58 NW14 Abstracts

IP1 Mathematics and Engineering Physics Stability and Synchrony in the Kuramoto Model University of Wisconsin, Madison [email protected] The phenomenon of the synchronization of weakly coupled oscillators is a old one, first been described by Huygens in his “Horoloquium Oscilatorium.’ Some other exam- IP4 ples from science and engineering include the cardiac pace- Coherent Structures in 2D Turbulence: Identifica- maker, the instability in the Millenium Bridge, and the tion and their Use synchronous flashing of fireflies. A canonical model is the Kuramoto model Ideas from dynamical systems have recently provided fresh insight into transitional fluid flows. Viewing such flows as dθ i = ω + γ sin(θ − θ ) a trajectory through a phase space littered with coherent dt i j i j structures (meaning ’exact’ solutions here) and their stable and unstable manifolds has proved a fruitful way of We describe the fully synchronized states of this model understanding such flows. Motivated by the challenge of together with the dimensions of their unstable manifolds. extending this success to turbulent flows, I will discuss how Along the way we will encounter a high dimensional poly- coherent structures can be extracted directly from long- tope, a Coxeter group, and a curious combinatorial iden- time simulations of body-forced turbulence on a 2D torus. tity. This leads to a proof of the existence of a phase tran- These will then be used to ’postdict’ certain statistics of sition in the case where the frequencies are chosen from as the turbulence. iid Random variables. Rich Kerswell Jared Bronski Department of Mathematics University of Illinois Urbana-Champaign University of Bristol, U.K. Department of Mathematics [email protected] [email protected]

IP5 IP2 A Neural Field Model of Binocular Rivalry Waves Integrability: Initial- Boundary Value Problems in 1+1, and Solitons in 3+1 Binocular rivalry is the phenomenon where perception switches back and forth between different images presented A review will be presented of the two most challenging to the two eyes. The resulting fluctuations in perceptual problems in the analysis of integrable nonlinear evolution dominance and suppression provide a basis for non-invasive equations: (a) initial- boundary value problems in 1+1, i.e. studies of the human visual system and the identification evolution equations in one spatial variable, and (b) the con- of possible neural mechanisms underlying conscious visual struction and solution of equations in 3+1. Regarding (a), awareness. In this talk we present a neural field model of it will be shown that the so- called unified transform, which binocular rivalry waves in visual cortex continuum neural provides the proper generalization of the inverse scattering fields are integro-differential equations that describe the transform, yields effective formulae for the large t- asymp- large-scale spatiotemporal dynamics of neuronal popula- totics of a variety of physically significant problems; this tions. We derive an analytical expression for the speed of includes problems on the half-line with t-periodic boundary a binocular rivalry wave as a function of various neurophys- conditions. Regarding (b), an integrable generalization of iological parameters, and show how properties of the wave the Davey-Stewardson equation 4+2, i.e. four spatial and are consistent with the wave-like propagation of perceptual two time variables, will be presented. A reduction of this dominance observed in recent psychophysical experiments. equation to an equation in 3+1 will also be discussed. Fur- We then analyze the effects of extrinsic noise on wave prop- thermore, one and two soliton solutions for both the 4+2 agation in a stochastic version of the neural field model. We and 3+1 Davey-Stewardson equations will be presented. end by describing recent work on rotating rivalry stimuli Thanasis Fokas and direction selectivity. Department of Applied Mathematics and Theoretical Physics Paul C. Bressloff University of Cambridge, UK University of Utah and University of Oxford, UK [email protected] Department of Mathematics bressloff@math.utah.edu

IP3 IP6 Minimal Models for Precipitating Convection Engineering Extreme Materials with Defects and Despite the importance of organized convection (e.g. squall Nonlinearity lines and hurricanes), numerical simulations remain a challenge. Comprehensive cloud resolving models would in- We study the fundamental dynamic response of discrete clude water vapor, liquid water and ice, and liquid wa- nonlinear systems and study the effects of defects in the ter would be separated into cloud water and rain water. energy localization and propagation. We exploit this un- Here we take a minimalist approach to cloud microphysics derstanding to create experimentally novel materials and by incorporating fast auto-conversion from cloud to rain devices at different scales (for example, for application in water, and fast condensation and evaporation. Numeri- energy absorption, acoustic imaging and energy harvest- cal and analytical results will be discussed, e.g., saturated ing). We use granular systems as a basic platform for test- regions have a linear stability boundary associated with ing, and control the constitutive behavior of the new ma- parcel (finite-amplitude) stability in unsaturated regions. terials selecting the particles geometry, their arrangement and materials properties. Ordered arrangements of par- Leslie Smith ticles exhibit a highly nonlinear dynamic response, which NW14 Abstracts 59

has opened the door to exciting fundamental physical ob- [email protected] servations (i.e., compact solitary waves, energy trapping phenomena, and acoustic rectiﬁcation). This talk will focus on energy localization and redirection in one- and two- CP1 dimensional systems. The Rogue Wave Solutions of the Kpi

Chiara Daraio Rogue wave solutions of integrable partial differential equa- ETH Zürich, Switzerland and California Institute of Tech tions are one kind of large-amplitude doubly localized ratio- [email protected] nal solutions. Recently, the first-order rogue wave solution of the NLS has been observed in water tanks and optical fibers. In this talk, we shall provide new rogue wave solutions of a well-known (2+1)- dimensional equation: the IP7 KPI, from the solutions two (1+1)-dimensional equations: Momentum Maps, Shape Analysis and Solitons NLS and complex mKdV. The main technique is the Dar- boux transformation. Much of this talk is based on work done with Jerry Mars- Jingsong He den (1942 - 2010) on shared geometric properties in the Ningbo University analysis of fluid flow and shape transformations. The talk [email protected] will discuss uses of geometric mechanics in the problem of registration of images, primarily in the example of planar closed curves. Many types of mathematics apply in this CP1 problem, including soliton theory and momentum maps. Mcc Type Strongly Nonlinear Internal Wave Mod- Some trade secrets will be revealed. els

DarrylD.Holm We discuss MCC type strongly nonlinear internal long Imperial College London wave models in two- and three-layer fluid systems in de- [email protected] tail. Their dispersion relations and stability properties comparing to coupled Euler equations will be presented. In addition, generalized strongly nonlinear models are also IP8 considered. Non-stationary time evolution for their performance tests will be presented as well. Symmetry, Modulation and Nonlinear Waves Tae-Chang Jo Modulation underpins many facets of the analysis of non- Inha University linear waves and patterns. A new perspective on modula- [email protected] tion combined with symmetry and conservation laws will be presented, which leads to a mechanism for the emergence of model wave PDEs such as KdV, KP, Boussinesq CP1 and others. This combination results in simple geomet- Laboratory Experiments and Simulations for Soli- ric formulae for the coefficients. Generalizations to multi- tary Internal Waves with Trapped Cores phase wavetrains, systems of conservation laws, Whitham equations with dispersion, and a mechanism for the forma- Under appropriate conditions, solitary internal waves in tion of multi-pulse planforms in pattern formation will also the ocean and atmosphere can develop recirculating re- be presented. gions, known as “trapped cores’. Fundamental properties, essential for constructing and testing theoretical models, remain to be quantified experimentally. These include core Tom J. Bridges circulation and density, as well as mass and energy trans- University of Surrey port, and Richardson number. By using a new technique, [email protected] we measure these properties in detail. To examine larger waves, we perform simulations. The corresponding theory agrees closely with our experiments. SP1 Paolo Luzzatto-Fegiz Martin D. Kruskal Prize Lecture - Nonlinear Waves University of Cambridge from Beaches to Photonics [email protected]

