librational-type waves are generated and the solutions should decay when |x| → ∞ [1]. We are particularly interested in a neighbourhood of a certain gradient catastrophe point that contains both modulated plane waves and localized structures or “spikes”. We aim to describe the solutions using special functions. Besides the gradient catastrophe point (we think of it as a more degenerate point than other generic locations of phase transition), we are also interested in describing phase transition in space-time as a boundary curve and the behaviours of the solutions nearby. These phase transitions exhibit universality in the sense that the solutions behave locally the same way in the asymptotic limit for diﬀerent initial data chosen from the class we consider; it is only the space-time location of the transition that depends on the initial data. We use the Deift-Zhou steepest descent method related to an approach of Bertola and Tovbis [2] to universality for the focusing nonlinear Schr¨odingerequation.

References

[1] Robert J. Buckingham and Peter D. Miller, The Sine-Gordon Equation in the Semi- classical Limit: Dynamics of Fluxon Condensates, Memoirs of the American Mathemat- ical Society, 225, (2013)

[2] Marco Bertola and Alexander Tovbis, Universality for the Focusing Nonlinear Schrodinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquee Solution to Painleve I, Communications on Pure and Applied Math- ematics, LXVI, (2013), 0678-0752

[3] Bingying Lu and Peter Miller, The Semi-Classical Sine-Gordon Equation, Universality at Phase Transition and the Gradient Catastrophe, In Preparation Maximal amplitude for the ﬁnite gap (quasiperiodic) solutions and its applications to large amplitude (rogue) waves for the focusing NLS

Marco Bertola Concordia University and SISSA

Alexander Tovbis∗ University of Central Florida [email protected]

Abstract

In this talk we show that, for the focusing NLS, the maximum of a ﬁnite gap solution with given (vertical) bands cannot exceed half of the total length of all the bands and, generically, does approach this value. In the particular case of genus two this result was obtained by O. Wright. A similar statement is true for the defocusing NLS. The obtained result allowed us to formulate a new concept of rogue waves within the class of ﬁnite gap solutions.

References

[1] M. Bertola and A. Tovbis, Maximal amplitudes of ﬁnite-gap solutions for the focusing nonlinear Schr¨odingerequation, arXiv:1601.00875.

[2] M. Bertola, G. El and A. Tovbis, Rogue waves in multiphase solutions of the focusing NLS equation, Proceedings of The Royal Society A (2016) DOI: 10.1098/rspa.2016.0340 (arXiv:1605.04713).

[3] O. Wright, Eﬀective integration of ultra-elliptic solutions of the focusing nonlinear Schrodinger equation, Physica D 321-322, 16 - 38, (2016) (arXiv:1505.03120). Nonlocal problems for linear evolution equations

David Smith Yale-NUS College, Singapore [email protected]

Abstract

Linear evolution equations, such as the heat equation, are commonly studied on finite spatial domains via initial-boundary value problems. In place of the boundary conditions, we consider “multipoint conditions”, where one specifies some linear combination of the solution and its derivative evaluated at internal points of the spatial domain, and “nonlocal” specification of the integral over space of the solution against some continuous weight.

References Global Existence and Asymptotics for the Derivative Non-Linear Schr¨odingerEquation in One Dimension

Robert Jenkins Department of Mathematics, University of Arizona Tucson, Arizona 85721–0089 U. S. A. [email protected]

Jiaqi Liu and Peter Perry∗ Department of Mathematics, University of Kentucky Lexington, Kentucky 40506–0027 U. S. A. jiaqi [email protected] and [email protected]

Catherine Sulem Department of Mathematics, University of Toronto Toronto M5S 2E4 Canada [email protected]

Abstract

The Derivative Non-Linear SchrödingerEquation (DNLS) is the dispersive nonlinear equa- 2 tion iut + uxx = iε (|u| u)x where ε = ±1. We use the completely integrable method to show that solutions to this equation exist for generic initial data in the Sobolev space H2,2(R) = {u ∈ L2(R): x2u, u00 ∈ L2(R)}. Our class includes initial data of arbitrary L2 norm, having no spectral singularities. Initial data that support finitely many solitons are included, and we obtain soliton resolution for such data using nonlinear steepest descent. Our work builds on previous work of Liu, Perry, and Sulem [1, 2]. This is the first of two minisymposium presentations on this work, and covers the global existence result. A second presentation, by Robert Jenkins, discusses our results on long-time asymptotics.

References

[1] J. Liu, P. Perry, C. Sulem. Global existence for the derivative nonlinear Schr¨odinger equation by the method of inverse scattering. Comm. Partial Diﬀerential Equations 41 (2016), no. 11, 1692–1760.

[2] J. Liu, P. Perry, C. Sulem. Long-time behavior of solutions to the derivative nonlinear Schrödingerequation for soliton-free initial data. Preprint, arxiv:1608.07659[Math.AP], submitted to Ann. Inst. Henri PoincaréC - Analyze non-linéaire. Global Existence and Asymptotics for the Derivative Non-Linear SchrödingerEquation in One Dimension: Part II

Robert Jenkins* Department of Mathematics, University of Arizona Tucson, Arizona 85721-0089 U.S.A [email protected]

Jiaqi Liu and Peter Perry Department of Mathematics, University of Kentucky Lexington, Kentucky 40506-0027 U.S.A [email protected] and [email protected]

Catherine Sulem Department of Mathematics, University of Toronto Toronto, Ontario MES 2E4 Canada [email protected]

Abstract

The Derivative Nonlinear SchrödingerEquation (DNLS) is the dispersive nonlinear equation 2 iut + uxx = iε(|u| u)x where ε = ±1. We use its complete integrability to show that solutions to this equation exist for generic initial data in the Sobolev space H2,2(R) = {u ∈ L2(R): x2u, u00 ∈ L2(R)}. Our class includes initial data of arbitrary L2(R) norm, having no spectral singularities. Initial data that supports finitely many solitons are included, and we obtain soliton resolution for such data using nonlinear steepest descent. Our work builds on the previous work of Liu, Perry and Sulem [2, 3] on DNLS; the DBAR steepest descent analysis is based on the work in [1]. This talk is the second of two minisymposium presentations on this work, and will discuss the long time asymptotic results. The first part, presented by Peter Perry, will discuss the global existence result.

References

[1] M. Borghese, R. Jenkins, K. McLaughlin. Long time asymptotic behavior of the focusing nonlinear Schrödingerequation. Preprint. arXiv:1604.07436[Math.PH], submitted to Ann. Inst. Henri PoincaréC - Analyze non-linéaire.

[2] J. Liu, P. Perry, C. Sulem. Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering. Comm. Partial Differential Eqiations, 41 (2016), no. 11, 1692–1760. [3] J. Liu, P. Perry, C. Sulem. Long-time behavior of solutions of the derivative nonlinear Schödingerequation for soliton-free initial data. Preprint. arXiv:1608.07659[Math.AP], submitted to Ann. Inst. Henri PoincaréC - Analyze non-linéaire. Numerical inverse scattering for the Toda lattice

Deniz Bilman University of Michigan 530 Church St, Ann Arbor, MI 48109 [email protected]

Thomas Trogdon University of California, Irvine Rowland Hall, Irvine, CA 92697 [email protected]

Abstract

We present a method to compute the inverse scattering transform (IST) for the famed Toda lattice by solving the associated Riemann–Hilbert (RH) problem numerically. Deformations for the RH problem are incorporated so that the IST can be evaluated in O(1) operations for arbitrary points in the (n, t)-domain, including short- and long-time regimes. No time- stepping is required to compute the solution because (n, t) appear as parameters in the associated RH problem. The solution of the Toda lattice is computed in long-time asymptotic regions where the asymptotics are not known rigorously. Spectral Stability in Reduced and Extended Systems

Richard Koll´ar Comenius University, Bratislava, Slovakia [email protected]

Abstract

Spectral stability captures behavior of a perturbed solution. It often determines nonlinear stability but it is limited to the exact form of the system. However, governing equations are often only an approximation. We show how are the stability of a solution in the reduced and full (extended) system related, particularly for ODEs (quasi-steady-state reduction) and a general embedding of into a larger system. A connection is drawn with geometric Krein signature that naturally to captures spectral properties under such extensions. Elliptic isomonodromy and the elliptic Painlev´eequation

Christopher Ormerod and Eric Rains 5752 Neville Hall, Room 322 , Orono, ME 04469 Mathematics 253-37, Caltech, Pasadena, CA 91125 [email protected] and [email protected]

Abstract

We present a linear system of diﬀerence equations whose entries are expressed in terms of theta functions. This linear system is singular at 4m + 12 points. We parameterize this linear system in terms a set of kernels at the singular points and regard the system of discrete isomonodromic deformations as an elliptic analogue of the Garnier system. When m = 1 we recover the elliptic Painleve equation and an associated Lax pair. Universality for eigenvalue algorithms

Thomas Trogdon∗ UC Irvine [email protected]

Percy Deift New York University [email protected]

Abstract

As shown in [2] (and the ref. therein), a class of classical integrable systems give rise to isospectral flows on symmetric/Hermitian matrices and these systems can be integrated to compute the eigenvalues. And as evidenced by Pfrang, Deift and Menon [7] (see also [3]) such numerical algorithms exhibit a universal runtime, or halting time, when they are applied to appropriate random matrices. Utilizing recent results from random matrix theory ([1, 6], for example), we prove universal limit theorems for specific halting times for the integrable Toda and QR algorithms and the power/inverse power methods [4, 5]. This work presents a confluence of the theory of integrable systems with the theory of integrable probability.

References

[1] A. Bloemendal, A. Knowles, H.-T. Yau, and J. Yin, On the principal components of sample covariance matrices, Probab. Theory Relat. Fields, 164 (2016), pp. 459–552. [2] P. Deift, T. Nanda, and C. Tomei, Ordinary diﬀerential equations and the symmetric eigenvalue problem, SIAM J. Numer. Anal., 20 (1983), 1–22. [3] P. A. Deift, G. Menon, S. Olver, and T. Trogdon, Universality in numerical computations with random data, Proc. Natl. Acad. Sci. U. S. A., 111 (2014), 14973–8. [4] P. Deift and T. Trogdon, Universality for eigenvalue algorithms on sample covariance matrices , arXiv Prepr. arXiv1604.07384, (2017). [5] P. Deift and T. Trogdon, Universality for the Toda algorithm to compute the eigenvalues of a random matrix, arXiv Prepr. arXiv1604.07384, (2016). [6] L. Erdos,˝ H.-T. Yau, and J. Yin, Rigidity of eigenvalues of generalized Wigner matrices, Adv. Math. (N. Y)., 229 (2012), 1435–1515. [7] C. W. Pfrang, P. Deift, and G. Menon, How long does it take to compute the eigenvalues of a random symmetric matrix?, Random matrix theory, Interact. Part. Syst. Integr. Syst. MSRI Publ., 65 (2014), 411–442. Semiclassical analysis for a 2D completely-integrable equation

Michael Music∗ 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043 [email protected]

Abstract

We solve the zero-energy Novikov-Veselov (NV) equation using the inverse scattering method (ISM) for a certain class of initial data. The NV equation is a (2+1)-dimensional generalization of the Korteweg-de Vries equation. We prove new existence and uniqueness results for the complex geometric optics solutions of the 2-dimensional stationery Schrodinger equation and then show that the method produces classical solutions to the NV equation [1]. We also present preliminary results on the semiclassical limit of the NV equation using the ISM.

References

[1] Music, M., Perry, P.: Global Solutions for the Zero-Energy Novikov-Veselov Equation by Inverse Scattering ArXiV:1502.02632 [math.AP]. Rational Mapping Factorization and Tau-functions of Discrete Painlev´eEquations

Adrian Stefan Carstea∗ National Institute of Physics and Nuclear Engineering, Bucharest, Romania [email protected]

Anton Dzhamay∗ School of Mathematical Sciences, The University of Northern Colorado Greeley, CO 80526, USA [email protected]

Tomoyuki Takenawa Faculty of Marine Technology, Tokyo University of Marine Science and Technology 2-1-6 Etchu-jima, Koto-ku, Tokyo, 135-8533, Japan [email protected]

Abstract

It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings [QRT], can be deautonomized to discrete Painlevé equations. In this project [CDT] we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber, clarifying its dependence on a choice of the fiber. Starting from a single autonomous mapping but varying the type of a fiber, we obtain different types of discrete Painlevéequations using this deautonomization procedure. We also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the Picard group, which allows us to obtain factorized expressions of discrete Painlevé equations, including the elliptic case. Imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painlevéequations, including examples whose symmetry groups do not appear explicitly in Sakai’s classification [Sak].

References [CDT] A. S. Carstea, A. Dzhamay, T. Takenawa, Fiber-dependent deautonomisation of integrable 2D mappings and simple expressions of discrete Painlevéequations (in preparation) [QRT] Quispel, G. R. W., Roberts, J., A. G. Thomson, C. J., Integrable mappings and soliton equations 2, Physica D34 (1989) 183–192 [Sak] Hidetaka Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevéequations, Comm. Math. Phys. 220 (2001), no. 1, 165–229. Free fermions at finite temperature and the MNS matrix model

Karl Liechty∗ DePaul University [email protected]

Dong Wang National University of Singapore [email protected]

Abstract

The Moshe–Neuberbger–Shapiro (MNS) matrix model is equivalent to one-dimensional fermions as ﬁnite temperature trapped in a quadratic well and was introduced by those authors in the mid-1990’s as an interpolant between random matrix-type statistics and Poissonian statistics [1]. Johansson [2] studied this crossover rigorously for a grand-canonical version of the ensemble, which is determinantal, and obtained kernels describing the limiting local behavior of the eigenvalues in the crossover regime both in the bulk and at the edge. I will discuss the analysis of the canonical MNS model, in which the number of particles is deterministic. This model is not determinantal, but correlation functions and gap probabilities can be written explicitly in a form amenable to asymptotic analysis. If time permits I will also discuss connections to interacting particle systems in the Kardar–Parisi–Zhang universality class.

References

[1] M. Moshe, H. Neuberger, and B. Shapiro, Generalized ensemble of random matrices, Phys. Rev. Lett., 73(11) (1994), 1497–1500.

[2] K. Johansson, From Gumbel to Tracy-Widom, Probab. Theory Related Fields, 138(1-2) (2007), 75–112. SESSION 16

Nonlocal and full-dispersion model equations in ﬂuid mechanics

Henrik Kalisch University of Bergen [email protected]

John D. Carter Seattle University [email protected]

Abstract

This session focuses on nonlocal and fully dispersive model equations for wave problems in ﬂuid mechanics. Some of these equations, such as the Benjamin-Ono and Whitham equations have very simple forms, but are more diﬃcult to analyze than equations which feature only pure derivative operators.

The session will feature a mix of rigorous mathematical results, numerical work, and modeling approaches. The topics under discussion will include existence and stability of traveling and solitary waves, well-posedness of the Cauchy problem and singularity formation, as well as numerical simulations of wave problems with the help of nonlocal models. Symmetry and decay of traveling waves to a nonlocal shallow water model

Gabriele Bruell, Mats Ehrnstr¨omand Long Pei Department of Mathematical Sciences, Norwegian Institute of Science and Technology, 7491 Trondheim, Norway. [email protected], [email protected] and [email protected]

Abstract

As an alternative for the Korteweg-de Vries (KdV) equation, G. B. Whitham put forward a model [5] for shallow water waves with small amplitude in 1967. This model is known as the Whitham equation now and has a nonlocal dispersive operator.

In the past a few years, a lot of study has been made on the Whitham equation from diﬀerent aspects, including the well-posedness in classical function spaces and the wavebreaking phenomena, although this talk focuses on traveling wave solutions. Recently, both numerical and rigorous investigations have been made on the existence [3, 2] and stability [4, 1] of traveling waves solutions to the Whitham equation.

We consider continuous, supercritical solitary wave solutions to the Whitham equation, which tend to zero at inﬁnity. We prove that all such solutions decay exponentially at inﬁnity and are symmetric, monotone on each side of the symmetry axis. Moreover, any classical symmetric solutions to the Whitham equation are steady. Paley-Wiener type theorems and the method of moving planes are used to prove the decay and symmetry of those solitary waves, respectively.

References [1] J. Cater, H. Kalisch, K. Kodama, N. Sanford, Stability of traveling wave solutions to the Whitham equation, Physics Letters A, 378 (30–31) (2014), 2100–2107. [2] M. Ehrnström,M. D. Groves, E. Wahlén,On the existence and stability of solitary–wave solutions to a class of evolution equations of Whitham type, Nonlinearity, 25 (2012), 1-34. [3] M. Ehrnström,H. Kalisch, Traveling waves for the Whitham equation, Differential and Integral Equations, 22 (11-12 (2009), 1193-1210. [4] V. M. Hur, M. A. Johnson, Modulational instability in the Whitham equation for water waves, Studies in Applied Mathematics, 134.1 (2015), 120-143. [5] G. W. Whitham, Variational Methods and Applications to Water Waves, Proc. R. Soc. Long., Ser. A, 299 (1967), 6-25. Spectral stability of periodic waves in the generalized reduced Ostrovsky equation

Anna Geyer1 and Dmitry E. Pelinovsky2 1 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria 2 Department of Mathematics, McMaster University, Hamilton, Ontario, Canada, L8S 4K1 [email protected] and [email protected]

Abstract

We consider stability of periodic travelling waves in the generalized reduced Ostrovsky equation with respect to co-periodic perturbations. Compared to the recent publications [1, 2], we give a simple argument that proves spectral stability of all smooth periodic travelling waves independently on the nonlinearity power. The argument is based on the energy con- vexity and does not use coordinate transformations of the reduced Ostrovsky equations to the semi-linear equations of the Klein–Gordon type.

References

[1] E.R. Johnson and D.E. Pelinovsky, “Orbital stability of periodic waves in the class of reduced Ostrovsky equations”, 261 (2016), 3268–3304.

[2] S. Hakkaev, M. Stanislavova, and A. Stefanov, “Spectral stability for classical periodic waves of the Ostrovsky and short pulse models”, arXiv: 1604.03024 (2016) Comparisons between experiments and various versions of the Whitham equation

John D. Carter Seattle University [email protected]

Abstract

In 1978, Hammack & Segur performed a series of tightly-controlled laboratory water-wave experiments. The experiments were conducted in a long, narrow tank with relatively shallow undisturbed water and a wave maker at one end. Among many other things, they showed that analytic and asymptotic results obtained from the KdV equation compared favorably with measurements from the experiments. In 2016, Trillo et al. conducted new experiments and demonstrated that the Whitham equation models those experiments more accurately than does the KdV equation.