Based on work of Kruskal et al the Korteweg-deVries equa- Karl Helfrich tion was found to have many remarkable properties. These Woods Hole Oceanographic Institution properties are shared by many other equations including [email protected] the 2D Kadomtsev-Petvishvili (KP) equation, which arises in the study of shallow water waves. Some solutions of KP and their relation to waves on shallow beaches will be dis- CP1 cussed. Nonlinear waves in photonic periodic lattices will InternalWavesIncidentUponAnInterface also be described; photonic graphene is a remarkable case. Since this is an evening lecture, equations and details will Recent results have shown that a vertical packet of inter- be kept to a minimum, and the lecture will be colloquium nal waves that are horizontally periodic will develop a dis- style. continuous mean flow at an interface. Here we consider a similar configuration where the waves are not horizon- Mark Ablowitz tally periodic, but instead exist within a wave packet that Dept. of Applied Mathematics is limited both horizontally and vertically. The basic state University of Colorado has constant stability N in two layers without a shear flow. 60 NW14 Abstracts

The horizontal limit of the wavepacket results in a much Istanbul Kültür University different wave-induced mean flow than the periodic case, Department of Mathematics and Computer Sciences as the mean flow is confined to the wavepacket and there- [email protected] fore must have a zero net flow across any vertical surface. The net effect is that gradients of the mean flow at the Murat Uzunca interface are stronger than the periodic case. Waves are Department of Mathematics & Institute of Applied treated with the nonlinear Schrodinger equations that are Mathematics solved numerically. Middle East Technical University, Turkey [email protected] John P. Mchugh University of New Hampshire [email protected] Bülent Karasözen Middle East Technical University Institute of Applied Mathematics CP1 [email protected] Nonlinear Harmonic Generation by Internal Wave Refraction CP2 Weakly nonlinear analysis is used to show how internal wave refraction through variable stratification generates Boundary Value Problems for Multidimensional harmonics. Refraction into a strongly stratified pycnocline Integrable Nonlinear Pdes layer will be analyzed, as ocurs for oceanic internal tides and topographically-generated beams. For large incidence The Fokas method, or Unified Transform method, is a sig- angles, the lowest harmonic is trapped, and its wavenum- nificant novel generalization of the classical inverse scat- ber selected by a resonance with the natural modes of the tering method, rendering it applicable to initial-boundary pycnocline. Analytic methods are used to calculate this value problems (IBVPs) for integrable nonlinear PDEs. We resonance condition, and the results compared to simula- present the implementation of the Fokas method to certain tion and experiment. IBVPs for multidimensional integrable nonlinear PDEs. In particular, we study the Davey-Stewartson equation (i) on Scott E. Wunsch the half-plane in the case of a time-periodic boundary con- Johns Hopkins University dition, (ii) on the quarter-plane, and (iii) on bounded do- [email protected] mains.

CP1 Iasonas Hitzazis Cylindrical Korteweg-De Vries Type Equation for Department of Mathematics Internal and Surface Ring Waves on a Shear Flow Univerity of Patras, Greece [email protected] We study the ring waves in a stratified fluid over a prescribed shear flow. A weakly-nonlinear 2+1-dimensional long wave model is derived from the full set of Euler equa- CP2 tions with background stratification and shear flow, written in cylindrical coordinates, subject to the boundary condi- Variable Separation Solutions and Interacting tions typical for the oceanographic applications. In the Waves of a Coupled System of the Modified KdV absence of the shear flow and stratification, the derived and Potential Blmp Equations equation reduces to the cylindrical Korteweg -de Vries type equations obtained by V.D. Lipovskii and R.S. Johnson. The features of the ring waves are then studied numeri- The multilinear variable separation approach (MLVSA) is cally using a finite-difference code for the derived model. applied to a coupled modified Korteweg de-Vries and potential Boiti-Leon-Manna-Pempinelli equations, as a result, the potential fields uy and vy are exactly the universal Xizheng Zhang, Karima Khusnutdinova quantity applicable to all multilinear variable separable Loughborough University, UK systems. The generalized MLVSA is also applied, and it [email protected], [email protected] is found uy (vy) is rightly the subtraction (addition) of two universal quantities with different parameters. Then interactions between periodic waves are discussed, for in- CP2 stance, the elastic interaction between two semi-periodic Model Order Reduction for Coupled Nonlinear waves and non-elastic interaction between two periodic in- Schr?dinger Equation stantons. An attractive phenomenon is observed that a domion moves along a semi-periodic wave. Proper orthogonal decomposition (POD) method is used to get numerical solution for coupled nonlinear Schrdinger Xiaoyan Tang (CNLS) equation in Hamiltonian form. The CNLS equa- East China Normal University tion is discretized in space by finite differences and is solved [email protected] in time by midpoint method. Numerical results for CNLS equation with different parameters indicate that the low- dimensional approximations obtained by POD is effective Ji Lin and reproduce very well the characteristic dynamics of the Zhejiang Normal University system, such as preservation of energy and phase space [email protected] structure of the CNLS equation. Zufeng Liang Canan Akkoyunlu Hangzou Normal University NW14 Abstracts 61

[email protected] solutions of an equation for surface water waves of moderate amplitude in the shallow water regime. Our approach is based on a method proposed by Grillakis, Shatah and CP2 Strauss in 1987, and relies on a reformulation of the evo- Explicit Construction of the Direct Scattering lution equation in Hamiltonian form. We deduce stability Transform for the Benjamin-Ono Equation with of solitary waves by proving the convexity of a scalar func- Rational Initial Conditions tion, which is based on two nonlinear functionals that are preserved under the flow. We propose a construction for the scattering data of the Benjamin-Ono equation with a rational initial condition, Anna Geyer under mild restrictions. The construction procedure con- University Autonoma de Barcelona sists in building the Jost function solutions explicitly and [email protected] use their analyticity properties to recover the reflection coefficient, eigenvalues, and phase constants. We finish by Nilay Duruk Mutlubas showing that this procedure validates certain well-known Kemerburgaz University, Istanbul, Turkey formal results obtained in the zero-dispersion limit by Y. [email protected] Matsuno.

Alfredo N. Wetzel, Peter D. Miller CP3 University of Michigan, Ann Arbor [email protected], [email protected] Velocity Field in Parametrically Excited Solitary Waves

CP3 Parametrically excited solitary waves are localized struc- Solitary Wave Perturbation Theory for Hamilto- tures in high-aspect-ratio free surfaces subject to vertical nian Systems vibrations. In this talk, we present the first experimental characterization of the hydrodynamics of these waves us- Solitary waves in Hamiltonian systems can be parame- ing Particle Image Velocimetry. Our results confirm the terized by conserved quantities and parameters associated accuracy of Hamiltonian models with added dissipation in with the system’s symmetries. When such systems are per- describing these structures. Our measurements also un- turbed, it is natural to consider slow dynamics constrained cover the onset of a streaming velocity field which is shown to the solitary wave solution manifold. A standard way to be as important as other nonlinear terms in the current to derive the modulation equations involves direct calcula- theory. tions of the perturbed conserved quantities. However, the evolution of the symmetry parameters containing higher Leonardo Gordillo order, but often important, information is not as readily Laboratoire Matiere et Systemes Complexes (MSC) obtained. In this talk, singular perturbation theory is em- Universite Paris 7 Diderot ployed to obtain the solitary wave modulation equations [email protected] for a general class of Hamiltonian systems. This methodology is applied to a six-parameter solitary wave solution Nicolas Mujica for the (2+1)D Landau-Lifshitz equation that models the Departamento de Fisica, FCFM magnetization of a thin, ferromagnetic material. Results Universidad de Chile are provided for recent experimental observations of soli- nmujica@dfi.uchile.cl tary waves in a magnetic layer where both damping and localized forcing are present. CP3 Lake Bookman,MarkA.Hoefer North Carolina State University Solitary Waves and an N-particle Algorithm for a [email protected], [email protected] Class of Euler-PoincaréEquations Nonlinearity arises generically in mathematical models of CP3 physical phenomena, and the interplay between nonlinearity and dispersion is thought to be responsible for many Resonance, Small Divisors and the Convergence of of these phenomena, such as the existence of traveling Newton’s Iteration to Solitons waves. We explore the relation between nonlinearity and Resonances create the small divisor problem for dispersion by studying the N-particle system of the Euler- Stokes/Poincare-Lindstedt series. We show through Poincaré differential equations, or the EPDiff equations. the Fifth-Orde KdV equation how the same cause makes In particular, we illustrate the existence and dynamics of Newton’s iteration stall at small residuals, and how to fix traveling wave solutions of the EPDiff equations. Solitary this to converge to a soliton or periodic traveling wave. waves for this class of equations can be made to correspond to interacting particles of a finite-degree-of-freedom Hamil- John P. Boyd tonian system. We analyze the dynamics of two-solitary University of Michigan wave interaction and show that two solitons can either scat- Ann Arbor, MI 48109-2143 USA ter or capture each other. The scattering or capture orbits [email protected] depend on the singularity level of the solitons, while singularity of a soliton is determined by the power of the linear elliptic operator associated with the EPDiff equations. CP3 Orbital Stability of Solitary Shallow Water Waves Long Lee of Moderate Amplitude Department of Mathematics University of Wyoming We study the orbital stability of solitary traveling wave [email protected] 62 NW14 Abstracts

Roberto Camassa ton which is perturbed perpendicular to its direction of University of North Carolina propagation. This gives us the Lagrangian density of the [email protected] coupled differential equations. The growth rate can be obtained by maximizing the Lagrangian density. By using Dongyang Kuang the spectral methods, the unstable planed soliton will be University of Wyoming shown a transformation to two-dimensional soliton-like so- [email protected] lution. Sarun Phibanchon CP3 Faculty of Art and Science, Burapha University The (1+2)-Dimensional Sine-Gordon Equation: Chanthaburi, Thailand Soliton Solutions and Spatially Extended Relativis- [email protected] tic Particles Michael Allen The (non-integrable) (1+2)-dimensional Sine-Gordon Physics Department, Mahidol University equation does have N-soliton solutions for all N = 1. Multi- Rama 6 Road, Bangkok, 10400, Thailand soliton solutions propagate rigidly at constant velocities [email protected] that are higher than or lower than the speed of light. A first integral of the equation vanishes on single-soliton solutions and maps multi-soliton solutions onto structures that are CP4 localized around soliton junctions. The slower-than-light On Rational Solutions to Multicomponent Nonlin- structures emulate spatially extended relativistic particles, ear Derivative Schrödinger Equation whose mass density obeys a source-driven wave equation. We consider multicomponent derivative nonlinear Yair Zarmi Schrödinger equations (DNLS) related to Hermitian Ben Gurion University of the Negev symmetric spaces. DNLS were firstly proposed by A. [email protected] Fordy as generalizations of Kaup-Newell’s equation. We present rational type solutions to DNLS. To do this we make use of Zakharov-Shabat’s dressing technique adapted CP4 for quadratic bundles. Spectral Band Gaps and Lattice Solitons for the Fourth Order Dispersive Nonlinear Schrödinger Tihomir I. Valchev Equation with Time Periodic Potential School of Mathematical Sciences, Dublin Institute of Technology Localization and dynamics of the one-dimensional bihar- [email protected] monic nonlinear Schrödinger equation in the presence of an external periodic potential is studied. The band gap structure is determined using the Floquet-Bloch theory CP4 and the shape of its dispersion/diffraction curves as a func- Long-Time Asymptotics for the Defocusing Inte- tion of the fourth order dispersion/diffraction coupling con- grable Discrete Nonlinear Schrödinger Equation stant β is discussed. Contrary to the classical nonlinear Schrödinger equation (with an external periodic potential) We investigate the long-time asymptotics for the defo- here it is found that for a fixed negative value of β, there cusing integrable discrete nonlinear Schrödinger equation 2 exists a nonzero threshold value of potential strength below idRn/dt+(Rn+1 −2Rn +Rn−1)−|Rn| (Rn+1 +Rn−1)=0 which there is no first band gap. For increasing values of introduced by Ablowitz-Ladik. We employ the inverse scat- potential amplitudes, the dispersion bands reorient them- tering transform and the Deift-Zhou nonlinear steepest de- selves leading to different soliton forms. Lattice solitons scent method. The leading part is a sum of two oscillatory corresponding to spectral eigenvalues lying the the semi- terms with decay of order t−1/2. Details will appear in infinite and first band gap are constructed. Stability prop- Journal of the Mathematical Society of Japan. erties of various localized lattice modes are analyzed and contrasted against direct numerical simulations. Hideshi Yamane Kwansei Gakuin University Justin Cole [email protected] Florida State University [email protected] CP5 Ziad Musslimani Pipes and Neurons Dept. of Mathematics Florida State University We exploit a surprising analogy between the subcritical [email protected] transition to turbulence and the dynamics action potentials in order to understand the onset of turbulence in pipe flow. The focus here is the transition from localized to CP4 expanding turbulence. Analysis of fronts connecting lami- Transverse Instability of a Soliton Solution to nar flow (quiescent state) to turbulent flow (excited state) Quadratic Nonlinear Schrödinger Equation gives the speeds of turbulent-laminar fronts. Combining with experiments and simulations we explain the various Madelungs fluid picture is applied to the quadratic nonlin- stages in the transition to turbulence. ear Schrödinger equation. This gives the travelling wave equation that describes ion acoustic wave in plasma with Dwight Barkley non-isothermal electrons. The variational method is ap- University of Warwick plied to determine the growth rate of instabilities of a soli- Mathematics Institute NW14 Abstracts 63

[email protected] protozoan predator and two different phytoplankton prey groups collected from Lake Constance. Baofang Song, Mukund Vasudevan Institute of Science and Technology Austria Sofia Piltz, Mason A. Porter [email protected], [email protected] University of Oxford sofi[email protected], [email protected] Marc Avila University of Erlangen-Nuremberg Philip K. Maini Germany Oxford University, Mathematical Inst. 24-29 St. Giles [email protected] Oxford. OX1 3LB [email protected] Bjoern Hof Institute of Science and Technology Austria CP5 [email protected] Self-Organisation in a Reaction-Diffusion System with a Non-Diffusing Component

CP5 Motivated by the regulation mechanism behind position- Optimal Control in a Mathematical Model of Low ing of flagellar basal bodies in bacteria, we study a model Grade Glioma system where a diffusion-driven instability leads to pattern formation. The model includes three species and its We address this research to a mathematical model for low- dynamics is described by a pair of reaction-diffusion equa- grade glioma treated with chemotherapy. We analyze the tions coupled to an ODE. We study how the presence of dynamics of the model and study the stability of the so- a non-diffusing species impacts the analytical properties of lutions. Besides, we characterize the optimal controls re- the solution and the resulting pattern. lated to drug therapy, using different strategies, including a quadratic control and a linear control. We establish the Peter Rashkov existence of the optimal control, and solve for the control Philipps-Universität Marburg in both the quadratic and linear case. Finally, from nu- [email protected] merical simulations, we discuss the optimal strategies from the clinical point of view CP5 Juan Belmonte-Beitia On a Volume-Surface Reaction-Diffusion Sys- Departamento de Matemáticas tem with Nonlinear Boundary Coupling: Well- Universidad de Castilla-La Mancha Posedness and Exponential Convergence to Equi- [email protected] librium