In this talk, we will compare measurements from the Hammack & Segur experiments with numerical simulations of the St. Venant, KdV, Serre, and Whitham equations. We will focus on various forms of the Whitham equation including the Whitham equation with surface tension and the bi-directional Whitham equation. Whitham-Boussinesq model for variable depth topography

Rosa Maria Vargas Maga˜na, Antonmaria Minzoni, Panayotis Panayotaros

IIMAS, Universidad Nacional Autńoma de México, Apdo. Postal 20-726, 01000 Cd. México, México

Abstract We propose a simpliﬁed long wave model combining a variable depth generalization of the exact nonlocal dispersion with the standard Boussinesq nonlinearity. The model relies on an approximate Dirichlet-Neumann operator that preserves some key structural properties of the exact operator and is simpler than alternative perturbative or implicit expressions. We examine the accuracy of this approximation by studying linear (2-D) normal modes and (3-D) longitudinal and Ursell modes for some geometries for which there are exact results. Solitary waves for a class of nonlocal Green-Naghdi systems

Vincent Duchˆene IRMAR University of Rennes 1, Campus de Beaulieu F-35042 Rennes cedex, France [email protected]

Dag Nilsson and Erik Wahl´en∗ Centre for Mathematical Sciences, Lund University Box 118, 22100 Lund, Sweden [email protected] and [email protected]

Abstract

A class of nonlocal modiﬁcations of the Green-Naghdi system for interfacial gravity waves in shallow water was recently introduced by Duchne, Israwi and Talhouk [1]. It is obtained by including some Fourier multipliers in the usual two-layer Green-Naghdi system and includes as special cases a system with the same linear dispersion relation as the original water wave problem and several well-posed systems (the two-layer Green-Naghdi system suﬀers from the Kelvin-Helmholtz instability in the absence of surface tension). I will outline a variational construction of solitary waves of the nonlocal Green-Naghdi systems. The proof relies on the concentration-compactness principle and, in some cases, a penalization argument. As a special case, one obtains the existence of solitary waves also for a nonlocal version of the classical one-layer Green-Naghdi system, which is well-posed even in the absence of surface tension. I will also present some numerical comparisons.

References

[1] V. Duchêne,S. Israwi, and R. Talhouk, A new class of two-layer Green-Naghdi systems with improved frequency dispersion, Stud. Appl. Math., 137 (2016), 356–415. The Whitham equation with capillarity Evgueni Dinvay and Henrik Kalisch Department of Mathematics, University of Bergen, Postbox 7800, 5020 Bergen, Norway. [email protected] and [email protected] Daulet Moldabayev Oceaneering Asset Integrity AS, PB 1228, Sluppen, 7462 Trondheim, Norway. [email protected] Denys Dutykh LAMA, UMR5127, CNRS - UniversitéSavoie Mont Blanc, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France. [email protected] Emilian Pãrãu School of Mathematics, University of East Anglia, Norwich Research Park, Norwich, NR4 7TJ, United Kingdom [email protected]

Abstract

The validity of the Whitham equation as a model for surface gravity waves of an inviscid incompressible fluid is under consideration. Making use of the Hamiltonian structure of the free-surface water-wave problem and the analyticity of the DirichletNeumann operator we derive a nonlocal Hamiltonian system. The equations describe long surface waves but are fully dispersive in the linear part. The model can be extended to include capillary effects and an elastic surface response. By restricting to one-way propagation, the system reduces to the modified Whitham equation taking into account capillarity or elasticity. The accuracy of unsteady solutions of the capillary Whitham equation and other model equations such as the KdV and Kawahara is compared to the Euler system solutions in different scaling regimes [1]. The performance of the elasticity Whitham model is carried out for steady solutions [2].

References

[1] Dinvay, E., Moldabayev, D., Dutykh, D., Kalisch, H. The Whitham equation with surface tension. Nonlinear Dyn (2017). doi:10.1007/s11071-016-3299-7 [2] Dinvay, E., Moldabayev, D., Kalisch, H., P˜ar˜au,E. The Whitham Equation for Hydroelastic Waves. Preprint. Two-dimensional bifurcation in the Whitham Equation with surface tension

M. Ehrnstr¨omand F. Remonato∗ Norwegian University of Science and Technology [email protected], ﬁ[email protected]

H. Kalisch University of Bergen [email protected]

M. Johnson University of Kansas [email protected]

Abstract

In this talk we show, analytically and numerically, the existence of local two-dimensional bifurcation in the capillary Whitham equation

2 ut + u + K ∗ u = 0, which is deﬁned with the convolution kernel given by s (1 + T ξ2) tanh(ξ) K(x) = F −1 (m (ξ)) (x), m (ξ) = . T T ξ

Here T ∈ R represents the capillarity and allows the Whitham equation to model small- amplitude shallow water waves in presence of surface tension.

In 2011, Ehrnstr¨om,Escher and Wahl´enproved the existence of small amplitude sheets of steady solutions bifurcating from a two-dimensional kernel for the full water-wave problem with vorticity. A similar result has been given for irrotational waves, but with surface tension, by Toland and Jones. After showing analogous results for the Whitham equation we will present several numerical plots uncovering its rich bifurcation structure. Solitary wave solutions to the full dispersion Kadomtsev-Petviashvili equation

Mark Groves FR Mathematik Universit¨atdes Saarlandes Postfach 151150 66041 Saarbr¨ucken Germany [email protected]

ABSTRACT

The KP-I equation ut + m(D)ux − 2uux = 0, where m(D) is the Fourier multiplier operator with multiplier

2 k2 1 1 2 m(k) = 1 + 2 + (β − 3 )k1, 2k1 2 arises as a weakly nonlinear model equation for gravity-capillary waves with strong sur- 1 face tension (Bond number β > 3 ). This equation admits – as an explicit solution – a ‘fully localised’ or ‘lump’ solitary wave which decays to zero in all spatial directions.

Recently there has been interest in the full dispersion KP-I equation

ut +m ˜ (D)ux − 2uux = 0 obtained by retaining the exact dispersion relation from the water-wave problem, that is, replacing m by 1/2 2 2 tanh |k| k2 m˜ (k) = (1 + β|k| ) 1 + 2 . |k| k1 In this talk I show that the full dispersion KP-I equation also has a fully localised solitary- wave solution. The existence theory is variational and perturbative in nature.

This project is joint work with Mats Ehrnstr¨om(NTNU, Norway). Oscillation Estimates for Eigenfunctions for Fractional Schrodinger Operators

Vera Mikyoung Hur University of Illinois at Urbana-Champaign [email protected]

Mathew A. Johnson University of Kansas [email protected]

Jeremy L. Martin University of Kansas [email protected]

ABSTRACT

In this talk, I will discuss a recent extension of the classical Sturm-Liouville theory to describe the oscillation of eigenfunctions associated with fractional Schrodinger operators. Our extension uses the characterization of the fractional Laplacian on the line as a Dirichlet-to- Neumann operator in the upper-half space and the combinatorics of non-crossing partitions. Surface Waves over Point-Vortices

Christopher W. Curtis and Henrik Kalisch San Diego State University University of Bergen [email protected], [email protected]

Abstract

The computation of surface and interfacial waves is a central problem in fluid mechanics. While much has been done, the effect of vorticity on surface wave propagation is still poorly understood. To address this, we study the problem of collections of irrotational point vortices underneath a free fluid surface. We present a derivation of a model and numerical scheme which allows for arbitrary numbers of vortices in a shallow-water limit.

While we are able to recreate much of the classical results for how surface waves form over two counter-propagating vortices, we go beyond this case and look at several examples involving four vortices. We discuss a wide range of unreported surface profiles and their associated energetics. Again, our approach allows for any number of vortices to be present, and so this work serves as a first step towards modeling the interactions of free surface waves with more general vortical profiles. The Cauchy problem for the fractionary Kadomtsev-Petviashvili equations

Jean-Claude Saut Laboratoire de Math´ematiques Universit´eParis-Sud [email protected]

Abstract

We will present various results on the Cauchy problem associated to a class of fractionary Kadomtsev-Petviashvili equations (including the KP version of the Benjamin-Ono and In- termediate Long Wave equations).

This is a joint work with Felipe Linares and Didier Pilod. On the symmetry of traveling-wave solutions to the Whitham equation

Gabriele Bruell∗ Norwegian University of Science and Technology [email protected]

Abstract

The Whitham equation is a nonlocal, nonlinear dispersive wave equation introduced by G. B. Whitham as an alternative wave model equation to the Korteweg-de Vries equation. We investigate questions regarding symmetry and decay of traveling-wave solutions. The method of moving planes is applied to obtain symmetry results for solitary and periodic traveling waves. Moreover, a rigorous study of the integral kernel allows to prove that any solitary traveling-wave solution decays exponentially. This is a joint work with Mats Ehrnstr¨omand Long Pei. SESSION 17

Analysis of numerical methods for dispersive and ﬂuid equations

David M. Ambrose and Gideon Simpson Department of Mathematics, Drexel University 3141 Chestnut St., Philadelphia, PA 19104 [email protected] and [email protected]

Abstract

While much progress has been made on analysis and computing for nonlinear waves, this session focuses on questions at the intersection of these areas, highlighting recent work in numerical analysis and computational methods. In their broadest sense, these topics raise a variety of interesting questions and challenges for dispersive equations, such as the nonlinear Schr¨odingerequation. Closely related challenges appear in ﬂuid dynamics, where dispersion appears in, for instance, free surface problems such as those with surface tension. Explicit high-order symplectic integration of nonseparable Hamiltonians, with a toy NLS example

Molei Tao School of Mathematics, Georgia Institute of Technology, USA

Symplectic integrators preserve the phase-space volume and have favorable per- formances in long time simulations of mechanical systems. Methods for an explicit symplectic integration have been extensively studied for separable Hamiltonians (i.e., H(q,p)=K(p)+V(q)), and they simultaneously achieve both accuracy and eﬃciency. However, nonseparable Hamiltonians also model important problems, such as non-Newtonian mechanics and nearly integrable systems in action-angle coordinates. Unfortunately, implicit methods had been the only available symplectic approach for general nonseparable systems. This talk will describe a recent progress that constructs explicit and arbitrary high- order symplectic integrators for arbitrary Hamiltonians. These integrators are based on a mechanical restraint that binds two copies of phase space together. Using backward error analysis, KAM theory, and some additional multiscale analysis, a pleasant error bound is established for integrable systems. For nonintegrable systems, some numerical experiments with the Nonlinear Schr¨odinger model proposed by Colliander and coauthors will be discussed.

1 On conservation laws of Navier-Stokes Galerkin discretizations

S. Charnyi, T. Heister and L. Rebholz∗ Department of Mathematical Sciences, Clemson University [email protected]

Maxim Olshanskii Department of Mathematics, University of Houston

Abstract

We study conservation properties of Galerkin methods for the incompressible Navier-Stokes equations, without the divergence constraint strongly enforced. In typical discretizations such as the mixed finite element method, the conservation of mass is enforced only weakly, and this leads to discrete solutions which may not conserve energy, momentum, angular momentum, helicity, or vorticity, even though the physics of the Navier-Stokes equations dictate that they should. We aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint, and doing so leads us to a new formulation that conserves each of energy, momentum, angular momentum, enstrophy in 2D, helicity and vorticity (for reference, the usual convec- tive formulation does not conserve most of these quantities). Several numerical experiments are performed, which verify the theory and test the new formulation. Trigonometric integrators for quasilinear wave equations Jeremy Marzuola Department of Mathematics CB#3250, Phillips Hall University of North Carolina, Chapel Hill Chapel Hill, NC 27599 [email protected] Abstract In joint work with Ludwig Gauckler, Jianfeng Lu, FrédéricRousset and Katharina Schratz, Trigonometric time integrators are introduced as a class of explicit numerical methods for quasilinear wave equations. Second-order convergence for the semi-discretization in time with these integrators is shown for a sufficiently regular exact solution. The time integrators are also combined with a Fourier spectral method into a fully discrete scheme, for which error bounds are provided without requiring any CFL-type coupling of the discretization parameters. The proofs of the error bounds are based on energy techniques and on the semiclassical G˚ardinginequality. For the considered and derived trigonometric integrators, we rigorously prove second-order convergence in time. We also prove convergence of a fully discrete method which is based on a combination with a spectral discretization in space, without requiring any CFL-type coupling of the discretization parameters. The methods and their analysis as presented in the present paper can be extended to higher spatial dimensions or to quasilinear Klein-Gordon equations with positive mass or to wave equations without the mass term that appears on the right hand side. Moreover, we could also only assume that a and g are smooth on an open subset and deal with smooth solutions that stay in this subset on the considered time interval. In this way, our scheme can be used to approximate the classical p-system of elasticity and gas dynamics (as long as the solutions are smooth and with no vacuum).

1 Adaptive Methods for Derivative Nonlinear Schrdinger Equations

Gideon Simpson Department of Mathematics, Drexel University Philadelphia, PA, USA [email protected]

Abstract

Numerical simulations of L2 supercritical derivative nonlinear Schrdinger equations suggest the existence of finite time singularities. Thus far, the numerical studies have relied upon either integration of the original equation or the dynamic rescaling method. The first is limited by the singularity, while the latter is limited by the hyperbolic nonlinearity. In both cases, simulations have been restricted to relatively supercritical cases. Using adaptive methods, we overcome prior difficulties, integrating closer to the singularity time. Convergence of a boundary integral method for 3D interfacial Darcy flow with surface tension

David M. Ambrose∗ Department of Mathematics, Drexel University Philadelphia, PA 19104 [email protected]

Abstract

A non-stiff boundary integral method for computing 2D interfacial flows with surface tension was developed by Hou, Lowengrub, and Shelley (HLS). Important elements of the HLS method are the choice of geometric dependent variables, the choice of a favorable parameter- ization, and the identification of leading-order terms to treat implicitly in the time-stepping. Furthermore, Hou and Ceniceros have proven convergence of such methods. We make a generalization of some aspects of these works to 3D: For a 3D interfacial flow with surface tension with fluid velocities given by Darcy’s Law, we develop a non-stiff boundary integral method, and we prove that a version of this method converges. This includes joint work with Yang Liu, Michael Siegel, and Svetlana Tlupova. SESSION 18

Nonlinear waves, dynamics of singularities, and turbulence in hydrodynamics, physical, and biological systems

Alexander O. Korotkevich Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, USA and Landau Institute for Theoretical Physics, Moscow, Russia [email protected]

Pavel M. Lushnikov Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, USA [email protected]

Abstract

Waves dynamics is one of the most interesting and appealing problems in applied mathematics and physics. We encounter waves in all areas of our everyday lives, from waves on the surface of a lake or a pool and sound waves to the electromagnetic waves propagation in ionosphere and plasma excitations on the sun. In wast majority of interesting cases the problem of wave propagation can be solved not only in the linear approximation but also with nonlinear effects taken into account, due to powerful tools of modern applied mathematics and theoretical physics. These approaches together with rapidly emerging computational power leads to new amazing advances in the study of waves dynamics in different media. Common approaches stimulate intensive interchange of ideas in the field which accelerates the development of the wave dynamics even further. Our minisymposium is devoted to new advances in the theory of waves and demonstrates vividly the similarity of approaches in a broad spectrum of important applications. Langmuir wave filamentation in the kinetic regime and multidimensional Vlasov simulations

Denis Silantyev∗ and Pavel Lushnikov University of New Mexico, Albuquerque, New Mexico 87131, USA [email protected] and [email protected]

Harvey A. Rose Theoretical Division, Los Alamos National Laboratory, MS-B213, Los Alamos, New Mexico 87545 New Mexico Consortium Inc., 4200 West Jemez Road Suite 301 Los Alamos, New Mexico 87544 [email protected]

Abstract

Nonlinear Langmuir wave in the kinetic regime kλD > 0.2 has a transverse instability, where k is the wavenumber and λD is the Debye length. The nonlinear stage of that instability development leads to the ﬁlamentation of Langmuir waves which in turn leads to the saturation of the stimulated Raman scattering in laser-plasma interaction experiments.

We study the linear stage of the transverse instability of the particular family [1] of Bernstein- Greene-Kruskal (BGK) modes [2] that is a bifurcation of the linear Langmuir wave. Per- forming direct 2 + 2D Vlasov-Poisson simulations of collisionless plasma we ﬁnd the growth rates of oblique modes of the electric ﬁeld as a function of BGK’s amplitude, wavenumber and the angle of the oblique mode’s wavevector relative to the BGK’s wavevector. Results of the simulations are compared to the theoretical predictions [3].

We also study the linear stage of transverse instability of dynamically prepared nonlinear electron plasma waves to ﬁnd the same instability growth rates suggesting the universal mechanism for the kinetic saturation of stimulated Raman scattering.

References

[1] Harvey A. Rose, David A. Russell, A self-consistent trapping model of driven electron plasma waves and limits on stimulated Raman scatter, Physics of Plasmas, 8 (11), 4784- 4799 (2011).

[2] Ira B. Bernstein, John M. Greene, and Martin D. Kruskal, Exact Nonlinear Plasma Oscillations, Physical Review, 108 (3), 546 (1957).

[3] Harvey A. Rose, Langmuir wave self-focusing versus decay instability, Physics of Plasmas, 12 (1), 012318 (2005). Metastability of the Nonlinear Wave Equation

Katie Newhall Dept. of Mathematics, UNC Chapel Hill [email protected]

Eric Vanden-Eijnden Courant Institute of Mathematical Sciences, NYU [email protected]

Abstract

I study the zero-damping limit of a stochastic Langevin equation describing a spatially extended system, namely a nonlinear wave equation in one-space dimension with stochastic initial data, 2 ′ utt − δ uxx + V (u)=0 x ∈ [0, 1] where δ > 0 is a parameter and V (u) is a potential bounded from below and growing at least like u2 as |u|→∞. For some cases, the solutions display metastability, in the sense that they spend long periods of time in one of two regions of phase-space and only rarely transition between them. In these regimes, the dynamics of the energy conserving wave equation are not fundamentally diﬀerent from that observed in its Langevin counterpart in which random noise and damping terms are added to the equation. Here, I quantify metastability by calculating exactly via Transition State Theory (TST) the mean frequency at which the solutions of the nonlinear wave equation with initial conditions drawn from its invariant measure cross a dividing surface lying in between the metastable sets. This work guides selecting the correct technique for determining the asymptotic scaling of transition times in the low-damping regime of the Langevin equation: TST for large domains, and stochastic averaging for small domains. Simulation of gas transportation networks. Comparison of dynamic and adiabatic approaches

Sergey A. Dyachenko Department of Mathematics at UIUC, 1409 W. Green St, Urbana, IL 61801 USA [email protected]

Alexander O. Korotkevich∗ Department of Mathematics and Statistics, MSC01 1115, 1 University of New Mexico, Albuquerque, NM 87131-0001 USA [email protected]

Michael Chertkov Physics of Condensed Matter and Complex Systems, T-4, Theoretical Division, Los Alamos National Laboratory Los Alamos, NM 87545 [email protected]

Abstract

We introduce a method for simulation of dynamic flows in natural gas transmission pipelines. The method solves a system of nonlinear hyperbolic partial differential equations (PDEs) on a metric graph that represents the pipeline network, subject to time-dependent boundary conditions and nodal controls. Gas dynamics along the pipe is described by gas density and gas velocity that evolve according to the equations of compressible hydrodynamics. The momentum losses on the 3D turbulent flow is modeled with phenomenological Darcy- Weisbach friction. Preliminary results can be found in [1].