We consider a model system consisting of two reaction- CP5 diffusion equations, where one specie diffuses in a volume Localized Structures and Their Dynamics in Bio- while the other specie diffuses on the surface which sur- convection of Euglena Gracilis rounds the volume. The two equations are coupled via nonlinear reversible Robin-type boundary conditions for We experimentally study localized patterns of suspension the volume specie and a matching reversible source term of a phototactic microorganism, Euglena gracilis, illumi- for the boundary specie. As a consequence the total mass nated from the bottom. By using annular container, two of the species is conserved. The considered system is mo- basic types of localized patterns were isolated. They are tivated by models for asymmetric stem cell division. We similar to those observed in thermal convection of binary first prove the existence of a unique weak solution via an it- fluid mixtures, although the governing equations for this erative method of converging upper and lower solutions to system is not known. We will also report their dynamics overcome the difficulties of the nonlinear boundary terms. and statistical analysis related to the bistability and pho- Secondly, we show explicit exponential convergence to equi- totactic responses of the microorganism. librium via an entropy method after deriving a suitable entropy entropy-dissipation estimate. Makoto Iima, Erika Shoji, Akinori Awazu, Hiraku Nishimori, Shunsuke Izumi Bao Q. Tang Hiroshima University Institute of Mathematics and Scientific Computing [email protected], m135752@hiroshima- University of Graz, Austria u.ac.jp, [email protected], nishimor@hiroshima- [email protected] u.ac.jp, [email protected] Klemens Fellner Institute for Mathematics and Scientific Computation CP5 University of Graz, Austria Models for Adaptive Feeding and Plankton Popu- [email protected] lations Evangelos Latos In this presentation, we develop (1) a piecewise-smooth Institute of Mathematics and Scientific Computing dynamical system and (2) reformulate it as a fast-slow University of Graz, Austria dynamical system to account for prey preference, flexi- [email protected] ble feeding behaviour or rapid evolution, and an ecologi- cal trade-off (all of which have been observed in plankton). We discuss connections between Filippov and fast-slow sys- CP6 tems, and compare our model predictions with data on Coherent Structures Inhibition and Destruction in 64 NW14 Abstracts

the Taylor-Couette System a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for pro- TaylorCouette flow remains one of the most widely stud- gressive waves with permanent form in finite water depth. ied problems. This is certainly due to the various impact- The UWM admits not only all smooth waves but also the ing fundamental and applied applications instilled by such peaked/cusped solitary waves. So, the UWM unifies the archetype flow.The Taylor vortices are coherent structures smooth and peaked/cusped solitary waves in finite water known to be omnipresent whatever is the flow regime. Sup- depth, for the first time. Some unusual characteristics of pression of these coherent structures is searched for sev- the peaked waves are found. For example, unlike smooth eral industrial processes such as crystal growth and os- waves, the phase speed of peaked solitary waves has noth- motic/photonic water purification. A control strategy is ing to do with the wave height. Besides, it is found that suggested to obliterate or inhibit the coherent structures in a cusped solitary wave is consist of an infinite number of a TaylorCouette flow. It consists in effecting minute radial peaked ones: this reveals the close relationship between pulsatile motion of the inner cylinder cross-section. The the peaked and cusped solitary waves. The UWM deep- superimposed modulations combined with the free surface ens and enriches our understandings about the smooth and dynamics suppress both the Ekman and Taylor coherent peaked/cusped waves as a whole. structures. When eliminated, fluid particles are no longer trapped within the Ekman or Taylor vortices. This yields Shijun Liao significant increase in the axial and azimuthal velocity fluc- Shanghai Jiaotong University tuations, which results in enhanced flow mixing. [email protected] Oualli Hamid EMP, Algiers CP6 [email protected] Large Amplitude Solitary Waves and Dispersive Shock Waves in Conduits of Viscous Liquids: Mekadem Mahmoud Model Derivation and Modulation Theory EMP, Algiers [email protected] The evolution of the interface separating a buoyant conduit of viscous fluid rising through a more viscous, exterior Bouabdallah Ahcene fluid is governed, counterintuitively, by the interplay be- USTHB, Algiers tween nonlinearity and dispersion, given sufficiently high [email protected] viscosity contrast. In this talk, a derivation of the scalar interfacial equation via an asymptotic, multiple scales procedure is presented. Perturbations about a state of ver- CP6 tically uniform pipe flow are considered in the context of Large Amplitude Solitary Waves and Dispersive the coupled fluid equations. The approximate model is Shock Waves in Conduits of Viscous Liquids: Ex- shown to be valid for long times and large amplitudes under perimental Investigation modest physical assumptions. We then apply Whitham-El modulation theory to derive analytical predictions for key The evolution of the interface separating a buoyant con- properties of dispersive shock wave solutions, as well as duit of viscous fluid rising through a more viscous, exterior demonstrate novel multiphase behavior which results from fluid is governed, counterintuitively, by the interplay be- a turning point in the dispersion relation. This talk estab- tween nonlinearity and dispersion, given sufficiently high lishes the theoretical background for quantitative experi- viscosity contrast. Here, we present the results of quantita- ments on solitons and DSWs in this simple, table-top fluid tive, experimental investigations into large amplitude soli- setting. tary wave interactions and dispersive shock waves (DSWs). Overtaking interactions of large amplitude solitary waves Nicholas K. Lowman are shown to exhibit nearly elastic collisions and universal Department of Mathematics interaction geometries according to the Lax categories for North Carolina State University KdV solitons. Further, we conduct experiments on well- [email protected] developed DSWs and demonstrate favorable comparison with Whitham-El modulation theory. We also report ob- Mark A. Hoefer servations of novel interaction experiments. Viscous liquid North Carolina State University conduits provide a well-understood, controlled, table-top [email protected] environment in which to study universal properties of dispersive hydrodynamics. CP6 Mark A. Hoefer Controlling the Position of Traveling Waves in North Carolina State University Reaction-Diffusion Systems [email protected] We present a method to control the position as a func- Nicholas K. Lowman tion of time of one-dimensional traveling wave solutions to Department of Mathematics reaction-diffusion systems according to a pre-specified pro- North Carolina State University tocol of motion. Given this protocol, the control function [email protected] is found as the solution of a perturbatively derived integral equation. We derive an analytical expression for the space and time dependent control function f (x, t)thatis CP6 valid for arbitrary protocols and many reaction-diffusion On the Unified Wave Model (uwm) for Smooth and systems. These results are close to numerically computed Peaked/cusped Solitary Waves optimal controls. The control can be expressed in terms of the uncontrolled wave profile and its propagation velocity, Based on the symmetry and fully nonlinear wave equations, rendering detailed knowledge of the reaction kinetics un- NW14 Abstracts 65

necessary. An extension to two higher dimensions allows a [email protected], [email protected] precise control of the shape of patterns of reaction-diffusion systems. CP7 Jakob Löber, Harald Engel New Exact Solution for Nonlinear Interaction of Institute of Theoretical Physics Two Pulsatory Waves of the Korteweg De Vries Technical University Berlin Equation in An Invariant Zigzag Structure [email protected], [email protected] berlin.de Interaction of two pulsatory waves of the KdV equation is treated by a novel method of solving nonlinear PDEs by invariant structures, which continues classical meth- CP6 ods of undetermined coefficients and separation of variables by developing experimental and theoretical comput- Interaction of a Weak Discontinuity Wave with El- ing. Using this approach a new solution for pulsatory ementary Waves of the Riemann Problem for Euler waves, which includes the two-solitons solution, is devel- Equations of Gasdynamics oped experimentally and proved theoretically by employing a zigzag, hyperbolic-tangent structure. Convergence, We investigate the effects of initial states and the shock tolerance, and summation of the structural approximation strength on the jumps in shock acceleration, and the am- are discussed. plitudes of reflected and transmitted waves. The test-data are considered for the three Riemann problems to analyze Victor A. Miroshnikov the situation when the initial discontinuity breaks up into College of Mount Saint Vincent two shocks and a characteristic shock. It is noticed that [email protected] for a weak shock, the jump in its acceleration due to an interaction with a weak discontinuity wave can be zero only when the incident wave belongs to the same family as the CP7 shock. Inertial Waves and Pattern Formation Inside a Ro- tating Cylinder Partially Filled with Liquid Vishnu D. Sharma Indian Institute of Tecnology Bombay We consider a rapidly rotating horizontal cylinder partially [email protected] filled with granular medium and/or fluid under gravity or vibration. In the rotating frame gravity oscillates and induces fluid oscillations, which take the form of inertial CP7 waves under resonance conditions. Dependently on the experimental conditions waves with different azimuthal and Oscillons in a Parametrically Forced PDE axial wavenumbers are excited. The inertial waves generate averaged fluid flows in the direction of propagation Spatially localized, time-periodic structures are common and bring to growth of regular patterns on the fluid-sand in pattern-forming systems, appearing in fluid mechanics, interface. chemical reactions, and granular media. We examine the existence of oscillatory localized states in a PDE model Denis A. Polezhaev, Victor Kozlov, Veronika Dyakova with single frequency time dependent forcing, where the The laboratory of vibrational hydromechanics primary pattern appears with non-zero wavenumber. Perm State Humanitarian Pedagogical University [email protected], [email protected], Alastair M. Rucklidge [email protected] Department of Applied Mathematics University of Leeds [email protected] CP7 Quasipatterns in Coupled Reaction-Diffusion Prob- Abeer Al-Nahdi, Jitse Niesen lems University of Leeds [email protected], [email protected] The nonlinear interaction between waves of different wavelengths is a mechanism that can stabilise 12-fold quasipatterns (two-dimensional patterns that are quasiperiodic ◦ CP7 in all directions but that have 30 rotation symmetry on average). Here, we investigate how coupling two pattern- Hopf Bifurcation from Fronts in the Cahn-Hilliard forming Turing systems can lead to these quasipatterns Equation being formed.

We study Hopf bifurcation from traveling-front solutions Alastair M. Rucklidge in the Cahn-Hilliard equation. Models of this form have Department of Applied Mathematics been used to study numerous physical phenomena, includ- University of Leeds ing pattern formation in chemical deposition and precipi- [email protected] tation processes. Technically we contribute a simple and direct functional analytic method to study bifurcation in the presence of essential spectrum. Our approach uses ex- CP7 ponential weights to recover Fredholm properties, spectral Vibrational Hydrodynamic Top Wave Instability ﬂow ideas to compute Fredholm indices, and mass conser- and Pattern Formation vation to account for negative index. Flows excited by light free sphere in cavity with liquid Ryan Goh, Arnd Scheel rotating around horizontal axis are experimentally inves- University of Minnesota tigated. Sphere commits diﬀerential rotation caused by 66 NW14 Abstracts

oscillating external force (vibrational hydrodynamic top). nian is indefinite a stable equilibria under a parameter Main flow has shape of 2D column. New instability is found change in a system may be subject to instability through azimuthal wave at column boundary and two-dimensional a Hamiltonian-Hopf (HH) bifurcation. HH bifurcations vortex system inside it. Stability threshold is one order of are generic if two purely imaginary eigenvalues of opposite magnitude lower compare to differential rotation of sphere Krein signature are close to each other. But it is possi- fixed on axis. Supercritical flows and transition sequences ble that under a particular perturbation eigenvalues pass are studied. each other and do not bifurcate; it happens when the perturbation is definite [Krein & Ljubarski] or if the system Stanislav Subbotin has an additional symmetry [Melbourne et al.]. Using the Perm State Humanitarian Pedagogical University graphical Krein signature we present a new type of mech- [email protected] anism that prevents HH bifurcations. Although such a mechanism is non generic it may commonly appear in ap- Victor Kozlov plications. Surprisingly the conditions appear to use infor- The laboratory of vibrational hydromechanics mation that may not be visible if one considers only the Perm State Humanitarian Pedagogical University typical form of the eigenvalue problem. [email protected] Richard Kollar Nikolay Kozlov Comenius University Department of General and Experimental Physics Department of Applied Mathematics and Statistics Perm State Pedagogical University, Perm, Russia [email protected] [email protected] Peter Miller University of Michigan CP8 [email protected] Complex Mode Dynamics of Coupled Wave Oscil- lators CP8 We show how nonlinear coherent waves localized in a few wells of a periodic potential can act analogously to a chain Persistence Results for Nonlocal Perturbations on of coupled oscillators. We identify the small-amplitude os- Unbounded Domains cillation modes of this system, demonstrate their extension to the large amplitude regime and reveal complex behavior The main focus of this talk will be to outline a, quite gen- such as the breakdown of Josephson-like oscillations, the eral, method to show persistence of coherent structures for destabilization of fundamental oscillation modes and the sufficiently small non-locality in spatially-extended PDEs. emergence of chaotic dynamics. As an example, the existence of stationary solutions for a nonlocal version of the Fisher-Kolmogorov-Petrovsky- Tristram J. Alexander Piskounov (FKPP) equation on the unbounded domain Rd UNSW Australia will be considered. It will be shown that only two bounded [email protected] non-negative stationary solutions of the nonlocal FKPP equation can exist for sufficiently small nonlocality. Dong Yan, P. Kevrekidis University of Massachusetts Christian Kuehn, Franz Achleitner [email protected], [email protected] Vienna University of Technology [email protected], [email protected]

CP8 Modulation Theory for the Fkdvb Equation Con- CP8 structing Periodic Solutions Pullback Attractor for the Non-Autonomous Stochastic Damped Wave Equations on Bounded We present a multiple-scale perturbation technique for ob- and Unbounded Domains taining solutions to the forced Korteweg de Vries-Burgers equation. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. From the This paper is devoted to dierential equations of the damped second order equation in the hierarchy, we find a system of wave equations with both non-autonomous deterministic odes governing the slow variation of the properties of the and random forcing terms. On unbounded domain, we cnoidal wave. This is achieved by imposing an extra condi- show the sufficient and necessary condition for existence tion at second order. We then construct periodic solutions of pullback attractors for the non-autonomous stochas- and examine their stability. tic damped wave equations on unbounded domain by establishing pullback absorbing set and pullback asymptotic Laura Hattam compactness of the cocycle in a certain parameter region, Monash University where the uniform estimates on the tails of solutions are [email protected] employed to overcome the non-compactness embeddings. On bounded domain, we prove the existence of pullback attractors for the non-autonomous stochastic damped wave CP8 equations by establishing pullback absorbing set and pull- New Criteria Preventing Hamiltonian-Hopf Bifur- back condition in a certain parameter region. At the same cations Using Graphical Krein Signature time, when the deterministic forcing terms is periodic, we obtain the pullback attractor is also periodic. Linear stability of equilibria in Hamiltonian systems is characterized by a restriction of the spectrum of the lin- Hongyan Li earized problem to the imaginary axis. If the Hamilto- Shanghai University of Engineering Science NW14 Abstracts 67

[email protected] University of Ottawa [email protected]

CP9 Abdolmajid Mohammadian Simulating Wall-Mode Convection: Numerical University of Ottawa Techniques and First Results [email protected]

We present simulations of strongly nonlinear wall-mode Martin Charron, Ayrton Zadra convection in a rapidly rotating container. Using the Environment Canada asymptotically reduced model by Vasil et al., we exam- [email protected], [email protected] ine the development of the wall-mode instability and subsequent pattern formation in isolation from bulk convection. This system includes nonlocal and nonlinear CP9 bounary conditions, and separate prognostic equations for Interaction of Atmospheric Vortex with A Land the barotropic and baroclinic components of the flow, which we implement using Dedalus, a new open-source We show that the complex behavior of the tropical cy- pseudospectral code. clone making landfall can be explained in the frame of two dimensional barotropic model obtained by averaging over Keaton Burns the hight of the primitive system of equations of the at- MIT Kavli Institute for Astrophysics and Space Research mosphere dynamics. In particular, this behavior includes [email protected] a significant track deflection, sudden decay and intensifi- cation. In contrast to other model, where first the addi- Geoffrey M. Vasil tional physically reasonable simplifications are made, we Theoretical Astrophysics Center deal with special classes of solutions to the full system. Astronomy Department, University of California, This allows us not to lose the symmetries of the model Berkeley and to catch the complicated features of the full model. geoff[email protected] Our theoretical considerations are confirmed by numerics made within a two dimensional barotropic model and are in a good compliance with experimental data. In particular, CP9 our method is able to explain the phenomenon of attraction Time-Averaging and Error Estimates for Reduced of the cyclone to the land and interaction of the cyclone Fluid Models. with an island.