As a faster and simpler alternative suitable for slower processes in gas transportation networks in a normal operation regime we consider so call adiabatic approach [2]. We introduced ﬂuctuations of consumption in the system and simulated development of ﬂuctuations of pressure in the frameworks of both methods. Results comparison was performed.

References

[1] . S.A. Dyachenko, A. Zlotnik, A.O. Korotkevich, and Michael Chertkov, Operator Split- ting Method for Simulation of Dynamic Flows in Natural Gas Pipeline Networks, submitted in SIAM Journal of Scientiﬁc Computing, (2016) arXiv: 1611.10008

[2] M. Herty, J. Mohring, V. Sachers, A new model for gas ﬂow in pipe networks, Math. Meth. Appl. Sci., 33 (2010) 845–855. Instability of Steep Ocean Waves and Whitecapping

Sergey Dyachenko1,∗ and Alan C. Newell2 1Institute for Computational and Experimental Research in Mathematics at Brown University, Providence, RI, 2Department of Mathematics, University of Arizona, Tucson, AZ sergey [email protected]

Abstract

Wave breaking in deep oceans is a challenge that still deﬁes complete scientiﬁc understanding. Sailors know that at wind speeds of approximately 5m/sec, the random looking windblown surface begins to develop patches of white foam [1] (whitecaps) near sharply angled wave crests. We idealize such a sea locally by a family of close to maximum amplitude Stokes waves [2, 3, 4] and show, using highly accurate simulation algorithms based on a conformal map representation, that perturbed Stokes waves develop the universal feature of an overturning plunging jet. We analyze both the cases when surface tension is absent and present. In the latter case, we show the plunging jet is regularized by capillary waves which rapidly become nonlinear Crapper waves in whose trough pockets whitecaps may be spawned.

References

[1] S.A. Dyachenko and A. C. Newell, Whitecapping, Studies in Applied Mathematics, 137(2), 199-213 (2016).

[2] S.A. Dyachenko, P.M. Lushnikov, and A.O. Korotkevich, Branch Cuts of Stokes Wave on Deep Water. Part I: Numerical Solution and Pad´eApproximation, Studies in Applied Mathematics, 137 (2016), 419-472.

[3] P.M. Lushnikov, Structure and location of branch point singularities for Stokes waves on deep water, Journal of Fluid Mechanics, 800, 557-594 (2016).

[4] G. G. Stokes, On the theory of oscillatory waves, Mathematical and Physical Papers 1, 197-229 (1880). Global Existence, Blowup and Scattering for large data Supercritical and other wave equations

Avy Soﬀer Department of Mathematics, Rutgers University soﬀ[email protected]

Abstract

I present a new approach to classify the asymptotic behavior of wave equations, supercritical and others, with large initial data. In some cases complete classiﬁcation of the solutions is given. This approach, developed jointly with M. Beceanu, is based on a new decomposition into incoming and outgoing waves for the wave equation, and positivity arguments. I will also present a new class of (quasi-normed) spaces that allow the extension of global existence results for ”large” data in the standard critical norms. On the Bogolubov-de Gennes Equations

Israel Michael Sigal Dept. of Mathematics, University of Toronto Toronto, ON Canada, M5S 2E4 [email protected]

Abstract

In this talk, I present recent results on the Bogolubov-de Gennes equations. These equations give an equivalent formulation of the BCS theory of superconductivity. I explain the connection with the BCS theory, discuss general features of the equations, describe key physical classes of solutions (normal, vortex and vortex lattice states) and present some recent results.

This talk is based on joint work with a PhD student, Li Chen. Stability of periodic waves of 1D nonlinear Schrdinger equations

Stephen Gustafson University of British Columbia [email protected]

Abstract

The cubic focusing and defocusing Schrdinger equations in one dimension admit periodic wave solutions given by snoidal, cnoidal, and dnoidal Jacobi elliptic functions. We present variational characterizations of these waves, and an approach to proving their spectral stability with respect to same-period perturbations which avoids (as much as possible) the use of complete integrability. Joint work with Stefan Le Coz and Tai-Peng Tsai. Nonlinear Schrodinger and Maxwell-Bloch systems with non-zero boundary conditions

Gregor Kovaˇciˇc∗ Rensselaer Polytechnic Institute [email protected]

Gino Biondini, Daniel Kraus, and Sitai Li University at Buffalo biondini@buffalo.edu, sitaili@buffalo.edu, and dkkraus@buffalo.edu

Abstract

The study of scalar and vector nonlinear Schrödinger(NLS) and Maxwell-Bloch (MB) systems with non-zero boundary conditions at infinity has received renewed interest recently. This talk will report on recent results on focusing scalar and vector NLS and MB equations with non-zero boundary conditions. It will be shown how the inverse scattering transform can be constructed in both cases, and a number of explicit soliton solutions will be discussed. (Unconditional) numerical instability of the split-step method in simulations of the soliton of the nonlinear Dirac equations Taras I. Lakoba Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05401 email: [email protected] ABSTRACT The Fourier split-step method (FSSM) is widely used for numerical solution of nonlinear evolution equations. Since the method is explicit, it can be only conditionally stable: for example, for the nonlinear Schrödingerequation (NLS), its stability threshold has been known to be 2 ∆tthresh = O(∆x ), where ∆t and ∆x are temporal and spatial step sizes. Recently, numerical studies [1, 2] addressed the verification of the theoretical prediction of stability of the soliton of the nonlinear Dirac equations. Below we reference only the version of those equations with scalar-scalar self interaction and with cubic nonlinearity: 2 2 2 2 (ut + vx) = −iu + ig(|u| − |v| )u, (vt + ux) = iv − ig(|u| − |v| )v, (1) where g = const. The (stationary) soliton solution of (1) has the form:

{u(x, t), v(x, t)} = {U(x),V (x)} exp[iωsolt], ωsol ∈ (0, 1). (2)

It was found that for ωsol below a certain threshold, the soliton becomes unstable. However, this may occur over very long time, which, moreover, depends on the length L of the computational domain. Therefore, it was concluded [2] that such an instability must be a numerical artifact. None of the methods used in [1, 2] employed the FSSM. We applied the FSSM to system (1) with two questions in mind. First, does the FSSM also ﬁnd an instability of the soliton for suﬃciently low ωsol and, if so, what causes that instability? Second, we wanted to derive a threshold for numerical instability, in analogy to that for the NLS.

Answering the first question, we used the FSSM with ∆t ≪ ∆tthresh (see below) and found that the observed numerical instability indeed had numerical origin: it is caused by radiation (generated by the numerical approximation of the soliton) re-entering the computational domain due to periodic boundary conditions. A simple “trick” of imposing absorbing boundary conditions eliminated that instability and rendered the soliton stable. Answering the second question, we obtained two unexpected findings. First, we found the counterpart of the threshold of the numerical instability for the NLS, which sets in near certain resonance wavenumbers. For Eqs. (1), this threshold is ∆tthresh = ∆x, which was expected given that the dispersion relation for (1) is linear, while for the NLS it is quadratic. The unexpected part was that for ∆t > ∆tthresh, this numerical instability occurs only if the simulated solution is truly, i.e., non-numerically, unstable. Second, we found two new mechanisms of numerical instability: one that occurs for all sufficiently high wavenumbers, and the other, which occurs only near the edges of the computational spectrum, for |k| . kmax. Most surprisingly, both these numerical instabilities may occur for arbitrarily small ∆t, but they depend on L (i.e., may periodically appear and disappear as one increases the length of the computational domain). In this talk we will focus on an analytical explanation of these new instability mechanisms.

References

[1] S. Shao, N.R. Quintero, F.G. Mertens, et. al., Phys. Rev. E 90, 032915 (2014). [2] J. Cuevas-Maraver, P.G. Kevrekidis, A. Saxena, et. al., in Ord. and Part. Diff. Eqs., Chap. 4. Nova Science, 2015. The Relatively Small Effective Nonlinearity of the Nonlinear SchrödingerEquation

Katelyn (Plaisier) Leisman∗ and Gregor Kovaˇciˇc Rensselaer Polytechnic Institute 110 8th Street, Amos Eaton, Troy, NY 12180 [email protected] and [email protected]

David Cai Shanghai Jiao Tong University, China Courant Institute of Mathematical Sciences, New York University, USA [email protected]

Abstract

The linear part of the NLS has quadratic dispersion relation. We have also found that when continuous radiation dominates solutions to the NLS, they exhibit an effective dispersion relation of a shifted parabola. This indicates some degree of linear behavior over long time average. We will first show that this effective dispersion relation minimizes the effective nonlinear behavior in both the pde and the Hamiltonian, and will conclude by showing that as we allow the solution to formally be more nonlinear, the relative effectively nonlinear part of the Hamiltonian actually decreases and the energy behaves effectively more linearly. Hamiltonian Integration Method for Nonlinear Schrödinger Equation

Anastassiya Semenova and Alexander O. Korotkevich Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM [email protected]

Abstract

The nonlinear Schrödinger equations (NLSE) are used to study different physical phenomena such as dynamics of water waves, Langmuir waves in hot plasmas, and light pulses propagation in fiber optics. Different methods can be used to solve these equations numerically. We compare second order split-step method [2] and Hamiltonian integration method introduced in [3]. In order to test these methods, we use focusing one dimensional NLSE which together with trivial integrals of motion (e.g. the Hamiltonian and wave action) allows us to study conservation of nontrivial integrals of motion [1]. Conservation of nontrivial integrals of motion shows whether a numerical method violates the integrability of the system significantly.

According to our simulations, we observe that Hamiltonian integration method provides better conservation of integrals of motion than split-step method. Our estimate shows that Hamiltonian integration method is twice as fast as second order split-step. So, it can be recommended for simulations of solitons interaction, etc.

References

[1] V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one- dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP, 34 (1972), 62–69.

[2] G.P. Agrawal, Nonlinear ﬁber optics (3rd ed.), Academic press,San Diego, CA, USA (2001), isbn:0-12-045143-3

[3] S. Dyachenko, A.C. Newell, A. Pushkarev, V.E. Zakharov, Optical turbulence: weak turbulence, condensates and collapsing ﬁlaments in the nonlinear Schr¨odinger equation, Physica D: Nonlinear Phenomena, 57 (1992), 96-160 Cnoidal wave solutions to the Lugiato-Lefever Equations with applications to microresonators

Zhen Qi and Curtis R. Menyuk UMBC, Computer Science and Electrical Engineering Dept., 1000 Hilltop Circle, Baltimore, MD 21250 [email protected]

Abstract

Comb generation in microresonators is governed by the Lugiato-Lefever equation. It has recently been established that broadband combs can be obtained using solitons. Solitons are a special case of cnoidal waves, and other cnoidal waves that correspond to narrowband combs can have important applications by making it possible to tailor the combs to speciﬁc applications. In this talk, we discuss the accessibility, stability, and methods for generating cnoidal wave solutions Ensemble dynamics and the emergence of correlations in wave turbulence in one and two dimensions

Thomas Y. Sheﬃeld and Benno Rumpf∗ Mathematics Department, Southern Methodist University Dallas, TX 75275, USA tsheﬃ[email protected] and [email protected]

Abstract

We investigate statistical properties of wave turbulence by monitoring the dynamics of ensembles of trajectories. The system under investigation [1] is a simpliﬁed model for surface gravity waves in one and two dimensions with a square-root dispersion and a four-wave interaction term. The simulations of decaying turbulence conﬁrm the Kolmogorov-Zakharov spectral power distribution of wave turbulence theory [2, 3]. Fourth order correlations are computed as ensemble-averages of numerically computed trajectories. The shape, scaling and time-evolution of the correlations agrees with the predictions by wave turbulence theory.

References

[1] A.J. Majda, D.W. McLaughlin, and E.G. Tabak, J. Nonlinear Sci. 7, 9 (1997)

[2] V.E. Zakharov, V. Lvov, G. Falkovich, Kolmogorov Spectra of Turbulence (Springer- Verlag, Berlin 1992)

[3] A.C. Newell, B. Rumpf, Annu. Rev. Fluid Mech. 43, 59 (2011) Instantons and the stochastic Burgers equation

Tobias Sch¨afer City University of New York [email protected]

Abstract

I will review the basics ideas of the instanon formalism for stochastic systems and its relationship to large deviation theory. As an example for the relevance on instantons in such systems, I will consider the stochastically driven Burgers equation and show that the computational instanton yields the correct prediction of the scaling of the probability distribution for large negative gradients of the velocity field. I will also comment on the potential impact of fluctuations around the instanton and further applications to fluid dynamics.

References

[1] T. Grafke, R. Grauer, T.Sch¨afer,and E. Vanden-Eijnden, Relevance of instantons in Burgers turbulence, Eurphysics Letters, 109 (2015) 34003. An Energy Based Discontinuous Galerkin Method for Hamiltonian Systems

Daniel Appel¨o∗ and Anastassiya Semenova Department of Mathematics and Statistics, University of New Mexico, Albuquerque, USA [email protected] and [email protected]

Thomas Hagstrom Department of Mathematics, Southern Methodist University, Dallas, USA [email protected]

Abstract

Fundamental to many models in physics is the Hamiltonian H[u] = R H[u]dx, where the solution u is governed by the equation: ∂u δH = J . ∂t δu Here J is a skew-adjoint Poisson operator, for example i, the imaginary number in Schr¨odinger’s ∂ equation, or simply the spatial derivative ∂x in the case of the scalar transport equation δH ut = ux. The notation δu denotes the functional derivative of the Hamiltonian. Inspired by our formulation of energy based discontinuous Galerkin (dG) methods for wave equations in second order form, [1], we propose a new energy based discontinuous Galerkin method for Hamiltonian systems. The talk will outline the general formulation and the analysis of it.

We will also present numerical experiments illustrating the properties of the method when applied to Korteveg de Vries equation and to the non-linear Schr¨odinger equation in one and two dimensions.

References [1] D. Appel¨oand T. Hagstrom. A new discontinuous Galerkin formulation for wave equations in second order form. SIAM Journal On Numerical Analysis, 53(6):2705–2726, 2015. SESSION 19

Functional analysis and PDEs

Runzhang Xu College of Science Harbin Engineering University, Harbin, P R China [email protected]

Abstract

This session mainly focuses on the functional analysis including variational methods and its applications to the partial diﬀerential equations. Many model equations will be illustrated in the talks, which will be but not limited to the high order nonlinear wave equations, Boussi- nesq equations, nonlinear Shr¨odingerequations, heat equations and so on. The interested problems like the global existence, non-global existence, long time behavior of the global solution will be discussed in this session. Global well-posedness of solutions for a class of fourth-order strongly damped nonlinear wave equations

Yang Yanbing∗ and Xu Runzhang College of Science, Harbin Engineering University, 150001, People’s Republic of China [email protected] and [email protected]

Abstract

This paper studies the initial boundary value problem for some fourth order strongly damped nonlinear wave equations. A finite time blow up result of solutions with low initial energy is showed by employing the concavity method. And also at critical initial energy level we prove the global existence, asymptotic behavior and blow up of solutions. Furthermore we give a sufficient condition on the initial data such that certain solutions with arbitrarily positive initial energy blow up in finite time. Global existence and blow up of solution for semi-linear hyperbolic equation with logarithmic nonlinearity

MD SALIK AHMED College of Science, Harbin Engineering University, 150001, People’s Republic of China. [email protected]

Abstract

We consider the semi-linear wave equation with logarithmic nonlinearity. By modifying and using potential well combined with Sobolev inequality, we gain the results of global existence and blow up at inﬁnity of solution at all initial energy level. The results in this article guide that the polynomial nonlinearity is a critical condition of blow up in ﬁnite time for the semi-linear wave equation.

This paper is funded by the International Exchange Program of Harbin Engineering Univer- sity for Innovation-oriented Talents Cultivation. Global existence and nonexistence for strongly damping wave equations with conical degeneration

Luo Yongbing College of Automation, Harbin Engineering University 150001, People’s Republic of China. [email protected]

Abstract

The existence and non-existence of the global solutions for the initial-boundary value problem for a class of semi-linear degenerate hyperbolic equations with strongly damping terms on the cone Sobolev spaces are obtained in the paper. And also high energy initial data for which the solutions blows up are constructed.

This paper is funded by the International Exchange Program of Harbin Engineering Univer- sity for Innovation-oriented Talents Cultivation. Global existence and blow up of solution for semi-linear edge-degenerate parabolic equations

Chen Yuxuan College of Science, Harbin Engineering University, 150001, People’s Republic of China chen [email protected]

Abstract

In this paper, we study the global existence and blow up of solution for a class of semilinear edge-degenerate parabolic equations. By introducing a family of potential wells and edge type Sobolev inequality, we derive a threshold of the existence of global solutions and the blow up in ﬁnite time with low initial energy and critical initial energy on degenerate edge. On the other hand, the blow up at +∞ with high initial energy is also discussed. Our result implies that the polynomial nonlinearity is important for the solutions of such kinds of semilinear parabolic equations to be blow up in ﬁnite time.

This paper is funded by the International Exchange Program of Harbin Engineering Univer- sity for Innovation-oriented Talents Cultivation. Global non-existence for nonlinear wave equations with conical degeneration with low initial energy

Lian Wei College of Science, Harbin Engineering University 150001, People’s Republic of China. [email protected]

Abstract

For the initial boundary value problem of nonlinear degenerate wave equations utt − ∆Bu = |u|p−1u, we analyze the behavior of the solutions on a manifold with conical singularity, where ∆B is Fuchsian type Laplace with totally characteristic on the boundary x1 = 0. By using a family of potential wells and concavity methods, we obtain non-existence results of global solutions.

This paper is funded by the International Exchange Program of Harbin Engineering Univer- sity for Innovation-oriented Talents Cultivation. SESSION 20

Stability properties for nonlinear dispersive equations

Mathieu Colin INRIA CARDAMOM 200 Avenue de la Vieille Tour, 33405 Talence, Cedex-France [email protected]

Tatsuya Watanabe Kyoto Sangyo University Motoyama, Kamigamo, Kita-ku, Kyoto-City, 603-8555, Japan [email protected]

Abstract

This session is focused on the existence and stability properties of special solutions to various type of nonlinear PDEs including Schr¨odingerequations or systems, water waves equations, asymptotic Boussinesq models and Witham equations. Special attention will be payed to solitary or stationary waves, starting from the existence and uniqueness theory and going up to stability. Numerical simulations will also be considered since in many situations, no theoretical stability results are available.The aim of this session is also to bring forward the state of the art concerning systems of PDEs, since the situation in this case can be drastically complicated compared to the case of single equations. Transverse stability of periodic waves in the KP-II equation

Mariana Haragus1, Jin Li2, and Dmitry E. Pelinovsky2 1 Institut FEMTO-ST & LMB, Univ. Bourgogne Franche–Comt´e,25030, Besan¸con,France 2 Department of Mathematics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada

Abstract

We present a general counting result for the unstable eigenvalues of linear operators of the form JL in which J and L are skew- and self-adjoint operators, respectively. Assuming that there exists a self-adjoint operator K such that the operators JL and JK commute, we prove that the number of unstable eigenvalues of JL is bounded by the number of nonpositive eigenvalues of K. As an application, we discuss the transverse stability of one-dimensional periodic traveling waves in the classical KP-II (Kadomtsev–Petviashvili) equation, which was earlier considered in [2]. We show that these one-dimensional periodic waves are transversely spectrally stable with respect to general two-dimensional bounded perturbations, including periodic and localized perturbations in either the longitudinal or the transverse direction, and that they are transversely linearly stable with respect to doubly periodic perturbations. The proof uses some algebraic manipulations from [1].