I will discuss the application of time-averaging in getting Olga S. Rozanova rigorous error estimates of some reduced fluid models, in- Moscow State University cluding the quasi-geostrophic approximation, incompress- Mathematics and Mechanics Faculty ible approximation and zonal flows. The spatial boundary [email protected] can be present as a non-penetrable solid wall. I will show a very recent (and somewhat surprising) result on the 2 accuracy of incompressible approximation of Euler equations, CP9 thanks to several decoupling properties. Atmospheric Moisture Transport: Stochastic Dy- namics of the Advection-Condensation Equation Bin Cheng (SoMSS) School of Mathematical and Statistical Sciences Transport of atmospheric moisture is studied using a Arizona State University stochastic advection-condensation model. Water vapor is [email protected] advected by a prescribed velocity field and condensation occurs as fluid parcels enter regions of low saturation hu- midity. In contrast to previous studies, we focus on two CP9 velocity models with spatial correlation: single-vortex stir- Optimal High-Order Diagonally-Implicit Runge- ring and the random sine flow. We predict the total mois- Kutta Schemes for Nonlinear Diffusive Systems on ture decay rate in the initial value problem and investigate Atmospheric Boundary Layer the spatial moisture distribution for the steady-state problem. Nonlinear diffusion equations are extensively applicable in diverse fields of science and engineering. Numerical stabil- Yue-Kin Tsang ity is a common concern in this class of equations. In the IREAP present study, a three-stage third-order diagonally-implicit University of Maryland Runge-Kutta (DIRK) scheme is introduced by optimizing [email protected] the error and linear stability analysis for a commonly used nonlinear diffusive system in atmospheric boundary layer. Jacques Vanneste The proposed scheme is stable for a wide range of time School of Mathematics steps and able to resolve different diffusive systems with University of Edinburgh diagnostic turbulence closures, or prognostic ones with a [email protected] diagnostic length scale, with enhanced accuracy and stability compared to current schemes. It maintains A-stability, which makes it appropriate for the solution of stiff prob- CP9 lems. The procedure implemented in this study is quite On the Asymptotics of Nonlinear Wall-Localized general and can be used in other diffusive systems as well. Thermal Convection Waves

Farshid Nazari We describe an asymptotically reduced model of multiscale University of Ottawa wall-mode convection in a rapidly rotating container. The 68 NW14 Abstracts

bulk-interior dynamics drives small-scale flow within side- [email protected] wall boundary layers, and the boundary layers feed back onto the interior via a nonlinear and non-local boundary conditions. These new PDE’s contain the results from pre- CP10 vious linear theory but in an elementary fashion and also Pattern Formation in a Circular Hydraulic Jump provide a new avenue for investigating several interesting phenomena in a strongly nonlinear regime. When a vertical jet of viscous fluid strikes a horizontal plate, a circular hydraulic jump occurs at some distance Geoffrey M. Vasil from the jet impact point. Under certain conditions, the Theoretical Astrophysics Center circular symmetry of the jump can break to give rise to sta- Astronomy Department, University of California, tionary or rotating polygonal patterns. The talk describes Berkeley experimental observations of the symmetry breaking and a geoff[email protected] model for the polygonal pattern formation. Aslan R. Kasimov Keaton Burns Division of Mathematical & Computer Sciences and MIT Kavli Institute for Astrophysics and Space Research Engineering [email protected] King Abdullah University of Science and Technology [email protected]

CP10 CP10 A Theory of Weakly Non-Linear Detonations Nonlinear Energy Transfer Between Fluid Sloshing and Vessel Motion We derive a new weakly non-linear asymptotic model of detonation waves that captures the rich dynamics observed This talk identifies a mechanism leading to an energy ex- in solutions of the reactive Euler equations. We investigate change between a sloshing fluid and the vessel containing the traveling wave solutions of the asymptotic model and the fluid. The theory is developed for Cooker’s pendulous their linear stability. The non-linear dynamics of the model sloshing experiment where a rectangular vessel is partially are investigated by direct numerical simulations. Predic- filled with fluid and suspended by two cables. Close to an tions of the asymptotic equations are compared quantita- internal 1 : 1 resonance, the energy transfer is manifested tively with the full reactive Euler equations. by a heteroclinic connection which connects the purely symmetric sloshing modes to the purely anti-symmetric Luiz Faria sloshing modes leading to slosh-induced destabilization. KAUST [email protected] Matthew R. Turner University of Brighton Aslan R. Kasimov [email protected] Division of Mathematical & Computer Sciences and Engineering Tom J. Bridges, Hamid Alemi Ardakani King Abdullah University of Science and Technology University of Surrey [email protected] [email protected], [email protected]

Rodolfo R. Rosales CP10 Massachusetts Inst of Tech Department of Mathematics Fluid Ratcheting by Oscillating Channel Walls with [email protected] Sawteeth Motions rectified by symmetry-breaking mechanisms in oscillating flows have been of great interest in biological lo- CP10 comotion and engineering applications. Inspired by an ex- Diffraction and Reflection of a Weak Shock at a periment that demonstrates that fluid can be pumped from Right-Angled Wedge in Real Fluids one end to the other in a narrow channel using the vibrational motions of the sawtooth channel walls, we put for- Shock reflection-diffraction problem is one of the most chal- ward here a theory describing the ratcheting effect of fluid. lenging problems for the mathematical theory of multidi- In a conformally transformed plane, the Stokes boundary mensional conservation law. Since at high pressure or low layer flow is analysed, revealing the nonlinear effects driv- temperature, the behavior of gases deviates from the ideal ing the rectified flow and its complex spatial structure. gas law and follows the van der Waals gas, we study here Whereas the wall sawtooth shape is a source of asymmetry, the reflection-diffraction phenomenon of a weak shock at the difference in entrance and exit flow conditions due to a right-angled wedge for the two-dimensional compressible the geometries at the channel ends is found to be a second Euler system, modeled by the van der Waals equation of source to break the left-right symmetry of the system. state. Using asymptotic expansions,some basic features of Jie Yu the phenomenon and the nature of flow pattern are ex- North Carolina State University plored within the context of a real gas. In this talk, we jie [email protected] discuss how the real gas effects influence the nature of the self-similar flow patterns, relative to what it would have been in the ideal gas case. CP11 Ocean Wave Energy Potential and Microseisms Neelam Gupta Indian Institute of technology Bombay Ocean gravity waves driven by wind and atmospheric pres- NW14 Abstracts 69

sure can generate pressure variations on the sea floor that Mathematical and Computer Sciences & Engineering are at the origin of microseisms. We will be concerned with King Abdullah University of Science & Technology secondary microseisms with a frequency twice that of the [email protected] causative wave and amplitude independent of the depth. We need to know the sea states that allow pressure varia- Manuel Quezada De Luna tions large enough to generate microseisms and we need to Texas A&M University understand how pressure variations vary in space and time [email protected] and how they are linked to the sea floor. We will show the conditions on different parameters to obtain pressure variation able to generate microseisms and we will study CP11 the pressure with respect to different parameters in order to illustrate our theoretical results. Interaction of Strongly Nonlinear Waves on the Free Surface of Non-Conducting Fluid under the Paul Christodoulides Action of Horizontal Electric Field Cyprus University of Technology Faculty of Engineering and Technology In the present work we consider the problem of free sur- [email protected] face motion of the fluid with high dielectric constant in the strong horizontal electric field. It is shown that nonlinear Lauranne Pellet surface waves can propagate nondispersively in the direc- Ecole Centrale Marseille tion of electric field . Moreover, the counter-propagating [email protected] solitary weakly nonlinear waves preserve the initial form after collision. The question arises as to whether the same Frédéric Dias tendency is observed for strongly nonlinear waves. We have University College Dublin shown that collisions of strongly nonlinear waves are elas- Ecole Normale SupérieuredeCachan tic, i.e., the energy of solitary wave does not change. How- [email protected] ever, results of our numerical simulations indicate that, in general case, the interaction of strongly nonlinear waves leads to the shape distortion and formation of regions with CP11 high surface steepness. On Periodic Solutions of a Model Equation for Sur- face Waves of Moderate Amplitude in Shallow Wa- Evgeny Kochurin ter Institute of Electrophysics, Ural Division of RAS [email protected] The work which will be presented is mainly on a nonlinear evolution equation for surface waves of moderate amplitude Nikolay Zubarev in shallow water given below: Institute of Electrophysics, Ural Division of RAS, P. N. Lebedev Physical Institute, RAS 3 3 2 2 3 3 3 u + u + εuu − ε u u + ε u u [email protected](1) t x 2 x 8 x 16 x μ 7ε + (uxxx − uxxt)+ μ(uuxxx +2uxuxx)=0,x∈ R, t > 0. 12 24 CP11 Here u(x, t) is the free surface elevation and ε and μ rep- High Order Hamiltonian Water Wave Models with resent the amplitude and shallowness parameters, respec- Wave-Breaking Mechanism tively. Solutions of the Cauchy problem corresponding to (1) which are spatially periodic of period 1 are investi- Based on the Hamiltonian formulation of water waves, we gated. Local well-posedness is attained by an approach derive higher order Hamiltonian equations by Taylor ex- due to Kato which is based on semigroup theory for quasi- pansions of the potential and the vertical velocity around linear equations. Moreover, it is shown that singularities the still water level. The polynomial expansion in wave for the model equation can occur only in the form of wave height is mixed with pseudo-differential operators that pre- breaking. serve the exact dispersion relation. The consistent approximate equations have inherited the Hamiltonian structure Nilay Duruk Mutlubas and give exact conservation of the approximate energy. In Istanbul Kemerburgaz University order to deal with breaking waves, we designed an eddy [email protected] viscosity model that is applicable for fully dispersive equations. As breaking trigger mechanism we use a kinematic CP11 criterion based on the quotient of horizontal fluid velocity at the crest and the crest speed. The performance Solitary Waves Without Dispersive Terms: the is illustrated by comparing simulations with experimental Shallow Water Equations over a Periodic Bottom data obtained at hydrodynamic laboratory. The comparison shows that the higher order models perform quite well; The shallow water equations are a non-dispersive hyper- the extension with the breaking wave mechanism improves bolic system and their solutions typically involve shock the simulations. waves. We show by computational experiment that in the presence of periodic bathymetry, solutions may be characterized instead by solitary wave formation. Unlike solitary Ruddy Kurnia wave solutions of the KdV or Boussinesq equations, these Applied Mathematics, University of Twente waves arise in the absence of any dispersive terms. In- [email protected] stead, they result from effective dispersion caused by wave diffraction and reflection. E. Van Groesen University of Twente David I. Ketcheson LabMath-Indonesia 70 NW14 Abstracts

[email protected] Four-Wave Mixing

Statistical mechanics of two coupled vector fields is stud- CP11 ied within the framework of the tight-binding model that The Higher-Order Boussinesq Equation with Peri- describes propagation of polarized light in discrete waveg- odic Boundary Conditions: Analytical and Numer- uides in the presence of the four-wave mixing. The equi- ical Results librium properties of the polarization state are described in terms of the Gibbs measure with positive temperature. We consider the higher-order Boussinesq (HBQ) equation The transition line T = ∞ is established beyond which proposed in [P. Rosenau, ”Dynamics of dense discrete sys- the discrete vector solitons are created. In the limit of the tems”, Prog. Theor. Phys. 79, 1028-1042, (1988)], [G. large nonlinearity an analytical expression for the distribu- Schneider, C. E. Wayne, ”Kawahara dynamics in dispersive tion of Stokes parameters is obtained which is found to be media”, Physica D 152-153, 384-394, (2001)], [N. Duruk, dependent only on the statistical properties of the initial A. Erkip, H. A. Erbay, ”A higher-order Boussinesq equa- polarization state and not on the strength of the nonlin- tion in locally non-linear theory of one-dimensional non- earity. The evolution of the system to the final equilibrium local elasticity”, IMA Journal of Applied Mathematics 74, state is shown to pass through the intermediate stage when 97-106, (2009)]. We propose a Fourier pseudo-spectral nu- the energy exchange between the waveveguides is still neg- merical method for the HBQ equation. We then prove the ligible. convergence of numerical scheme in the energy space. We consider three test problems concerning the propagation Stanislav Derevyanko of a single solitary wave, the interaction of two solitary Weizmann Institute of Science waves and a solution that blows up in finite time. The [email protected] numerical results show that the Fourier pseudo-spectral method provides highly accurate results. This work has been supported by the Scientific and Technological Re- CP12 search Council of Turkey (TUBITAK) under the project Dynamics of Bright-Dark Solitons and Their Col- MFAG-113F114. lisions in the General Multi-Component Yajima- Oikawa System Goksu Topkarci, Gulcin M. Muslu Department of Mathematics, Istanbul Technical We consider a general multi-component Yajima-Oikawa University system governing the dynamics of nonlinear resonance in- [email protected], [email protected] teraction of multiple short waves with a long wave in one dimension. We construct the exact bright-dark (mixed) soli- Handan Borluk ton solution by using the Hirota’s bilinearization method. Department of Mathematics, Isik Universty By performing an asymptotic analysis, we unravel two [email protected] types of fascinating energy sharing collisions and also the standard elastic collision of bright-dark solitons. The dynamics of soliton bound states is also explored. CP12 A Multiscale Continuation Algorithm for Binary Sakkaravarthi Karuppaiya Rydberg-Dressed Bose-Einstein Condensates Bishop Heber College [email protected] We study a multiscale continuation algorithm for binary Rydberg-dressed Bose-Einstein condensates which is Kanna Thambithurai governed by a system of the Gross-Pitaevskii equations Bishop Heber College, Tiruchirappalli–620 017 (GPEs). The proposed four-parameter continuation al- kanna [email protected] gorithm has the advantage that we can obtain the con- tours of the wave functions ψ1 and ψ2 for various values ∗ ∗ of μ11 = μ22 ∈ [0,μ1]andμ12 ∈ [0,μ2],whereweonly CP12 need to trace the ground state solution curve once. Our Vortex Clustering and Negative Temperature numerical experiments show that the proposed algorithm States in a 2D Bose-Einstein Condensate outperforms the classical continuation algorithm, and is very competitive compared to other numerical methods. In this work we investigate the question of clustering of like signed vortices in a two-dimensional Bose-Einstein con- Cheng-Sheng Chien densate. Such clustering can be understood in terms of Ching Yin University negative temperature states of a vortex gas. Due to the Jungli, Taiwan long-range nature of the Coloumb-like interactions in point [email protected] vortex flows, these negative temperature states strongly depend on the shape of the geometry in which this clustering Sirilak Sriburadet phenomena is considered. We analyze the problem of clus- National Chung Hsing University tering of point vortices in a number of different regions. [email protected] We present a theory to uncover the regimes for which clustering of like signed vortices can occur and compare our Yun-Shih Wang predictions with numerical simulations of a point vortex Chien Hsin University of Science and Technology, Taiwan gas. We also extend our results to the Gross-Pitaevskii [email protected] model of a Bose gas by performing numerical simulations for a range of vortex configurations using parameters that are relevant to current experiments. CP12 The Structure of the Equilibrium Field Distribu- Hayder Salman, Davide Maestrini tion in a Discrete Optical Waveguide System with University of East Anglia NW14 Abstracts 71

[email protected], [email protected] can be obtained via simplifying assumptions. This method can be applied to other similar dissipative equations.