References

[1] B. Deconinck and T. Kapitula, “The orbital stability of the cnoidal waves of the Korteweg de Vries equation”, Phys. Lett. A 374 (2010), 4018–4022.

[2] M. Haragus, “Transverse spectral stability of small periodic traveling waves for the KP equation”, Stud. Appl. Math. 126 (2010), 157–185. Solitons in quadratic media

Mathieu Colin∗ INRIA CARDAMOM and Bordeaux INP, France [email protected]

Laurent Di Menza University of Reims [email protected]

Jean-Claude Saut University of Orsay [email protected]

Abstract

In this talk, we investigate the properties of solitonic structures arising in quadratic media. More precisely, we look for stationary states in the context of normal or anomalous dispersion regimes, that lead us to either elliptic or nonelliptic systems of Schr¨odingertype and we address the problem of orbital stability. Finally, we present some numerical experiments in order to compute localized states for several regimes.

References

[1] M. Colin, L. Di Menza and J.-C. Saut Solitons in quadratic media Nonlinearity, 29(3) (2016), 1000-1035. Standing waves for the nonlinear Schr¨odingerequation coupled with the Maxwell equation

Mathieu Colin INRIA CARDAMOM 200 Avenue de la Vieille Tour, 33405 Talence, Cedex-France [email protected]

Tatsuya Watanabe Kyoto Sangyo University Motoyama, Kamigamo, Kita-ku, Kyoto-City, 603-8555, Japan [email protected]

Abstract

In this talk, we consider stability issues of solitary waves for the following nonlinear Schr¨odingerequation coupled with Maxwell equation:

2 2 p−1 iψt + ∆ψ = eϕψ + e |A| ψ + ieψ divA + 2ie∇ψ · A − |ψ| ψ. (1) ¯ 2 2 Att − ∆A = e Im(ψ∇ψ) − e |ψ| A − ∇ϕt − ∇divA. (2) e −∆ϕ = |ψ|2 + divA . (3) 2 t where ψ : R × R3 → C, A : R × R3 → R3, ϕ : R × R3 → R, e > 0, 1 < p < 5 and i denotes the unit complex number. System (1)-(3) describes the interaction of the Schr¨odingerwave function ψ with the gauge potential (A, ϕ) for the magnetic ﬁeld. The constant e describes the strength of the interaction and plays an important role in our analysis.

For ω > 0, we consider the standing wave for (1)-(3) of the form:

ψ(t, x) = exp(iωt)u(x), A(t, x) = 0 and ϕ(t, x) = ϕ(x). (4)

Then one has the following non-local elliptic problem:

−∆u + ωu + e2S(u)u = |u|p−1u in R3, (5)

where u : R3 → C and ∫ 1 1 1 |u(y)|2 S(u)(x) := (−∆)−1|u(x)|2 = ∗ |u|2 = dy. 2 8π|x| 8π R3 |x − y|

First we study the existence and the uniqueness of minimizers with prescribed charge of (5). Then by adapting the Cazenave-Loins type stability result to (1)-(3), we are able to obtain the orbital stability of standing waves of (4) for the quadratic nonlinearity. Instability of solitary waves for a generalized derivative nonlinear Schr¨odingerequation in a borderline case

Noriyoshi Fukaya Department of Mathematics, Graduate School of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan [email protected]

Abstract

We study the following generalized derivative nonlinear Schr¨odingerequation.

− 2 − | |2σ ∈ 1+1 i∂tu = ∂xu i u ∂xu, (t, x) R , where σ > 0. This equation has a two-parameter family of solitary waves of the form iωt e ϕω,c(x−ct). Liu, Simpson and Sulem [1] investigated orbital stability of the solitary waves, and showed that when 1 < σ < 2, its stability or instability depends on the parameter (ω, c). However, they did not treat the case where the parameter (ω, c) is in the borderline between stability and instability regions. In this talk, we treat the borderline case. By extending the abstract theory of Ohta [2] to two-parameter cases, we give a suﬃcient condition for instability in a degenerate case. Moreover, when 7/6 < σ < 2, we verify this condition.

References

[1] X. Liu, G. Simpson and C. Sulem, Stability of solitary waves for a generalized derivative nonlinear Schr¨odingerequation, J. Nonlinear Sci., 23 (2013), 557-583.

[2] M. Ohta, Instability of bound states for abstract nonlinear Schr¨odingerequations, J. Funct. Anal., 261 (2011), 90-110. Existence of a highest wave in a fully dispersive two-way shallow water model

Kyle M. Claassen and Mathew A. Johnson Department of Mathematics, University of Kansas, Lawrence, KS 66045 USA [email protected] and [email protected]

Mats Ehrnstrom∗ Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, 7491 Trondheim, Norway [email protected]

Abstract

We consider the existence of periodic traveling waves in a bidirectional Whitham equation, combining the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity. Of particular interest is the existence of a highest, cusped, traveling wave solution, which we obtain as a limiting case at the end of the main bifurcation branch of 2π-periodic traveling wave solutions. Unlike the unidirectional Whitham equation, containing only one branch of the full Euler dispersion relation, where such a highest wave behaves like |x|1/2 near its crest, the cusped waves obtained here behave like |x log |x||. Although the linear operator involved in this equation can be easily represented in terms of an integral operator, it maps continuous functions out of the H¨olderand Lipschitz scales of function spaces by introducing logarithmic singularities. Since the nonlinearity is also of higher order than that of the unidirectional Whitham equation, several parts of our proofs and results deviate from those of the corresponding unidirectional equation, with the analysis of the logarithmic singularity being the most subtle component. This paper is part of a longer research programme for understanding the interplay between nonlinearities and dispersion in the formation of large-amplitude waves and their singularities. Global dynamics above the ground state energy for a class of nonlinear Schr¨odingerequations with critical growth

Takafumi Akahori and Slim Ibrahim Faculty of Engineering, Shizuoka University Department of Mathematics and Statistics, University of Victoria [email protected] and [email protected]

Hiroaki Kikuchi∗ Department of Mathematics, Tsuda College [email protected]

Hayato Nawa Department of Mathematics, School of Science and Technology, Meiji University [email protected]

Abstract

We consider the following nonlinear Schr¨odingerequation:

∂ψ p−1 4 i + ∆ψ + |ψ| ψ + |ψ| d−2 ψ = 0, (1) ∂t where ψ = ψ(x, t) is a complex-valued function on Rd ×R (d ≥ 3), ∆ is the Laplace operator on Rd and p satisﬁes that 4 4 2 := 2 + < p + 1 < 2∗ := 2 + . (2) ∗ d d − 2 In our previous result, we proved the scattering/blowup dichotomy below a ground state threshold for this equation in the four and higher dimensions, in the spirit of of Kenig and Merle [1]. In this talk, for the ground state of small frequency, we study the solutions slighly above the ground state threshold. Our aim is to derive the “nine-set theory” developed by Nakanishi and Schlag [2].

References

[1] Kenig, C.E. and Merle, F., Global well-posedness, scattering and blow-up for the energy- critical, focusing, non-linear Schr¨odingerequation in the radial case. Invent. Math., 166 (2006), 645–675. [2] Nakanishi, K. and Schlag, W., Global dynamics above the ground state energy for the cubic NLS equation in 3D. Calc. Var. and PDE., 44 (2012) 1–45. SESSION 21

Nonlinear Waves: Mathematical Methods and Applications

Vassilis M Rothos Department of Mechanical Engineering & Lab of Nonlinear Mathematics Faculty of Engineering, Aristotle University of Thessaloniki GR54124 Thessaloniki, Greece [email protected]

Efstathios Charalampidis Department of Mathematics and Statistics Lederle Graduate Research Tower University of Massachusetts Amherst, MA 01003-9305 [email protected]

Abstract he goal of this session is to survey recently developed methods and novel results on the subjects of the formation and the dynamics of Nonlinear Waves and related applications. A wide range of physical systems will be discussed including optics and photonics, metama- terials, Bose-Einstein condensates. In mathematics, these objects are realized as solutions of nonlinear partial diﬀerential equations and nonlinear lattice equations. The speakers in this special session will present their results on the existence, stability, dynamical properties, and bifurcations of these solutions in systems obtained using analytical and numerical techniques. Stability of gap solitons in the presence of a weak nonlocality

I. K. Mylonas and V. M. Rothos Department of Mechanical Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece [email protected] and

A. K. Rossides Oceanography Center, University of Cyprus, Nicosia, Cyprus [email protected]

Abstract

In this work, we study the stability and internal modes of one-dimensional gap solitons employing the modified nonlinear Schrödingerequation with a sinusoidal potential together with the presence of a weak nonlocality [1]. Using an analytical theory, it is proved that two soliton families bifurcate out from every Bloch-band edge under self-focusing or self- defocusing nonlinearity, and one of these is always unstable. Also we study the oscillatory instabilities and internal modes of the modified nonlinear Schrödingerequation.

References

[1] W. Krolikowski, and O. Bang, Phys. Rev. E, 63, (2001) 016610 Bounded ultra-elliptic solutions of the defocusing nonlinear Schr¨odingerequation

Otis C. Wright, III Department of Science and Mathematics Cedarville University 251 N. Main St. Cedarville, OH 45314 e-mail: [email protected]

Abstract

The bounded ultra-elliptic (two-phase) solutions of the defocusing cubic nonlinear Schr¨odingerequation are explicitly constructed by solving a Jacobi inversion problem on an invariant spectral curve. Elementary algebraic arguments are used to ﬁnd simple formulas for both the maximum and the minimum of the solution. Simple algebraic conditions on the branch points of the invariant curve are found for the presence of cavitation, i.e., a point where the solution has zero amplitude. Single and double pulses in the FitzHugh–Nagumo system

Paul Carter Department of Mathematics University of Arizona, Tucson, AZ 85721 [email protected]

Bj¨ornSandstede Division of Applied Mathematics Brown University, Providence, RI 02912 bjorn [email protected]

Abstract

It is well known that the FitzHugh-Nagumo system exhibits stable, spatially monotone traveling pulses, as well as traveling pulses with oscillatory tails. We discuss analytical results regarding the existence and stability of such pulses using geometric blow-up techniques and singular perturbation theory, and we outline a mechanism that explains the transition from single to double pulses that was observed in earlier numerical studies. We also discuss implications for traveling waves and canard orbits in the discrete FitzHugh–Nagumo equations. Inverse scattering transform for a square matrix nonlinear Schr¨odingerequation with nonzero boundary conditions

Barbara Prinari Department of Mathematics - University of Colorado Colorado Springs [email protected]

Abstract

In this talk we discuss the Inverse Scattering Transform (IST) under nonzero boundary conditions for a square matrix nonlinear Schr¨odingerequation which has been proposed as a model to describe hyperﬁne spin F = 1 spinor Bose-Einstein condensates with either repulsive interatomic interactions and anti-ferromagnetic spin-exchange interactions, or attractive interatomic interactions and ferromagnetic spin-exchange interactions.

We formulate the IST in terms of a suitable uniformization variable, which allows to deﬁne the direct and inverse problems on the complex plane, instead of a two-sheeted Riemann surface or the cut plane with discontinuities along the cuts; and we discuss the soliton solutions. Reduced-order prediction of rogue waves in two-dimensional deep-water waves

Mohammad Farazmand and Themis Sapsis Department of Mechanical Engineering, Massachusetts Institute of Technology 77 Massachusetts Av., Cambridge, MA 02139-4307 [email protected] and [email protected]

Abstract

We consider the problem of large wave prediction in two-dimensional water waves. Such waves form due to the synergistic effect of dispersive mixing of smaller wave groups and the action of localized nonlinear wave interactions that leads to focusing. Instead of a direct simulation approach, we rely on the decomposition of the wave field into a discrete set of localized wave groups with optimal length scales and amplitudes. Due to the short-term character of the prediction, these wave groups do not interact and therefore their dynamics can be characterized individually. Using direct numerical simulations of the governing envelope equations we precompute the expected maximum elevation for each of those wave groups. The combination of the wave field decomposition algorithm, which provides information about the statistics of the system, and the precomputed map for the expected wave group elevation, which encodes dynamical information, allows (i) for understanding of how the probability of occurrence of rogue waves changes as the spectrum parameters vary, (ii) the computation of a critical length scale characterizing wave groups with high probability of evolving to rogue waves, and (iii) the formulation of a robust and parsimonious reduced- order prediction scheme for large waves. We assess the validity of this scheme in several cases of ocean wave spectra. Whitham modulation theory for the Kadomtsev-Petviashvili and two dimensional Benjamin-Ono equations

Qiao Wang⇤ and Gino Biondini Department of Mathematics, State University of New York at Bu↵alo Bu↵alo, NY, 14260 qiaowang@bu↵alo.edu and biondini@bu↵alo.edu

Mark Ablowitz University of Colorado, Boulder, CO, 80303 [email protected]

Abstract

A (2+1)-dimensional generalization of the Whitham modulation theory is presented and used to derive the genus-1 Whitham modulation equations for the Kadomtsev-Petviashvili (KP) and two dimensional Benjamin-Ono (2DBO) equations, namely the KP-Whitham system and 2DBO-Whitham system. Basic properties of both Whitham systems are discussed. The Whitham systems are used to study the stability of the genus-1 solutions for the KP and 2DBO equations. Second Order Maxwell-Bloch equation as an inﬁnite dimensional dynamical system

Vassilis M Rothos Department of Mechanical Engineering & Lab of Nonlinear Mathematics Faculty of Engineering, Aristotle University of Thessaloniki GR54124 Thessaloniki, Greece [email protected]

Panagiotis Vassilopoulos and Antonis Charalambopoulos School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou Campus 15780 Athens,Zografou [email protected]

Abstract

Lasers are very rich dynamical systems, which exhibit various time-dependent phenomena characteristic of nonlinear systems, such as phase and mode locking, self-pulsing and breath- ing, and, generally, spatiotemporal pattern formation and dynamical chaos. Almost all these effects can be understood and quantitatively studied using the semiclassical laser theory in the form of Maxwell-Bloch (MB) equations, a set of coupled nonlinear equations for the spaceand time-dependent electric field amplitude E(r, t), and the polarization and inversion of the gain medium P (r, t) and D(r, t). We study the second order Maxwell -Bloch equations governing a two level laser in a ring cavity by proving the persistence of an arbitrarily smooth slow manifold under an unbounded perturbation. The proof is obtained by a modified graph transform method. SESSION 22

Partial diﬀerential equations of quantum physics and their topological solutions

Stephen Gustafson Dept. of Mathematics, University of British Columbia Vancouver, BC, Canada gustaﬂ@math.ubc.ca

Israel Michael Sigal Dept. of Mathematics, University of Toronto Toronto, ON, Canada [email protected]

Avy Soﬀer Dept. of Mathematics, Rutgers University New Brunswick, NJ, USA soﬀ[email protected]

Abstract

This session is dedicated mostly to partial diﬀerential equations of quantum physics (quantum PDEs). Examples of such equations are the Schr¨odinger,Ginzburg-Landau, Yang-Mills, Hartree-Fock, Landau-Lifshitz, Gross-Pitaevski equations.

Though such equations share many features with classical ones and the techniques developed for the latter ones work nicely for the former, there are important distinctions in features and the questions asked. Among the features we mention the gauge invariance and the enhanced role of geometry (diﬀerential and algebraic) and topology. Among the questions one asks are the binding, interaction with the environment (or measuring apparatus), existence of topological defects/excitations, symmetry breaking and its implications.

In this session we review recent progress on the quantum PDEs, the new techniques, the problems and perspectives. Dynamics of singularities in 2D free surface hydrodynamics

Pavel M. Lushnikov1,∗, Sergey A. Dyachenko2 and Denis A. Silantyev1 1Department of Mathematics and Statistics, University of New Mexico, USA 2Institute for Computational and Experimental Research in Mathematics at Brown University, Providence, RI, USA plushnik [a-t] math.unm.edu

Abstract

2D hydrodynamics of ideal fluid with free surface is considered. A time-dependent conformal transformation is used which maps a free fluid surface into the real line with fluid domain mapped into the lower complex half-plane. The fluid dynamics is fully characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. The initially flat surface with the pole in the complex velocity turns over arbitrary small time into the branch cut connecting two square root branch points. Without gravity one of these branch points approaches the fluid surface with the approximate exponential law corresponding to the formation of the fluid jet. The addition of gravity results in wavebreaking in the form of plunging of the jet into the water surface. The use of the additional conformal transformation to resolve the dynamics near branch points allows to analyze wavebreaking in details. The formation of multiple Crapper capillary solutions is observed during overturning of the wave contributing to the turbulence of surface wave. Another possible way for the wavebreaking is the slow increase of Stokes wave amplitude through nonlinear interactions until the limiting Stokes wave forms with subsequent wavebreaking. For non-limiting Stokes wave the only singularity in the physical sheet of Riemann surface is the square-root branch point located. The corresponding branch cut defines the second sheet of the Riemann surface if one crosses the branch cut. The infinite number of pairs of square root singularities is found corresponding to infinite number of non-physical sheets of Riemann surface. Each pair belongs to its own non-physical sheet of Riemann surface. Increase of the steepness of the Stokes wave means that all these singularities simultaneously approach the real line from different sheets of Riemann surface and merge together forming 2/3 power law singularity of the limiting Stokes wave. We found that that non-limiting Stokes wave at the leading order consists of the infinite product of nested square root singularities which form the infinite number of sheets of Riemann surface [1,2].

References

[1] P.M. Lushnikov. Structure and location of branch point singularities for Stokes waves on deep water. Journal of Fluid Mechanics, 800, 557-594 (2016).

[2] S.A. Dyachenko, P.M. Lushnikov, and A.O. Korotkevich. Branch cuts of Stokes wave on deep water. Part I: Numerical solution and Pade approximation. Studies in Applied Mathematics, 137, 419-472 (2016). PDE Models of Ginzburg-Landau Type for Defect Formation in Pattern-Forming Systems

Nicholas M. Ercolani∗ Department of Mathematics, University of Arizona, Tucson, AZ 85721

[email protected]

Abstract

The topic of this talk concerns understanding defect formation in spatially extended pattern- forming systems that are far from equilibrium. We view these defects as coherent structures or, in stat-mech terms, non-equilibrium steady states. Using the Swift-Hohenberg equation as a canonical model, this talk will focus on the transition between grain boundaries and concave-convex disclination pairs. Some of the more recent work described is joint with J. Lega (University of Arizona) and N. Kamburov (Pontiﬁcia Universidad Catlica de Chile). Miguel Ballesteros Indirect Measurements and Quantum Trajectories [email protected] IIMAS, Universidad Nacional Autonoma de Mexico (UNAM)

The Tenth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory March 29 - April 1, 2017

Abstract

We consider a quantum dot in a semi-conductor device P (or a quantum cavity) and a train of probes sent to it. Each probe is suﬃciently far away to any other in order to ensure they do not interact within each other and at most one of them interacts with the quantum dot at any given time. A projective measurement, corresponding to an operator X, is applied to each probe, just after it interacts with P . This produces a discrete density-matrices-valued stochastic process

{ρn}n∈N, where ρn represents the state in P after n measurements occurred. Suppose that N is an observable in P whose Heisenberg evolution varies slowly as the time goes. We prove that after some threshold value N0, the density matrices ρn, for n ≥ N0, are pure on the spectrum of N , up to a tiny error. Our results can be used to mathematically understand situations in the spirit of experiments of Haroche and Wineland, where P consists in a quantum cavity and the environment and N is the photon-number operator. It is possible to construct continuous versions of the discrete stochastic process {ρn}n∈N, which lead to stochastic evolution equations. This is a next step of our future research program. This is a joint work with Martin Fraas (KU Leuven), JürgFröhlich (ETH, Zurich) and Baptiste Schubnel (SBB). Gravitational analog of the Aharonov-Bohm effect

Gregory Eskin,

Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA [email protected]

Abstract In 1959 seminal paper Aharonov and Bohm considered the Schrödinger equation with magnetic potential A(x) = (A1(x),A2()) in a plane domain with an obstacle. They showed that if A(1) and A(2) are two magnetic potential such that their correponding magnetic fields are zero outside the obstacle but the fluxes of A(1) and A(2) are not equal modulo 2πn, n is an integer, then A(1) and A(2) have a different physical impact. This phenomenon is called Aharonov-Bohm (AB) effect. Wu and Yang use the notion of the gauge equivalence to give a more general formulation of AB effect: If A(1) and A(2) are not gauge equivalent then they have a different physical impact. Their formulation of AB effect can be applied to the more general situations such as the case of several obstacles in two and three dimensions, time-dependent electric and magnetic potentials, etc. We shall study the gravitational analog of the AB effect. It was first considered by Stachel who proved it in some special case. We shall consider the general case. We shall show that if g is a stationary metric that is locally static but is not globally static, i.e. g is not isomorphic globally to a static metric g0 then g and g0 have different physical impacts. We shall explain a close analogy between the gravitational AB effect and the quantum mechanical AB effect.

References [1] G. Eskin, Aharonov-Bohm eﬀect revisited, Rev. in Math. Phys., vol. 22, no. 2 (2015), 1530001

1 The Nonlinear Schr¨odingerequation with a potential

Pierre Germain Courant Institute of Mathematical Sciences, 251 Mercer Street, New York 10012-1185 NY, USA. [email protected]

Fabio Pusateri∗ Department of Mathematics, Princeton University, Washington Road, Princeton 08540 NJ, USA. [email protected]

FrédéricRousset Laboratoire de Mathématiquesd’Orsay (UMR 8628), UniversitéParis-Sud, 91405 Orsay cedex, France, et Institut Universitaire de France. [email protected]

Abstract

We consider the cubic nonlinear Schr¨odingerequation with a potential in one space dimension: 2 2 i∂tu − ∂xu + V u = |u| u. (1) We assume that the potential V is generic, suﬃciently localized, with no bound states, and study the long-time asymptotic behavior of small solutions to the Cauchy problem for (1).

We prove that, as time goes to inﬁnity, solutions exhibits modiﬁed scattering with a nonlinear phase correction that depends on the scattering matrix associated to the potential.

The proof of our result is based on the use of the distorted Fourier transform, a precise understanding of the “nonlinear spectral measure” associated to the equation, and various nonlinear stationary phase arguments. Finite gap (multiphase) solutions of the focusing 1D NLS equation and large amplitude (rogue) waves

G. El, M. Bertola and A. Tovbis

Abstract

• Rogue waves appearing on deep water or in optical fibres are often modeled by certain breather solutions of the focusing nonlinear Schrödinger (fNLS) equation which are referred to as solitons on finite background (SFBs).

• A more general modeling of rogue waves can be achieved via the consideration of multiphase, or ﬁnite-band, fNLS solutions of whom the standard SFBs and the structures forming due to their collisions represent particular, degenerate, cases.

• A generalised rogue wave notion then naturally enters as a large-amplitude localised coherent structure occurring within a ﬁnite-band fNLS solution. In this talk, we use the winding of real tori to show the mechanism of the appearance of such generalized rogue waves and derive an analytical criterion distinguishing ﬁnite-band potentials of the fNLS equation that exhibit generalised rogue waves.

References

M. Bertola, G. El and A. Tovbis, Rogue waves in multiphase solutions of the focusing NLS equation, Proceedings of The Royal Society A (2016) DOI: 10.1098/rspa.2016.0340. (see also arXiv:1605.04713).

1 Fluctuation dynamics around Bose-Einstein condensates

Volker Bach Institut fuer Analysis und Algebra, Carl-Friedrich-Gauss-Fakult¨at,TU Braunschweig, 38106 Braunschweig, Germany [email protected]

Sebastien Breteaux BCAM - Basque Center for Applied Mathematics, 48009 Bilbao, Basque-Country, Spain [email protected]

Thomas Chen∗ Department of Mathematics, University of Texas at Austin, Austin TX 78712, USA [email protected]

JürgFröhlich Institut fürTheoretische Physik, ETH Hönggerberg, CH-8093 Zürich, Switzerland [email protected]

Israel Michael Sigal Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada [email protected]

Abstract

In this talk, we discuss an extension to the Hartree equation, which describes thermal ﬂuc- tuations around the Bose-Einstein condensate. Using quasifree reduction, we derive the Hartree-Fock-Bogoliubov (HFB) equations, and discuss the well-posedness of the corresponding Cauchy problem. In particular, the emergence of Bose-Einstein condensates at positive temperature via a self-consistent Gibbs state is addressed. This is based on joint work with V. Bach, S. Breteaux, J. Fr¨ohlich, and I.M. Sigal, [1].

References

[1] V. Bach, S. Breteaux, T. Chen, J. Fr¨ohlich, I.M. Sigal, The time-dependent Hartree- Fock-Bogoliubov equations for Bosons, https://arxiv.org/abs/1602.05171. Quantum Master Equations in Kinetic Theory

Maria Carvalho Dept. of Mathematics, Rutgers University Piscatawy, NJ, USA [email protected]

Abstract

We present recent results on models for quantum systems of N particles undergoing random binary collisions, focusing on the rate of convergence to equilibrium and the propagation of chaos. These questions arise from the work of Mark Kac and his investigation into the probabilistic structure underlying the Boltzmann equation. Recently, the quantum mechanical variation on Kac’s question has begun to be investigated. In this case, the Kac Master equation becomes an evolution equation of Lindblad type, while the corresponding Boltz- mann equation is a novel sort of non-linear evolution equation for a density matrix. The treatment departs from the classical treatment because in quantum mechanics, conditional probability is not always well deﬁned. Nonetheless, a substantial quantum analog of the Kac program can be carried out, and it leads to an interesting and novel class of quantum kinetic equations.

This is joint work with Eric Carlen and Michael Loss.

Nonlinear evolution equations of quantum physics and their topological solutions

Eric Carlen Rutgers University [email protected]

Abstract

We present joint work with Jan Maas showing that Quantum Markov semigroups satisfying a detailed balance condition are gradient flow for quantum relative entropy, and use this prove some conjectured inequalities arising in quantum information theory.

SESSION 23

Waves and Instabilities in Vlasov Plasmas

B. A. Shadwick Department of Physics and Astronomy University of Nebraska–Lincoln [email protected]

Bedros Afeyan Polymath Research [email protected]

Antoine Cerfon Courant Institute of Mathematical Sciences, NYU [email protected]

Abstract

Nonlinear self-organized structures, especially in response to coherent energy injection, con- stitute a fascinating subject in all of many-body physics. In particular, in plasmas, they play a crucial role in stemming chaotic loss of control. To redirect energy or store it in a plasma with the ability to subsequently retrieve it often necessitates the reliance on nonlinear self-organization. This is manifest in astrophysical plasmas, plasma-based accelerators, laser-plasma instability physics, and many other areas. Within this context, we will explore topics ranging from plasma wave echoes to BGK modes to KEEN waves, and reduced phase space descriptions of various flavors, with insights into non-linear plasma self-organization. These sessions include invited presentations on the theory of and computational challenges in tackling nonlinear, self-organized structures from single to multi-mode, from stationary to inherently non-stationary, from continuum or mean field models to particle based modeling schemes, from non-relativistic to relativistic regimes. These papers span analytical theory all the way to massively parallel computations with Vlasov-Fokker-Planck and PIC codes. These sessions, in the setting of Nonlinear Waves and Coherent Structures are an excellent venue to socialize plasma physics results with the rest of the community including crucial links with fluid dynamics. Implicit, charge and energy-conserving particle-in-cell multidimensional algorithms for low-frequency plasma kinetic simulations in curvilinear geometries

G. Chen∗, L. Chac´on Los Alamos National Laboratory Los Alamos, NM 87545 [email protected], [email protected]

Abstract

Classical particle-in-cell (PIC) algorithms employ an explicit approach (e.g. leap-frog) to advance the Vlasov-Maxwell/Poisson system using particles coupled to a grid. Explicit PIC is subject to both temporal stability constraints (either light-wave or plasma-wave CFL) and spatial stability constraints (so-called ﬁnite-grid instability), which makes it unsuitable for engineering-scale kinetic simulations, even with modern massively parallel computers.

Implicit PIC algorithms may eliminate both spatial and temporal stability constraints of explicit PIC, thus potentially becoming orders of magnitude more efficient than explicit ones. This has motivated much exploration of these algorithms in the literature since the 1970’s. However, the lack of efficient nonlinear solvers for a very large system of particle-field equations required approximations that resulted in intolerable accumulation of numerical errors in long-term simulations.

In this presentation, we discuss a multi-dimensional, nonlinearly implicit electromagnetic PIC algorithm [1]. We focus our implementation on the low-frequency Darwin approximation to Maxwell’s equations, but we have recently extended the formulation to the full set of Maxwell equations. The approach conserves exactly total energy, local charge, canonical momentum in ignorable directions, and it automatically preserves involutions such as the Coulomb gauge and Poisson’s equation. Key to the performance of the algorithm is a moment-based preconditioner, featuring the correct asymptotic limits. The formulation has been extended to curvilinear meshes [2], opening the possibility of accurate body-ﬁtted and/or spatially adaptive PIC simulations. The superior accuracy and eﬃciency properties of the scheme will be demonstrated with paradigmatic numerical examples.

References

[1] G. Chen, L. Chac´on, Comput. Phys. Commun. 197 (2015), 73–87 .

[2] L. Chac´on,G. Chen, J. Comput. Phys., 316 (2016), 578–597 Properties of nonlinear electron plasma waves driven by intense lasers

Carl B. Schroeder, Carlo Benedetti, Eric Esarey, and Wim Leemans Lawrence Berkeley National Laboratory, Berkeley, CA 94720 USA [email protected]

Abstract

The coupled evolution of an intense laser propagating in an underdense plasma and excitation of nonlinear Langmuir waves, driven by the ponderomotive force of the laser, are described by the Vlasov-Maxwell equations. The properties of these nonlinear laser-driven plasma waves are of interest for plasma-based particle acceleration applications. Characteristic of this physical system is a separation in time scales between the laser frequency and the electron plasma frequency, such that the response of the plasma to the laser ﬁeld is determined by the laser-frequency-averaged ponderomotive force, and the laser evolves constrained by an adiabatic invariant (wave action conservation). In addition, the evolution of the laser driver and plasma wave are slow compared to the plasma period, allowing the quasi-static approximation, greatly simplifying the coupled equations.

Although the plasma motion in response to the laser field is relativistic, the phase-space distribution is narrow in momentum space. This allows the Vlasov equation to be replaced by fluid equations with closure obtained from a warm (or near cold) approximation [1]. The resulting quasi-static fluid equations driven by the laser ponderomotive force have been extensively studied to determine the properties of electron plasma waves excited by an intense laser pulse. For example, the maximum possible amplitude of the nonlinear longitudinal plasma wave may be calculated [1]. Of particular interest to plasma-based particle acceleration concepts is the value of the phase velocity of the nonlinear plasma wave. This quantity is determined by the evolution of the laser in response to the presence of the nonlinear plasma wave. We present solutions to the coupled nonlinear laser field and plasma wave evolution equations to derive the nonlinear phase velocity of the laser-driven plasma waves. Although the plasma wave phase velocity is typically assumed to be approximately the laser group velocity, we show that the nonlinear wave phase velocity is significantly lower than the laser group velocity and further decreases as the pulse propagates [2].

References [1] C. B. Schroeder and E. Esarey, Relativistic warm plasma theory of nonlinear laser-driven electron plasma waves, Phys. Rev. E, 81 (2010), 056403. [2] C. B. Schroeder, C. Benedetti, E. Esarey and W. P. Leemans, Nonlinear pulse propagation and phase velocity of laser-driven plasma waves, Phys. Rev. Lett., 106 (2011), 135002. Sparse grids for PIC simulations of kinetic plasmas

Lee F. Ricketson Lawrence Livermore National Laboratory, L-637, P.O. Box 808, Livermore, CA 94511-0808, USA [email protected]

Antoine J. Cerfon∗ Courant Institute of Mathematical Sciences, NYU, 251 Mercer St, New York, NY 10012, USA [email protected]

Abstract

The Particle-In-Cell (PIC) method is a popular scheme for the simulation of kinetic plasmas. It is conceptually simple, applicable to a wide variety of physical systems, quite robust and often easier to parallelize than continuum methods. However, PIC simulations of many processes in plasma physics remain prohibitively expensive, requiring several hours on massively parallel architectures. The foremost reason for the large computational cost is the statistical error inherent to particle based methods. High statistical resolution requires a large number of particle per cell. When a large number of cells is needed to resolve a physical process, this leads to a large total number of particles, and thus large CPU and memory costs.

We propose a new method to reduce the computational complexity of PIC simulations. Our scheme combines the PIC method with the sparse grids combination technique [1]. In this approach, the charge density and the current density are computed on a sequence of grids which is such that each grid has a diﬀerent resolution in each coordinate direction and a much lower resolution than the grid one would use for a standard PIC simulation. By a clever combination of the approximations on each grid and an appropriate choice of the sequence of grids, one can achieve an accuracy comparable to the standard PIC simulation, at a much reduced computational cost.

The beneﬁts of applying sparse grids to the PIC method can be explained as follows. Since each sparse grid in the sequence has fewer cells than in a standard PIC simulation, for a ﬁxed number of particles there are more particles per cell in our sparse grids implementation than in standard PIC. Conversely, for a target statistical resolution, our sparse grids scheme requires fewer particles than a standard PIC scheme, thereby reducing the number of particle operations, which usually determine the overall computational complexity of a PIC solver.

References [1] M. Griebel, M. Schneider and C. Zenger, A combination technique for the solution of sparse grid problems, Iterative Methods in Linear Algebra ed. R. Bequwens and P. de Groen (Amsterdam: Elsevier) (1990) 263 - 81. Hamiltonian ﬂuid reductions of kinetic equations in plasma physics

Maxime Perin University of Nebraska-Lincoln [email protected]

Abstract

Solving kinetic equations used in plasma physics is a challenging task due to the high complexity of both their analysis and computation. Here we investigate fluid models derived from kinetic equations. These models have a lower numerical cost and are usually more tan- gible than their kinetic counterpart as they describe the time evolution of physical quantities such as density, fluid velocity, pressure, etc. However, all the fluid moments are dynamically coupled such that there is a need for a closure of the resulting infinite hierarchy of fluid equations, which can be based on various physical assumptions.

Here, we present a strategy for building fluid models from kinetic equations while preserving their Hamiltonian structure. This ensures that the reduction does not introduce any non- physical dissipation. Starting with the one dimensional Vlasov-Poisson equation, we derive a fluid model for the first four moments of the distribution function (density, fluid velocity, pressure and heat flux). We propose a closure based on a dimensional analysis argument and discuss the linear stability of the resulting system of equations.

References

[1] M. Perin, C. Chandre, P.J. Morrison and E. Tassi, Hamiltonian closures for ﬂuid models with four moments by dimensional analysis, J. Phys. A, 48 (2015), 275501 An Eulerian Discontinuous Galerkin Scheme for the Fully Kinetic Vlasov-Maxwell System

J. M. TenBarge, J. Juno IREAP, University of Maryland, MD, USA [email protected] and [email protected]

A. Hakim PPPL, Princeton, NJ, USA [email protected]

Abstract ollisionless or weakly collisional plasmas are ubiquitous in the universe, necessitating a kinetic description. The Vlasov-Maxwell system of equations describes the evolution of the probability distribution function of the particles subject to self-consistently generated electromagnetic ﬁelds. Typically, the Vlasov-Maxwell system is solved in a Lagrangian framework using the particle-in-cell method; however, we focus on the development of a fully kinetic (ions and electrons) Eulerian approach, which is ideal for studying problems requiring low amplitude and high sensitivity, such as energy dissipation in turbulence and magnetic re- connection. In particular, we describe the development and performance of a high-order discontinuous Galerkin scheme in 6D phase space. The resulting system is applied to multiscale turbulence studies with an emphasis on wave-particle resonance interactions and other forms of energy dissipation. The existence of stable BGK waves

Zhiwu Lin School of Mathematics Georgia Institute of Technology Atlanta, GA 30332 email: [email protected]

ABSTRACT

The 1D Vlasov-Poisson system is the simplest kinetic model for describing an electrostatic collisonless plasma. BGK waves are exact steady solutions of Vlasov-Poisson, which play an important role on the long time dynamics of a collisionless plasma as potential ”ﬁnal states” or ”attractors”, thanks to many numerical simulations and observations. Despite their importance, the existence of stable BGK waves has been an open problem since their discovery in 1957. Joint with Yan Guo, we constructed linearly stable BGK waves near homogeneous states. Fourier-Vlasov Simulations in Non-Inertial Reference Frames and Nonlinear Evolution of Electromagnetic Cyclotron Waves

R.D. Sydora and I. Silin Department of Physics, University of Alberta, Canada [email protected] and [email protected]

Abstract

Some useful extensions of the spectral Fourier-Vlasov algorithm [1] for simulations of interactions of collisionless plasmas with ion beams are presented. For many practical applications the relative drifts of various particle populations require high resolution of particle distribution functions (PDFs) or the use of large phase space domain, which makes the simulations extremely memory- and time-consuming. We propose using non-inertial reference frames moving in the velocity dimensions for the beam particle distribution functions [2]. As a result, it is possible to simulate plasma-beam interactions at a much lower resolution. This method is particularly suitable for simulations of fast particle beams and plasmas with heavy ion species or cold particle populations. In addition, for simulations of strongly nonlinear instabilities which cause strong plasma heating, the adaptive mesh refinement and phase space reduction are proposed. Contrary to the Vlasov simulation in the real velocity space, plasma heating in the Fourier inverted space leads to the PDF profile shrinking. Thus, instead of having to extrapolate the PDF into the regions where it was previously undefined, in the Fourier-space, it is sufficient to interpolate the pre-existing solution.

In order to investigate low frequency electromagnetic ion cylcotron waves driven unstable by low density ion beams or ion thermal anisotropy, the Fourier-Vlasov approach is combined with the hybrid Vlasov model where electrons are treated as a massless ﬂuid. The combined algorithm is used to investigate the nonlinear evolution of electromagnetic ion cyclotron waves in multi-ion species plasmas [3]. These waves play an important role in energy transport and heating in collisionless space plasmas.

References

[1] B. Eliasson, Outﬂow boundary conditions for the Fourier transformed three-dimensional Vlasov-Maxwell system, J. Comput. Phys., 225 (2007), 1508-1532.

[2] I. Silin and R. D. Sydora, Hybrid Fourier-Vlasov simulation in non-inertial reference frames, Comput. Phys. Communications, 182 (2011), 2508-2518.

[3] I. Silin, R. D. Sydora, I.R. Mann, K. Sauer, and R.L. Mace, Nonlinear evolution of electromagnetic ion cyclotron waves, Phys. Plasmas, 18 (2011), 042108. Nonlinear echoes and Landau damping

Jacob Bedrossian 4176 Campus Drive College Park, MD 20742 [email protected]

Abstract

In this talk we will discuss the construction of solutions to the Vlasov-Poisson equations with periodic boundary conditions which are initially arbitrarily close to homogeneous equilibrium in Sobolev spaces but which display arbitrarily long sequences of plasma echoes [2]. The oscillations show that Landau’s linear analysis is inaccurate for long times in Sobolev regularity, although it was proved by Mouhot and Villani [1] that the linear analysis correctly predicts the nonlinear behavior for all time in suﬃciently high regularity.

In addition, we also provide an estimate of the smallest amount of collisions necessary to suppress the echoes and restore the approximate validity of linear theory in Sobolev regularity. Time permitting, we may discuss some connections with the subcritical transition threshold of shear ﬂows in the 3D Navier-Stokes equations [4, 5, 6].

References

[1] C. Mouhot and C. Villani. On Landau damping, Acta Math., 207.1 (2011), 29-201.

[2] J. Bedrossian. Nonlinear echoes and Landau damping with insuﬃcient regularity, Preprint; arXiv:1605.06841, (2016).

[3] J. Bedrossian. Suppression of plasma echoes and Landau damping in Sobolev spaces by weak collisions in a Vlasov-Fokker-Planck equation, Preprint, (2017).

[4] J. Bedrossian, P. Germain, N. Masmoudi, Dynamics near the subcritical transition of the 3D Couette ﬂow I: Below threshold case. Preprint; arXiv:1506.03720, (2015).

[5] J. Bedrossian, P. Germain, N. Masmoudi, Dynamics near the subcritical transition of the 3D Couette ﬂow OI: Above threshold case. Preprint; arXiv:1506.03721, (2015).

[6] J. Bedrossian, P. Germain, N. Masmoudi, On the stability threshold for the 3D Couette ﬂow in Sobolev regularity, To appear in Annals of Math.; arXiv:1511.01373, (2015). KEEN and KEEPN Waves in Vlasov Plasmas

Bedros Afeyan Polymath Research Inc. Pleasanton, CA, 94566 [email protected]

Bradley Shadwick∗ University of Nebraska, Lincoln [email protected]

David Larson Lawrence Livermore National Laboratory [email protected]

Abstract

KEEN (Kinetic Electrostatic Electron Nonlinear) Waves and their pair plasma cousins, KEEPN waves, in the setting of the Vlasov-Poisson set of equations, will be discussed. These are non-stationary, multi-mode, far from equilibrium states of plasma self-organization which are driven, for example, by the ponderomotive force of crossing laser beams in high energy density plasmas. We show the scaling of these waves with primitive drive parameters such as the frequency, wavenumber, duration and amplitude of the driving field. These self- sustaining structures much after the drive has been turned off introduce fascinating states of nested phase space structures that help each other maintain a coherent, multi-mode coherent field. KEEN and KEEPN waves challenge the ubiquity assigned stationary, single-mode structures, such as BGK modes, or simpler structures such as echoes and linear Landau damping notions, as viable long time behaviors of plasmas. How phase space partitions are formed, their scaling with drive parameter changes and initial distribution functions will be explored.

Trapped to untrapped particle ratios in the segregated partitions of phase space, each having its own unique dynamical signatures will be given. Ways of using KEEN and KEEPN waves for phase space sculpting purposes will be broached. Their inﬂuence on traditional linearly resonant and single mode structures such as electron plasma waves (EPW) which do form BGK modes time asymptotically, will also be given. Nonlocal disruption or enhancement of modes in phase space due to such interactions between KEEN waves and EPW will be highlighted. Diﬀerent novel simulation tools used to extract these results will be reviewed. Future improvements to these simulation tools currently being pursued will also be mentioned.

Work supported by the Air Force Oﬃce of Scientiﬁc Research (AFOSR) PEEP program. Simulation of Longitudinal and Transverse Instability of Ion Acoustic Waves using the Grid Based Continuum Code LOKI

J. A. Hittinger, T. Chapman, R. L. Berger, B. I. Cohen Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, California 94551, USA [email protected], [email protected], [email protected], [email protected]

J. W. Banks2 Rensselaer Polytechnic Institute, Department of Mathematical Sciences, Troy, NY 12180 [email protected]

S. Brunner Centre de Recherches en Physique des Plasmas, Association EURATOM-ConfédérationSuisse, Ecole Polytechnique Fédéralede Lausanne, CRPP-PPB, CH-1015 Lausanne, Switzerland stephan.brunner@epfl.ch

Abstract

Kinetic simulation of multi-dimensional plasma waves through direct discretization of the Vlasov equation is a useful tool to study fundamental plasma physics and is particularly attractive for situations where low numerical noise levels are desired, e.g., when measuring growth or decay rates of plasma wave instabilities. Direct discretization of high-dimensional phase space is computationally expensive, and as a result, there are few examples of published results using Vlasov codes in more than a single conﬁguration space dimension. To address this situation, we have developed the scalable Eulerian (mesh-based) kinetic code LOKI that evolves the Vlasov-Poisson system in 2D+2V phase space. High-order-accurate conservative methods are used to reduce the cost of phase-space computation, while retaining excellent parallel scalability to make eﬃcient use of large scale computing resources.

This talk will overview the algorithms used in the code as well as large-scale simulation results of the fully kinetic (both ion and electron) evolution of ion acoustic wave decay processes. Ion acoustic waves are shown to be susceptible to at least two such distinct decay processes; the decay channel where daughter modes propagate parallel to the mother mode is found to dominate at larger amplitudes, while the decay channel where the daughter modes propagate at angles to the mother mode dominates at smaller amplitudes. Both decay processes may occur simultaneously and, due to nonlinear trapping, with onset thresholds below those suggested by ﬂuid theory. Eventually, there is a multi-dimensional collapse of the mother mode into a turbulent state analogous to the state that has been shown to produce ion solitons in a single dimension.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-

07NA27344 and funded by the Laboratory Research and Development Program at LLNL under project tracking code 15-ERD-038. LLNL-ABS-720253 Large Amplitude Plasma Waves for Particle Acceleration

B. A. Shadwick Department of Physics and Astronomy University of Nebraska–Lincoln [email protected]

Abstract

Plasmas are capable of supporting coherent, very large amplitude collective oscillations with electric fields in excess of 100 GeV/m, a characteristic that is very attractive for particle acceleration. When a laser pulse propagates in a low density plasma, the transverse oscillations of laser converted into a compressional density wave that supports a large electric field (the so-called wakefield) in the direction of laser propagation. The coupling of drive laser pulse results in excitation of the plasma wave and a loss of energy and momentum by the laser pulse. Thus as the pulse propagates, its frequency is reduced, while the nonlinear optical response of the plasma leads to evolution of the pulse envelope. One consequence of this energy depletion mechanism is the need for surprisingly high spatial resolution to accurately capture the associated physics computationally. We will discuss the general phenomenology of large amplitude wakefields in the context of particle acceleration. We also discuss a semi-analytical theory rate of change of the laser pulse energy and average wave number and develop universal scaling laws [1]. In addition we will touch on thermal effects in laser-driven wakefield accelerators [2] and show that the related phenomenology can be exploited to greatly reduce the computational cost of Eulerian Vlasov simulations [3] of these systems.

Supported by the National Science Foundation under contract no. PHY-1535678

References

[1] B. A. Shadwick, C. B. Schroeder, and E. Esarey, “Nonlinear Laser Energy Depletion in Laser-Plasma Accelerators,” Phys. Plasmas 16, 056704 (2009).

[2] B. A. Shadwick, G. M. Tarkenton, and E. H. Esarey, “Hamiltonian Description of Low- Temperature Relativistic Plasmas,” Phys. Rev. Lett. 93, 175002 (2004).

[3] B. A. Shadwick, G. M. Tarkenton, E. Esarey, and C. B. Schroeder, “Fluid and Vlasov Models of Low-Temperature, Collisionless, Relativistic Plasma Interactions,” Phys. Plas- mas 12, 056710 (2005). SESSION 24

Nonlinear internal waves and their interaction with surface waves

Wooyoung Choi New Jersey Institute of Technology Newark, New Jersey 07102-1982, USA [email protected]

Ricardo Barros Loughborough University Loughborough, Leicestershire LE11 3TU, UK [email protected]

Abstract

Large amplitude internal waves have been observed frequently in coastal oceans, and are believed to be responsible for much of the mixing vital to maintain the grand network of ocean currents that carries heat around the globe. This session will focus on the mathematical modeling of interfacial waves in ﬂuids, especially when wave amplitudes are large and strongly nonlinear eﬀects are important.

Among the topics to be discussed here is the presence of multiple interfaces. In the context of oceanic internal solitary waves (ISW), two-layer models may eﬀectively be used to describe these waves, provided the density transition layer is sharp enough. In most theoretical studies, it is common to assume that the top surface is rigid under the so-called rigid- lid approximation. However, it is important to allow the top surface to be free when the interaction of internal solitary waves with relatively short surface waves is investigated to better understand their surface expression appearing on satellite images.

In addition, two-layer models can only capture the prevalent form of ISW observed in the ocean – known as the ﬁrst baroclinic mode (mode-1) ISW – and fail to describe another equally important form of observed waves. These are known as the second baroclinic mode (mode-2) ISW and, to describe them, two-layer models must be extended and include an intermediate layer.

A blend of analytical, numerical, and experimental work will be featured in this session. Long wave approximation with hodograph transformation for periodic internal waves in a two-ﬂuid system

Sunao Murashige∗ Department of Mathematics and Informatics, Ibaraki University Mito, Ibaraki, 310-8512, Japan [email protected]

Abstract

This work considers long wave approximation with hodograph transformation for irrotational motion of periodic waves propagating in permanent form with constant speed at the interface between two immiscible inviscid fluids bounded by rigid top and bottom walls, on the assumption that the thickness-to-wavelength ratio is small in each layer. The wave motion is two-dimensional in the vertical cross-section along the direction of wave propagation. This irrotational plane flow is steady in a frame of reference moving with waves, and the flow domain for one period in each layer can be conformally mapped onto a rectangular region with small aspect ratio in the corresponding complex potential plane, respectively.

Suitable choice of variables in these conformally mapped domains, namely the hodograph transformation, allows us to derive some nonlinear models for periodic internal waves using smallness of the aspect ratio of each domain and the idea of long wave approximation. The merit of this approach is that one of the independent variables is taken along streamlines and, even for overhanging wave solutions, the interface can be represented by a single- valued function. Numerical examples demonstrate that a strongly nonlinear model can catch broadening and overhanging of wave proﬁles with increase of wave amplitude. Large amplitude internal waves in three-layer ﬂows

Ricardo Barros Loughborough University Loughborough, Leicestershire LE11 3TU, UK [email protected]

Wooyoung Choi New Jersey Institute of Technology Newark, NJ 07102-1982, USA [email protected]

Paul Milewski University of Bath Claverton Down, Bath BA2 7AY, UK [email protected]

Abstract

Large amplitude internal waves in a three-layer ﬂow conﬁned between two rigid walls will be examined in this talk. The mathematical model under consideration arises as a particular case of the multi-layer model proposed by Choi [2] and is an extension of the two-layer MCC (Miyata-Choi-Camassa) model [4, 3]. The model can be derived without imposing any smallness assumption on the wave amplitudes and is well-suited to describe internal waves within a strongly nonlinear regime. Solitary-wave solutions will be investigated and some of their properties will be unveiled by carrying out a detailed critical point analysis of the underlying dynamical system, as in [1]. We will also address the role played by criticality on the polarity of interfacial waves and highlight some shortcomings of the Boussinesq approximation.

References [1] R. Barros, Remarks on a strongly nonlinear model for two-layer ﬂows with a top free surface, Stud. Appl. Math., 136 (2016), 263–287. [2] W. Choi, Modeling of strongly nonlinear internal waves in a multilayer system, in Proc. of the Fourth Int. Conf. on Hydrodynamics (Y. Goda, M. Ikehata, K. Suzuki Eds.), (2000), 453–458. [3] W. Choi and R. Camassa, Fully nonlinear internal waves in a two-ﬂuid system, J. Fluid Mech., 396 (1999), 1–36. [4] M. Miyata, Long internal waves of large amplitude, in: H. Horikawa, H. Maruo, (Eds.), Proc. of the IUTAM Symp. on Nonlinear Water Waves, (1988), 399–406. A mathematical model for spilling breakers

Gavrilyuk, S. L. ∗ Aix-Marseille Universit´e,UMR CNRS 7343, IUSTI, 5 rue E. Fermi, Marseille, France [email protected]

Liapidevskii, V. Yu. and Chesnokov, A. A. Novosibirsk State University, 2 Pirogova street, 630090 Novosibirsk, Russia [email protected] and [email protected]

Abstract

A two-layer long-wave approximation of the homogeneous Euler equations for a free- surface flow evolving over mild slopes is derived. The upper layer is turbulent and is described by depth-averaged equations for the layer thickness, average fluid velocity and fluid turbulent energy. The lower layer is almost potential and can be described by Serre–Su–Gardner– Green–Naghdi equations (a second-order shallow water approximation with respect to the parameter H/L, where H is a characteristic water depth and L is a characteristic wavelength). A simple model for vertical turbulent mixing is proposed governing the interaction between these layers. Stationary supercritical solutions to this model are first constructed, containing, in particular a local turbulent subcritical zone at the forward slope of the wave. The non-stationary model was then numerically solved and compared with experimental data.

References

[1] S. L. Gavrilyuk, V. Yu. Liapidevskii and A. A. Chesnokov, Spilling breakers in shallow water: applications to Favre waves and to the shoaling and breaking of solitary waves, J.Fluid Mech., 808 (2016), 441-468. On modeling nonlinear surface and internal waves

Wooyoung Choi Department of Mathematical Sciences New Jersey Institute of Technology Newark, NJ 07102, USA [email protected]

Abstract

We revisit an asymptotic formulation based on unsteady Stokes expansion for nonlinear surface waves in water of ﬁnite depth. Our focus is on spectral models for wave ﬁelds with either continuous or discrete spectrum and their applications to traveling and standing waves are discussed. The formulation is then generalized to a two-layer system to describe the interaction between surface and internal waves. Nonlinear evolution of Faraday waves in a rectangular container

William Batson and Wooyoung Choi New Jersey Institute of Technology Department of Mathematical Sciences [email protected] and [email protected]

Abstract

The aim of this work is to study interaction between internal and surface waves driven by parametric excitation[1]. Before considering the requisite two-layer problem, here we analyze the simpler one-layer case. Starting with a model based on the work of West et al.[2], we ﬁrst derive a complex nonlinear ODE that governs the evolution of the primary instability. Then, the ﬁxed points and steady states of this equation will be discussed.

References

[1] J. Miles and D. Henderson, Parametrically forced surface waves, Annu. Rev. Fluid Mech., 22 (1990): 143-165.

[2] B. J. West, K. A. Brueckner, R. S. Janda, D. M. Milder and R. L. Milton, A new numerical method for surface hydrodynamics,J. Geophys. Res., 92 (1987), 11803-11824. Internal Waves in the Ocean in the Presence of Shear: wave turbulence perspective.

Yuri V Lvov Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy NY 12180 [email protected]

Abstract

In our previous studies we have investigated wave turbulence of internal waves in the ocean and had shed light on how internal wave energy spectrum might be formed. In this study we present preliminary results about wave turbulence of internal waves in the presence of horizontally uniform shear. We derive kinetic equation for spectral energy density of internal waves in the presence of shear, and analyze its structure, properties and possible solutions. SESSION 25 Nonlinear dynamics in Mathematical Biology and Neuroscience

Andrea Barreiro Southern Methodist University P.O. Box 750156 Dallas, TX 75275 [email protected]

Katie Newhall University of North Carolina at Chapel Hill 120 E Cameron Avenue Chapel Hill, NC 27514 [email protected]

Remus Osan Georgia State University 30 Pryor Street Atlanta GA, 30303 [email protected]

Pamela B. Pyzza Ohio Wesleyan University 61 S. Sandusky Street Delaware, OH 43015 [email protected]

Abstract

Nonlinear dynamics arise throughout mathematical biology and neuroscience. While some systems can be analyzed with familiar techniques used to analyze diﬀerential equations in physics, others have novel properties which require novel techniques. The speakers in this session use computational, analytical, and experimental tools to address problems in the following areas of mathematical biology and neuroscience: (1) Mathematical Advances in Molecular Biology, (2) Dynamics of Sleep, and (3) Coherent Structures, Symmetries, and Correlations in Neuronal Networks Effective dynamics of multiple molecular motors

Joe Klobusicky* and Peter Kramer Rensselaer Polytechnic Institute [email protected] and [email protected]

Abstract

The transport of cargo attached to multiple motors may be modeled as a system of stochastic differential equations. Motors may switch between attached and detached states, each of which having separate equations for determining motor positions. In this talk, we derive effective velocities and diffusions for such motor systems. This involves limit theorems taken from renewal theory and multiscale averaging techniques. Identical motor systems and nonidentical systems, both cooperative and tug of war, will be considered. Synchronizing cortical dynamics via electrotonic junctions between excitatory neurons

Jennifer Crodelle∗ and Gregor Kovacic Rensselaer Polytechnic Institute 110 8th street, Amos Eaton Troy, NY 12180 [email protected], [email protected]

David Cai Shanghai Jiao Tong University, China Courant Institute of Mathematical Sciences, New York University, USA [email protected]

Abstract

Synchronized neuronal activity has been shown to contribute to cognitive processes such as learning and memory formation [1]. Though much is still unknown about the mechanism behind the generation of this synchrony, researchers believe that electric coupling through sites called gap junctions may facilitate its emergence.

In this paper, we use a modiﬁed version of the Hodgkin-Huxley equations to construct a detailed model with both synaptic and electric coupling and examine the resulting emergence of synchrony.

References

[1] Y. Wang, A. Barakat, and H. Zhou. Electrotonic coupling between pyramidal neurons in the neocortex. PLoS One, 5(4):e10253, 2010. An artiﬁcial neural network approach to automated particle tracking analysis of 2D and 3D microscopy videos

Jay Newby Department of Mathematics and Applied Physical Sciences, University of North Carolina–Chapel Hill, Chapel Hill, NC 27599 [email protected]

Abstract racking of microscopic species is one of the most utilized experimental technologies in materials science, biophysics, tissue engineering and nanomedicine. The goal is to analyze particle paths to elucidate the underlying transport mechanism. Routinely tracked particles include viruses, bacteria, microbeads, and nano-sized drug carriers. The relevant biological fluids create imaging environments with exceedingly poor signal-to-noise ratios. To overcome these shortcomings, we develop a new approach for particle identification and tracking, based on deep learning and convolutional neural networks. Modeling the Dynamics of the Human Sleep/Wake Cycle Shelby Wilson Morehouse College 830 Westview Drive, Atlanta, GA 30311 [email protected] Selenne Bañuelos California State University-Channel Islands, Camarillo, CA [email protected] Janet Best The Ohio State University, Columbus, OH [email protected] Gemma Huguet Universitat Politècnicade Catalunya, Barcelona, Spain [email protected] Alicia Prieto-Langarica Youngstown State University, Youngstown, OH [email protected] Pamela Pyzza Ohio Wesleyan University, Deleware, OH [email protected] Markus H. Schmidt Ohio Sleep Medicine Institute, Dublin, OH [email protected]

Abstract Here, we present a nonlinear, Morris-Lecar type, ODE model of human sleep-wake regulation including REM/NonREM dynamics. This model demonstrates several features observed in humans during sleep such as elongation of duration and number of REM bouts across the night. We will also discuss the dynamics associated with the model. This model qualitatively agrees with experimental data which and therefore could be used as a foundation for experimental simulations pertaining to jet lag, sleep deprivation, and temperature eﬀects on sleep. Pattern formation in a nonlocal mathematical model for the multiple roles of the TGF-β pathway in tumour dynamics.

Raluca Eftimie Division of Mathematics, University of Dundee Dundee, DD14HN, UK [email protected]

Mathieu Perez Institut National des Sciences Appliqu´eesde Rouen 76801 Saint-Etienne´ du Rouvray Cedex, France [email protected]

Pietro-Luciano Buono∗ Faculty of Science, University of Ontario Institute of Technology Oshawa, ONT, L1H 7K4, Canada [email protected]

Abstract

This talk introduces a partial diﬀerential equation (PDE) model to investigate the complex roles of TGF-β signalling pathways on the inhibition and growth of tumours, as well as on the epithelial-to-mesenchimal transition involved in the metastasis of tumour cells. TGF-β is one of the most investigated signalling pathways in oncology since it can regulate multiple aspects of cell behaviour: cell proliferation and apoptosis.

I will be discussing a pair of nonlocal first-order partial differential equations in one spatial dimension describing repulsive and adhesive cell-cell interactions. The model is coupled with a reaction-diffusion equation for the growth and spread of TGF-β molecules. By imposing periodic boundary conditions, the PDE model is symmetric with respect to the group O(2) of all translations on a circle and a special reflection. I will present a few analytical results, but mostly numerical simulations which show that this model can explain the formation of new tumour cell aggregations at positions in space that are further away from the main aggregation and also explore the relation between the tumour size and its metastatic spread.

References

[1] R. Eftimie, G. de Vries, M.A. Lewis, and F. Lutscher, Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol., 69 (2007), 1537–1566. Stochastic Modeling of Motor-driven DNA Origami

John Fricks School of Mathematical and Statistical Sciences Arizona State University [email protected]

Abstract any cell types, neurons in particular, require active transport to get cargos to where they are needed using ATP-driven molecular motors along microtubules. However, the observed features of the motors in an in vitro environment using straightforward models fail to capture the behavior of living cells. To resolve this paradox, we study motor-cargo complexes using DNA origami as cargo, where interactions with the molecular motors can be specified. Stochastic models of these systems along with statistical analysis will be presented. Modeling the Effects of Temperature on Sleep Patterns Pamela B. Pyzza Ohio Wesleyan University 61 S. Sandusky Street, Delaware, OH 43015 [email protected] Selenne Bañuelos California State University-Channel Islands, Camarillo, CA [email protected] Janet Best The Ohio State University, Columbus, OH [email protected] Gemma Huguet Universitat Politècnicade Catalunya, Barcelona, Spain [email protected] Alicia Prieto-Langarica Youngstown State University, Youngstown, OH [email protected] Markus H. Schmidt Ohio Sleep Medicine Institute, Dublin, OH [email protected] Shelby Wilson Morehouse College, Atlanta, GA [email protected]

Abstract Experimental work and prior models suggest that changes in ambient temperature can affect sleep patterns in humans. This is further supported by the observation that the neurons responsible for sleep behavior and many temperature-sensitive neuron populations are both located in the hypothalamus. We have constructed a mathematical model of human sleep–wake behavior incorporating thermoregulation and ambient temperature responses. Simulations of this model show features previously presented in experimental data such as elongation of duration of REM bouts and number of REM bouts during the night, REM latency at the onset of sleep, and the appearance of awakenings due to deviations in body temperature from thermoneutrality. This model qualitatively agrees with experimental data which suggests that humans experience more awakenings during the night when sleeping at extreme ambient temperatures. We will further investigate the how sleep history may play a role in sleep behavior. Wave Patterns in an Excitable Neuronal Network

Christina Lee∗ Department of Mathematics and Computer Science Oxford College of Emory University 180 Few Circle Oxford, GA 30054 [email protected]

Gregor Kovaˇciˇc Mathematical Sciences Department Rensselaer Polytechnic Institute 110 8th Street Troy, NY 12180 [email protected]

Abstract

This talk describes a study of spiral- and target-like waves traveling in a two-dimensional network of integrate-and-ﬁre neurons with close-neighbor coupling. The individual neurons are driven by Poisson trains of incoming spikes. Each wave is a result of a ﬂuctuation in the drive. It begins as a target or a spiral, and eventually evolves into a straight “zebra”-like grating. Some of the waves contain defects arising from collisions with other waves. The wavelength and wave speed of the patterns were investigated, as were the temporal power spectra of the oscillations experienced by the individual neurons as waves were passing through them. Firing Rate Heterogeneity and Consequences for Coding in Feedforward Circuits

Cheng Ly∗ Virginia Commonwealth University Richmond, VA 23284 [email protected]

Gary Marsat West Virginia University [email protected]

Keywords: ﬁring rate heterogeneity, electric ﬁsh, neural coding.

Topics: Session 25, Nonlinear dynamics in mathematical biology and neuroscience

Abstract

Heterogeneity of neural attributes is recognized as a crucial feature in neural processing. We present recently developed theoretical methods to characterize how the ﬁring rate distribution of neurons changes with intrinsics and network heterogeneity in a generic recurrent spiking neural network model. The relationship between intrinsic and network heterogeneity can lead to various levels of ﬁring rate heterogeneity, depending on regime. We employ dimension reduction methods and asymptotic analysis to derive compact expressions to describe the phenomena.

Next we adapt our work to a delayed feedforward network model of the electrosensory system of electric ﬁsh. Experimental recordings indicate that feedforward network input can mediate response heterogeneity of pyramidal cells. We use the theory to demonstrate struc- tured connectivity rules can lead to qualitatively similar statistics as the experimental data. The stimulus tuning of particular cells is related to the eﬀective network architecture or connectivity. Thus, the model demonstrates that intrinsic and network attributes do not interact in a linear manner but rather in a complex stimulus-dependent fashion to increase or decrease neural heterogeneity and thus shape population codes.

We also present some preliminary work based on electric fish data with noisy stimuli where we find that firing rate heterogeneity is a signature of optimal (Bayesian) stimulus estimation. Dimension reduction for stochastic conductance based neural models with time scale separation

Deena R. Schmidt∗ Department of Mathematics and Statistics University of Nevada, Reno [email protected]

Roberto F. Gal´an1 and Peter J. Thomas2 Department of Electrical Engineering and Computer Science1 Department of Mathematics, Applied Mathematics and Statistics2 Case Western Reserve University [email protected] and [email protected]

Abstract

Markov process are used throughout cell biology and neuroscience to model the random dynamics of processes transitioning among multiple states. Complexity reduction for such models aims to capture the essential dynamics via a simpler representation, with minimal loss of accuracy. Classical approaches, such as aggregation of nodes and adiabatic elimi- nation of fast variables, lead to reduced models that are no longer Markovian. Stochastic shielding provides an alternative approach by simplifying the description of the noise driving the process, while preserving the Markov property, by removing from the model those ﬂuctuations that are not directly observable [1].

We previously applied the stochastic shielding approximation to several Markov processes arising in neuroscience and processes on random graphs [2]. In this talk, we explore the range of validity of the stochastic shielding approximation for processes with nonuniform stationary probabilities and multiple timescales, including ion channel models with “bursty” dynamics. We show that stochastic shielding is robust to the introduction of time scale separation for a class of simple networks, but it can break down for more complex systems with three distinct time scales. We also show that our related edge importance measure remains a valid tool for analysis for arbitrary networks regardless of multiple time scales.

References [1] N.T. Schmandt and R.F. Gal´an.Stochastic-shielding approximation of Markov chains and its application to eﬃciently simulate random ion-channel gating, Phys. Rev. Lett., 109 (2012), 118101. [2] D.R. Schmidt and P.J. Thomas. Measuring edge importance: A quantitative analysis of the stochastic shielding approximation for random processes on graphs, J. of Math. Neurosci., 4 (2014), 1-52. Piecewise smooth maps for the circadian modulation of sleep-wake dynamics

Victoria Booth and Cecilia Diniz Behn Departments of Mathematics and Anesthesiology, University of Michigan Department of Applied Mathematics and Statistics, Colorado School of Mines [email protected] and [email protected]

Abstract

The timing of human sleep is strongly modulated by the 24 h circadian rhythm. To investigate the dynamics of circadian modulation of sleep, we developed a one-dimensional map for a physiologically-based, sleep-wake regulatory network model for human sleep [1]. The piecewise continuous map reveals changes in sleep patterning, including REM sleep behavior, at diﬀerent circadian phases. Using the map, we analyze model bifurcations to understand how variations in REM sleep propensity and the homeostatic sleep drive aﬀect human sleep patterning.

References

[1] V. Booth, I. Xique and C. G. Diniz Behn, One-dimensional map for the circadian modulation of sleep in a sleep-wake regulatory network model for human sleep, SIAM J Appl Dyn Sys, accepted for publication (2017). Edge-correlations and synchrony in neuronal networks

Duane Q. Nykamp∗ School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA [email protected]

Abstract

To examine how network microstructure and global structure can inﬂuence synchronization in a neuronal network, we developed a network model that combines edge correlations of the SONET model [1] with heterogeneous ﬁrst order connectivity statistics. The second order connectivity statistics, i.e., the edge-covariances that determine the frequency of convergent, divergent, chain, and reciprocal connections, remain homogeneous across the network. We investigate how the SONET parameters as well as the global parameters predict synchrony across the network.

References

[1] L. Zhao, B. Beverlin II, T. Netoﬀ and D. Q. Nykamp, Synchronization from second order network connectivity statistics Frontiers Comp. Neurosci., 5 (2011), 28. Stochasticity and the Neural Sleep-Wake Architecture

Tom Dinitz* and Janet Best Department of Mathematics Ohio State University [email protected] and [email protected]

Abstract

In several mammalian species, while sleep bouts are exponentially distributed, an intermediate power law has been observed in wake bouts [1]. These distributions suggest stochasticity plays an important role in the timing of sleep-wake transitions. However, it remains unclear which elements of the neural system drive the stochasticity observed in bout distibutions. Interestingly, during the sleep-wake cycle, REM-promoting neurons have greater variability in ﬁring rates than REM-inhibiting or wake-promoting neurons.

In this talk, I introduce a new model for the sleep-wake network utilizing doubly stochastic Poisson processes (Cox processes). I then show that a simpliﬁed model can exhibit power-law phenomenon similar to that observed in the distribution of wake bouts. Finally, I analyze a toy model in order to better understand how the stochastic elements result in an intermediate power-law distribution.

References

[1] Lo, Chung-Chuan, et al. Common scale-invariant patterns of sleepwake transitions across mammalian species. Proceedings of the National Academy of Sciences of the United States of America, 101.50 (2004), 17545-17548. Constraining neural networks with spiking statistics

Andrea K. Barreiro∗ Department of Mathematics Southern Methodist University [email protected]

Shree Hari Gautam and Woodrow L. Shew Department of Physics University of Arkansas [email protected] and [email protected]

Cheng Ly Department of Statistical Sciences and Operations Research Virginia Commonwealth University [email protected]

Abstract

As experimental tools advance, measuring whole-brain dynamics with single-neuron resolution is becoming closer to reality. However, a task that remains technically elusive is to measure the interactions within and across brain regions that govern such system-wide dynamics. We propose a procedure to derive constraints on hard-to-measure network attributes — such as inter-region synaptic strengths — using easy-to-measure quantities such as ﬁring rates and pairwise correlations.

As a test case, we studied interactions in the olfactory system [1]. We used two micro- electrode arrays to simultaneously record from olfactory bulb (OB) and anterior piriform cortex (PC) of anesthetized rats who were exposed to several odors. We predicted that i) inhibition within the aﬀerent region (OB) has to be less than the inhibition in PC, ii) excitation from PC to OB is often stronger than excitation from OB to PC, iii) excitation from PC to OB and inhibition within PC have to both be relatively strong compared to presynaptic inputs from OB. These predictions are validated in a full spiking (leaky integrate- and-ﬁre) neural network model of the OB–PC pathway.

References

[1] A.K. Barreiro, S.H. Gautam, W.L. Shew, and C. Ly. A theoretical framework for analyzing coupled neuronal networks: Application to the olfactory system. Submitted, 2017. Traveling waves in one-dimensional Hodgkin Huxley neuronal networks

Ricardo Erazo∗ Neuroscience Institute Georgia State University [email protected]

William Barnett, Gennady Cymbalyuk, Remus Osan Neuroscience Institute, Department of Mathematics and Statistics Georgia State University [email protected]

Gennady Cymbalyuk Neuroscience Institute Georgia State University [email protected]

Remus Osan Department of Mathematics and Statistics Georgia State University [email protected]

Abstract

The present work explores propagation of traveling waves in one-dimensional excitatory neuronal networks of biophysically inspired Hodgkin Huxley-type cells. The model relies on a codimension-2 bifurcation to exhibit bistability, quiescence, tonic spiking and bursting [1]. Our data show that for all cases of asymmetrical connections, traveling waves are front-propagating and reach a constant speed solution, in agreement with integrate and ﬁre models [2]. On the other hand, symmetrical connections produce front-propagating and back-propagating waves.

References

[1] W.H. Barnett, G.S. Cymbalyuk (2017) A Codimension-2 Bifurcation Controlling En- dogenous Bursting Activity and Pulse-Triggered Responses of a Neuron Model. PLoS ONE 9(1): e85451. doi:10.1371/journal.pone.0085451 [2] J Zhang, R. Osan, (2016) Analytically tractable studies of traveling waves of activity in integrate-and-ﬁre neural networks, Phys. Rev. E 93, 052228 When two wrongs make a right: synchronized neuronal bursting from combined inhibitory and electrical coupling

Reimbay Reimbayev, Kevin Daley, and Igor Belykh∗ Department of Mathematics and Statistics, Georgia State University, 30 Pryor Street, Atlanta, GA 30303, USA [email protected]

Abstract

Synchronized cortical activities in the central nervous systems of mammals are crucial for sensory perception, coordination, and locomotory function. The neuronal mechanisms that generate synchronous synaptic inputs in the neocortex are far from being fully understood. In this paper, we study the emergence of synchronization in networks of bursting neurons as a highly nontrivial, combined eﬀect of electrical and inhibitory connections. We report a counterintuitive ﬁnd that combined electrical and inhibitory coupling can synergistically induce robust synchronization in a range of parameters where electrical coupling alone pro- motes anti-phase spiking and inhibition induces anti-phase bursting. We reveal the underlying mechanism which uses a balance between hidden properties of electrical and inhibitory coupling to act together to synchronize neuronal bursting. We show that this balance is controlled by the duty cycle of the self-coupled system which governs the synchronized bursting rhythm [1].

Our study reinforces previous work [2], in which it was shown that the addition of strong pairwise repulsive inhibition to excitatory networks of bursting neurons can induce synchrony due to the transition between diﬀerent types of bursting. Our studies of neuronal synchronization form a basis for understanding the counter-intuitive dynamics of small-scale bursting networks which may yield meaningful insight into the phenomenon of pathological synchrony in epileptic networks. Epileptic seizures are strongly associated with a synchronized state of certain brain networks. Our results together with [2] suggest that promoting normally repulsive inhibition in an attempt to prevent seizures can have an unintended eﬀect of inducing pathological synchrony.

References

[1] R. Reimbayev, K. Daley, and I. Belykh, When two wrongs make a right: synchronized neuronal bursting from combined electrical and inhibitory coupling, Philosophical Trans- actions of the Royal Society A, (2017) (in press).

[2] I. Belykh, R. Reimbayev, K. Zhao, Synergistic eﬀect of repulsive inhibition in synchronization of excitatory networks. Phys. Rev. E, 91 (2015) 062919.

ABSTRACTS for CONTRIBUTED PAPERS

Scattering of nonlinear bulk strain waves in delaminated bars

M. R. Tranter∗ and K. R. Khusnutdinova Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK [email protected], [email protected]

Abstract

In this talk we will consider the modelling of long longitudinal bulk strain solitary waves in delaminated elastic bars. Firstly we describe the dynamics of a symmetric perfectly bonded layered bar with delamination. The numerical results for the full scattering problem and for a semi-analytical technique are in good agreement and show ﬁssion of an incident solitary wave in the delaminated region [1].

We extend our approaches to the modelling of a layered bar with a soft bonding layer, described by a system of coupled Boussinesq equations [2], and again consider the behaviour of the waves in a delaminated region of the waveguide. Numerical results are compared for the full scattering problem and the semi-analytical approach, as well as considering the case where we have a delamination of ﬁnite length.

References

[1] K. R. Khusnutdinova and M. R. Tranter, Modelling of nonlinear wave scattering in a delaminated elastic bar, Proc. Roy. Soc. A, 471 (2015), 20150584.

[2] K. R. Khusnutdinova, A. M. Samsonov and A. S. Zakharov, Nonlinear layered lattice model and generalized solitary waves in imperfectly bonded structures, Phys. Rev. E, 79 (2009), 056606. Interaction of ocean waves of nearly equal frequencies and the effect on pressure

Paul Christodoulides Faculty of Engineering and Technology, Cyprus University of Technology, Limassol, Cyprus [email protected]

Lauranne Pellet Ecole Centrale Marseille, Marseille, France [email protected]

Sarah Donne School of Earth Sciences, University College Dublin, Dublin, Ireland [email protected]

Chris Bean School of Cosmic Physics, Dublin Institute for Advanced Studies, Dublin, Ireland [email protected]

Frédéric Dias School of Mathematics and Statistics, University College Dublin, Dublin, Ireland [email protected]

Abstract

We study the superposition of a train of freely traveling waves in a form that includes the possibility for each wave of complex amplitude An to have a ‘sister’ wave of complex amplitude Bn with equal frequency and opposite direction. For an ideal, incompressible and homogeneous fluid, we consider three-dimensional flows that are irrotational and space- periodic. Through a weakly nonlinear analysis we obtain full second-order expressions for the free-surface elevation, the velocity potential and the dynamic pressure. Then we generalize and unify all related expressions in the literature, without any assumption on the water depth. When the frequencies of the surface waves of nearly opposite directions are nearly equal, a second-order pressure can be felt all the way to the sea bottom. Hence, in particular, we apply a theoretical analysis on the dynamic pressure obtained, and we quantify the degree of nearness in amplitude, frequency and incidence angle that must be reached to observe the phenomenon. Such phenomena of the second-order pressure, independent of the depth, have been supposed to be at the origin of so-called secondary microseisms. A comparison with real data for pressure induced by waves in the ocean is also presented.

Groundwater flow and Ground Heat Exchangers

Lazaros Lazari Department of Mechanical Engineering and Material Science and Engineering, Cyprus University of Technology, Limassol, Cyprus [email protected]

Lazaros Aresti, Georgios Florides and Paul Christodoulides Faculty of Engineering and Technology, Cyprus University of Technology, Limassol, Cyprus [email protected], [email protected] and [email protected]

Abstract

The flow of groundwater in multiple ground layers can play a significant role on the cooling or heating of vertical heat columns and Ground Heat Exchangers (GHEs), and hence on the construction of the latter. The heat distribution over time is described by the general heat transfer equation based on the energy balance. Thus the three-dimensional conservation of the transient heat equation for an incompressible fluid is applied in COMSOL Multiphysics. Heat transfer in porous media, Darcy’s velocity and seepage velocity are introduced by taking typical values of hydraulic conductivity, along with average borehole surface temperatures on every ground layer. The model parameters are validated against experimental values and multiple boreholes are examined. Although the key for an overall capital cost reduction for a GHE is known to be the borehole length, the numerical results here indicate that using the groundwater available, construction of shallow GHE systems can be achieved with an increase of the coefficient of performance (COP).

Instbilities of Two-Stratiﬁed Fluids Under Linear Shear

Katie Oliveras∗ Mathematics Department Seattle University Seattle, WA [email protected]

Christopher Curtis Department of Mathematics San Diego State University San Diego, CA [email protected]

Abstract

Euler’s equations describe the dynamics of gravity waves on the surface of an ideal fluid with arbitrary depth. In this talk, we discuss the stability of periodic traveling wave solutions describing the interface between two fluids of varying density and vorticity trapped between to rigid lids. Using a generalization of a non-local formulation of the water wave problem due to Ablowitz, et al. [1], and Ashton & Fokas [2], we determine the spectral stability for the periodic traveling wave solution by extending Fourier-Floquet analysis to apply to this non-local problem. We develop a numerical scheme to determine traveling wave solutions by exploiting the bifurcation structure of the non-trivial periodic solutions. Next, we determine numerically the spectral stability for the periodic traveling wave solution by extending Fourier-Floquet analysis to apply to the non-local problem. We can generate the full spectra for all traveling wave solutions. We discuss Kelvin-Helmholtz and Benjamin-Feir instabilities, as well as explore the suppression or amplification of such instabilities as a function of shear strength, density stratification, and the ratio of depths between the fluids.

References

[1] Ablowitz, M. J., A. S. Fokas, and Z. H. Musslimani. “On a new non-local formulation of water waves.” Journal of Fluid Mechanics, 562 (2006), 313-343.

[2] ACL Ashton and A. S. Fokas. “A non-local formulation of rotational water waves.” Journal of Fluid Mechanics, 689 (2011), 129-148. Eigenvalue asymptotics for Zakharov-Shabat systems with long-range potentials

Martin Klaus Department of Mathematics, Virginia Tech Blacksburg, Virgina, USA [email protected] (540) 231 6533

Abstract

We consider Zakharov-Shabat systems of the form

0 0 v1 = −iξv1 + q(t)v2, v2 = −q(t)v1 + iξv2 where ξ is a complex spectral parameter and q(t) is a real potential satisfying additional assumptions. We are interested in the discrete spectrum of this system when q(t) is not absolutely integrable but rather has slow decay, typically like ∼ |t|−γ with 0 < γ ≤ 1. For such potentials, there may be inﬁnitely many eigenvalues on the imaginary axis accumulating at zero. If N(s) denotes the number of purely imaginary eigenvalues ξ with Im ξ > s (s > 0), then one can ask about the asymptotic behavior of N(s) as s → 0. We will answer this question for suitable potentials of long range. In the process we will obtain a new result which guarantees that eigenvalues near zero are algebraically simple.

1 Steady three-dimensional ideal ﬂows with nonvanishing vorticity in domains with edges

Douglas Svensson Seth∗ and Erik Wahl´en Centre for Mathematical Sciences, Lund University Box 118, 221 00 Lund, Sweden Tel. +4646 222 86 08 douglas.svensson [email protected] and [email protected]

Abstract

We consider steady, inviscid, incompressible flows governed by the Euler equations. In two dimensions these can be studied using a stream function and three-dimensional irrotational flows can be understood through the introduction of a harmonic velocity potential. However neither of these approaches are possible if we seek a three-dimensional flow with nonvanishing vorticity.

An earlier result by Alber [1] shows the existence of such flows with H3 regularity in domains with a smooth boundary. These flows are found by perturbing a given reference flow using boundary conditions on the vorticity. The aim of this paper is to extend Alber’s result to a type of generalized cylinders, which are domains bounded by three smooth surfaces meeting at right angles along two edges (for example a finite cylinder with a general smooth cross section). A similar existence and regularity result for the generalized cylinders requires some additional compatibility conditions to be imposed on the given data. These conditions are given explicitly.

References

[1] H. D. Alber, Existence of threedimensional, steady, inviscid, incompressible ﬂows with nonvanishing vorticity, Math. Ann., 292 (1992), 493-528. On the Theory of Nonlinear Shock Waves and Supersonic Flow

Youssef R. Driss UNC Wilmington, 601 S College Rd, Wilmington, NC 28403 [email protected]

Abstract

In this talk we will relate nonlinear wave propagation with supersonic flow to understand sonic booms. We explore the formation of shock waves described by the N-wave solution to the viscous Burgers’ equation. The sonic boom phenomena is also a by-product of N-waves. Then, we will dive into the theory of supersonic flow by first introducing the Whitham’s F-curve and present an example for a particular supersonic fuselage. Evolution Equations in Topological Vector Spaces

Zhivko S. Athanassov Address:Institute f Mathematics, Bulgarian Academy of Sciences Bulgaria email: [email protected]

ABSTRACT

Let V be a topological vector space, A : X → X a closed operator, B : F → X a continuous operator, where F is the space of continuous X-valued functions on R+.Given f ∈ F , we obtain conditions for existence and uniqueness of solutions to the equation x˙(t)=Ax(t)+f(t) which satisfy Bx = x0 for x0 ∈ X. We point out that these conditions depend on the topology. The conditions in the given topology are obtained, and then, the conditions in a weaker topology.

ABSTRACTS

for

POSTERS

Comparisons Between Mathematical Models and Experiments of Waves on Deep Water

Isabelle Butterﬁeld and Dr. John Carter 901 12th Ave, Seattle, WA 98122 [email protected] and [email protected]

Abstract

We have written MATLAB codes that solve a variety of partial differential equations (PDEs) that model the evolution of wave trains on deep water. We use these codes to compare the PDE predictions with data from physical experiments conducted by Diane Henderson at Penn State University. The classical, cubic nonlinear Schrodinger (NLS) equation was derived in hopes of accurately modeling experiments of this sort. We use the NLS equation and a number of its generalizations as our base models. These include the Dysthe [1], viscous Dys- the (vDysthe) [2], dissipative NLS (DNLS) [3], Schober [4], and Gordon [5] equations. The codes solve these PDEs using operator splitting methods, and plot the predictions against the experimental data. Additional MATLAB codes compare the difference between each prediction and the experimental data to determine which PDE is best predicting the experimental behavior. In examining the first two sets of experimental data, we have found that the viscous Dysthe system does the best job modeling the experimental data in comparison to our selected generalizations of NLS, optimized over their free parameters.

References

[1] K. B. Dysthe. Note on a modiﬁcation to the nonlinear Schr¨odingerequation for application to deep water waves. Proc. Roy. Soc. London A, 369:105-114, 1979.

[2] J. D. Carter and A. Govan. Frequency downshift in a viscous ﬂuid. Eur. J. Mech. B- Fluids, 59:177-185, 2016.

[3] F. Dias and A. I. Dyachenko and V. E. Zakharov. Theory of weakly damped free-surface ﬂows: a new formulation based on potential ﬂow solutions. Phys. Lett. A, 372:1297-1302, 2008.

[4] A. Islas and C. M. Schober. Rogue waves and downshifting in the presence of damping. Nat. Hazards Earth Syst. Sci., 11:383-399, 2011.

[5] J. P. Gordon. Theory of the soliton self-frequency shift. Opt. Lett., 11:662-664, 1986. Second-order elliptic equations with Wentzel and transmission boundary conditions and applications

Hung Le University of Missouri, Columbia [email protected]

Abstract

In this poster, we present results about the existence and uniqueness of classical solutions for elliptic equations with transmission and Wentzel boundary conditions, and an application to steady water waves in the presence of wind. The ﬁrst part is proved by using ideas from Ladyzhenskaya and Uraltseva’s results [2] on elliptic equations with transmission condition, tpgether with the work of Luo and Trudinger on elliptic equations with Wentzel boundary conditions [3].

The application focuses on developing existence theory for small-amplitude two-dimensional traveling waves in an air–water system with surface tension. The water region is assumed to be irrotational and of ﬁnite depth, and we permit a general distribution of vorticity in the atmosphere. This result generalizes the paper of B¨uhler, Shatah [1], and Walsh who studied the same system but without surface tension.

References

[1] O. B¨uhler,J. Shatah, S. Walsh, Steady water waves in the presence of wind, SIAM Journal Mathematical Analysis, 45 (2013), 2182-2227.

[2] O. A. Ladyzhenskaya, N. N. Uraltseva, Linear and Quasilinear Elliptic Equations, 46, (1968).

[3] Y. Luo, N. S. Trudinger, Linear second order elliptic equations with Venttsel boundary conditions, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 118(3-4) (2001), 193207. A new perspective on steady ﬂow over bathymetry

Daniel Ferguson∗ and Katie Oliveras Seattle University, Mathematics Department, Seattle, WA, USA [email protected] and [email protected]

Vishal Vasan International Centre for Theoretical Sciences, Bangalore, India [email protected]

Abstract

Using a modification of the work in [1] we derive a relationship between the bathymetry, free- surface, and pressure at the bottom of a fluid for steady flow. For example, given the shape of the bathymetry, we recover both the pressure along the bathymetry as well as the shape of the free surface. We also derive asymptotic relationships between the bathymetry, pressure, and free surface. The asymptotic models generated are contrasted with other models for steady flow over bathymetry. Qualitative comparisons with physical experiments are also performed.

References

[1] K. L. Oliveras, V. Vasan, B. Deconinck, and D. Henderson, Recovering the Water- Wave Proﬁle from Pressure Measurements, SIAM Journal on Applied Mathematics, 72:3 (2012), 897-918. Asymptotics of the ﬁnite Toda lattice

Robert Jenkins and Kyle Pounder∗ Department of Mathematics, University of Arizona [email protected] and [email protected]

Kenneth T.-R. McLaughlin Department of Mathematics, Colorado State University [email protected]

Abstract

Flaschka’s variables associate the finite N-particle Toda lattice with N × N Jacobi matrices J(t) (i.e., symmetric, tridiagonal matrices with positive off-diagonal entries). The Toda flow is known to leave the spectrum of the Jacobi matrices J(t) invariant. Moreover, the flow is sorting [1], i.e., at large times J(t) approaches a diagonal matrix with ordered eigenvalues on its diagonal. Recently there has been interest in studying how long it takes for the Toda flow to diagonalize a Jacobi matrix [2]. Using Riemann-Hilbert techniques we study the evolution of the Jacobi matrices under the Toda flow in detail. Our results give a sharper version of the results in [1], and also describe the evolution of nearly singular Jacobi matrices (i.e., whose off-diagonal entries are very close to zero) which take comparatively longer to reach their sorted diagonalization.

References

[1] J. Moser, Finitely many mass points on the line under the inﬂuence of an exponential potential – an integrable system, Dynamical systems, theory and applications, (1975).

[2] C. W. Pfrang, P. Deift, & G. Menon, How long does it take to compute the eigenvalues of a random symmetric matrix, Random Matrix Theory, Interacting Particle Systems, and Integrable Systems, Math. Sci. Res. Inst. Publ, 65 (2013), 411-442. Numerical inverse scattering for the sine-Gordon equation

Bernard Deconinck and Xin Yang Department of Applied Mathematics University of Washington, Seattle 98105 [email protected] and [email protected]

Thomas Trogdon Department of Mathematics University of California, Irvine [email protected]

Abstract

The sine-Gordon equation is a nonlinear hyperbolic PDE which appears in diﬀerential geometry, superconductivity and a number of other applications. It is known to be integrable and solved by Kaup [1] using the Inverse Scattering Transform(IST). In 2012, Trogdon, Olver and Deconinck [2] implement the Inverse Scattering Transform for the Korteweg-de Vries and modiﬁed Korteweg-de Vries equations. The same idea is applied to the sine-Gordon equation with some treatment of the extra singularity appeared in the IST for the sine-Gordon equation.

In this poster we implement the Inverse Scattering Transform for the sine-Gordon equation using the ISTPackage developed by Trogdon and RHPackage by Olver [3]. Our numerical experiments have shown that the method is spectrally accurate. Since x and t are parameters in the method, one computes only at the point of interest and time stepping is not required as opposed to traditional numerical method for time evolution equation.

References

[1] D. J. Kaup, Method for solving the sine-Gordon equation in laboratory coordinates, Stud. Appl. Math., 54 (1975), 165-179.

[2] T. Trogdon, S. Olver and B. Deconinck, Numerical inverse scattering for the Korteweg-de Vries and modiﬁed Korteweg-de Vries equations, Physica D., 241 (2012), 1003-1025.

[3] T. Trogdon and S. Olver, Riemann-Hilbert problems, their numerical solution and the computation of nonlinear special functions, SIAM (2015). Eﬃcient Moving mesh simulation of fourth order PDES in 2D Modeling of elastic-electrostatic deﬂections

Kelsey DiPietro and Alan E. Lindsay Department Applied Computational Mathematics and Statistics, University of Notre Dame 153 Hurley Hall, Notre Dame, IN 46556 [email protected] and [email protected]

Abstract

We develop a robust moving mesh finite difference method for the simulation of fourth order nonlinear PDEs describing elastic-electrostatic interactions in two dimensions. We use and extend the r-adaptive methods developed by [1] to solve two fourth order PDEs with different evolutionary dynamics.

The ﬁrst PDE displays ﬁnite time quenching singularities that form at discrete spatial location(s). In this case, the moving mesh method must detect temporally forming singularities and dynamically resolve them to proper length scales.

The second PDE is a regularized model which considers the elastic-electrostatic interactions that occur after the quenching singularity occurs. The solution to this PDE has initial curved interface starting around singularity point that propagates through the domain until it is pinned at the boundaries. Unlike the quenching case, the interface location and curvature characteristics are known at the beginning of the simulation, so the role of the moving mesh method is to dynamically shift mesh points to track the interface and to accommodate for its growing perimeter.

The method utilizes the self-similar and boundary layer structures of the two PDES to locate and dynamically resolve the desired properties of each equation to high accuracy. We compare our numerical approximation to the asymptotic approximations of the solutions developed in [3],[2].

References

[1] C.J. Budd and J.F. Williams. Moving mesh generation using the parabolic monge-ampere equation. SIAM Journal on Scientiﬁc Computing, 31(5):3438-3465, 2009. [2] A.E. Lindsay and J. Lega, Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor. SIAM Journal On Applied Mathematics, 72(3):935-958, 2012. [3] A.E. Lindsay, J. Lega, K.B. Glasner. Regularized model of post-touchdown conﬁgura- tions in electrostatic mems: interface dynamics. IMA Journal of Applied Mathematics, 80(6):1635-1663, 2015. On the integrability of long and short wave interaction models

Bernard Deconinck and Jeremy Upsal∗ Department of Applied Mathematics, University of Washington Seattle, WA 98195-2420, USA [email protected] and [email protected]

Abstract

We examine the integrability of two models used for the interaction of long and short waves in dispersive media. One is more classical but can arguably not be derived from the underlying water wave equations, while the other one was recently derived in [1]. We use the method of Zakharov and Shulman [2] to attempt to construct conserved quantities for these systems at different orders in the magnitude of the solutions. The coupled KdV-NLS model is shown to be nonintegrable, due to the presence of fourth-order resonances. A coupled real KdV - complex KdV system is shown to suffer the same fate, except for three special choices of the coefficients, where higher-order calculations or a different approach are necessary to conclude integrability or the absence thereof.

References

[1] B. Deconinck, N.V. Nguyen, and B.L. Segal, The interaction of long and short waves in dispersive media, Journal of Physics A Mathematical General, 49 (2016), 415501

[2] V.E. Zakharov and E.I. Schulman, Integrability of nonlinear systems and perturbation theory, What is integrability?, (1991), 185–250. Volume Bounds for the Synchronization Region in the Kuramoto Model

Timothy Ferguson 1409 West Green Street, 250 Altgeld Hall Urbana, IL 61801 [email protected]

Abstract

The Kuramoto model is an important model used to study synchronization of a network of coupled oscillators. It is well known that if the natural frequencies of the individual oscillators are similar to each other that the oscillators synchronize, whereas if the natural frequencies are very different the oscillators don’t. We consider the problem of determining the volume of the set of such frequencies. We obtain both upper and lower bound in terms of spanning trees for general graphs.