CP12 Stefan C. Mancas Pulse Interactions and Bound-State Formation in Embry-Riddle Aeronautical University, Daytona Beach Non-Local Active-Dissipative Equations [email protected]

We analyse coherent structures in active-dissipative equations that in addition contain non-local contributions in the CP13 form of pseudo-differential operators. Such equations arise Phase Wave Equation e.g. in the modelling of a liquid film flow in the presence of various external effects. We develop a weak-interaction I consider coupled oscillators which exhibit frequency syn- theory of pulses in such equations and demonstrate that chronization without phase synchronization. The separa- pulse interactions and bound-state formation are strongly tion of frequency and phase synchronizations leads to a influenced by non-locality and are characterised by several new wave equation which can be called a phase wave equa- features that are not present in local equations. tion. I first explain the basic properties of the phase wave equation. Then I demonstrate that the phase wave equa- Dmitri Tseluiko tion is generic and can support various kinds of wave which Department of Mathematical Sciences include barbers pole, solitary wave, crawling motion etc. [email protected] Takashi Odagaki Tokyo Denki University Te-Sheng Lin [email protected] Department of Mathematical Sciences Loughborough University [email protected] CP13 Chiral Solitary Waves in Galilean Covariant Fermi Marc Pradas Field Theories with Self-Interaction Department of Mathematical Sciences [email protected] A generalization of the Levy-Leblond equations for a (3 + 1)-dimensional non-relativistic Fermi field for the case of Serafim Kalliadasis the non-zero rest energy is considered. The self-interaction Department of Chemical Engineering of the field is included and its rest mass is assumed to Imperial College London differ from the inertial one. A positive chirality solitary 3 [email protected] wave solution with space oscillations in the x -coordinate is constructed, the amplitude of the solution depending on the rest energy. The wave’s energy and mass transferred Demetrios Papageorgiou 1 2 Department of Mathematics through the unit area in the (x ,x )-plane are calculated. Imperial College London, UK Stability of the solution is discussed. [email protected] Fuad Saradzhev Centre for Science, Athabasca University CP12 Athabasca, Alberta, Canada [email protected] Ground State Solutions of Coupled IR/UV Station- ary Filaments CP13 The ground state solutions of coupled infrared (IR) / ul- traviolet (UV) stationary filaments have been studied both Ultrashort Pulse Propagation in Periodic Media - analytically and numerically. The UV pulse have exact Analysis and Novel Applications Townes solition power for the small IR pulse. And the We present a novel technique for manipulating short pulses eigenstates of the UV Townes branch where the IR fila- in periodically modulated media. Specifically, we show how ment is no longer small are also found numerically. to perform time-reversal, extreme pulse compression, ultrafast switching and slow light. Our study includes an unusu- Danhua Wang ally sophisticated multiple-scales analysis which is corrob- University of Vermont orated numerically with a full electrodynamic model and [email protected] a simplified coupled mode theory. The technique can find applications in any wave systems, especially in optics and Alejandro Aceves acoustics. Finally, we review experimental tests of these Southern Methodist University ideas. [email protected] Yonatan Sivan Tel Aviv University CP13 [email protected] Weierstrass Traveling Wave Solutions for Dissipa- tive Equations CP13 In this talk the effect of a small dissipation on waves is in- Transmission and Reflection of Airy Beams at An cluded to find exact solutions to the modified BBM equa- Interface of Dielectric Media tion. We will use a general formalism based on Ince’s transformation to write the general solution in terms of Weier- Transmission and Reflection of Airy beams at an interface strass ℘ functions, from which all the other known solutions between two diffusive linear-nonlinear dielectric media are 72 NW14 Abstracts

studied numerically and analytically. The beam dynam- both variables. ics is simulated by the beam propagation method. In this study, the interesting cases of critical incidence of Brew- Alex Himonas ster and total-internal-reflection angels are considered. It University of Notre Dame, USA is observed that the reflected beam nearly unaffected but [email protected] shifted. Some interesting results will be presented in this talk. MS1 Rajah P. Varatharajah On Rigorous Aspects of the Unified Transform North Carolina A&T State University Method [email protected] The unified transform method, introduced by Fokas in 1997, produces novel solution formulae to boundary value CP13 problems for linear as well as integrable nonlinear equa- Absolute Stabilization for the Axially Moving tions. For linear problems, in particular, this method Kirchhoff String with Nonlinear Boundary Feed- can be employed even for non-separable or non-self-adjoint back Control problems, where the classical solution techniques fail. We will discuss some rigorous aspects of these novel formulae In this work, we consider the stabilization problem of the in the case that the prescribed initial and boundary data nonlinear axially moving Kirchhoff string using nonlinear belong to appropriate Sobolev spaces. boundary control. The nonlinear boundary control is the negative feedback of the transverse velocity of the string at Dionyssis Mantzavinos one end, which satisfies a polynomial type sector constrain. University of Notre Dame Employing the multiplier method, we establish explicit ex- [email protected] ponential and polynomial stability for the Kirchhoff string. Simulation examples are presented based on finite element method with Lagrange hat basis. MS1 The Fokas Method for Interface Problems Yuhu Wu, Tielong Shen Sophia University Problems related to interfaces in different media arise fre- [email protected], [email protected] quently in applications and often are not solvable analytically using classical approaches. In my talk I will show that by adapting and expanding the Method of Fokas, solving MS1 many types of interface problems for the heat equation is The Unified Transform for the NLS with t-Periodic now possible. I will also show how to use these methods Boundary Conditions for an interface problem involving the linear Schrödinger equation and expand to mixed interface problems where The most challenging problem in the implementation of the different sides of the interface are governed by differing so-called unified transform to the analysis of the nonlinear equations. Schrödinger equation on the half-line is the characterization of the unknown boundary value in terms of the given Natalie E. Sheils initial and boundary conditions. For the so-called lineariz- Department of Applied Mathematics able boundary conditions, this problem can be solved ex- University of Washington plicitly. Furthermore, for non-linearizable boundary con- [email protected] ditions which decay for large t, this problem can be largely bypassed in the sense that the unified transform yields useful asymptotic information for the large t behavior of the MS2 solution. However, for the physically important case of pe- Motion in a Bose-Einstein Condensate: Vortices, riodic boundary conditions, it is necessary to characterize Turbulence and Wakes the unknown boundary value. Recent progress towards the solution of this problem will be presented. We show that an elliptical obstacle moving in a 2- dimensional Bose-Einstein condensate induces a wake re- Athanassios S. Fokas sembling that seen in ordinary classical flows past a cylin- University of Cambridge der. The wake consists of clusters of like-signed quantum [email protected] vortices which are nucleated at the boundary of the obstacle. Vortex singularities in the inviscid superfluid thus mimic classical vortex patterns typical of viscous flows. MS1 Transitions from symmetric wake to time dependent Kar- The Initial Value Problem in Analytic Spaces man vortex street to turbulent wake are observed. Finally, of Nonlinear Evolution Equations with Traveling we discuss these effects in view of current experiments in Wave Solutions which oscillating wires, forks and grids are used to generate turbulence in superfluid helium. We shall consider dispersive and weakly dispersive evolution equations, which in addition to being integrable they Carlo F. Barenghi all have interesting traveling wave solutions and discuss Applied Mathematics Group their Cauchy problem when the initial data are analytic. Newcastle University For dispersive equations, like KdV and cubic NLS, the solu- [email protected] tion to the initial value problem with analytic initial data is analytic in the space variable and Gevrey in the time George Stagg, Nick G. Parker variable. However, for weakly dispersive equations, like Newcastle University Camassa-Holm type equations, the solution is analytic in [email protected], NW14 Abstracts 73

[email protected] through solving an equation similar to the Schrödinger equation but with an imaginary term representing loss. We discuss how some transport properties, such as the quan- MS2 tum Hall effect, show themselves in such driven-dissipative Intrinsic Photoconductivity of Ultracold Fermions systems. in An Optical Lattice Tomoki Ozawa We report on the experimental observation of an analog to Dipartimento di Fisica a persistent alternating photocurrent in an ultracold gas of UniversitàdiTrento fermionic atoms in an optical lattice. The dynamics is in- [email protected] duced and sustained by an external harmonic confinement. While particles in the excited band exhibit long-lived oscillations with a momentum-dependent frequency, a strik- MS3 ingly different behavior is observed for holes in the lowest Asymptotic Stability for Nls with Nonlinear Point band. An initial fast collapse is followed by subsequent Interactions periodic revivals. Both observations are fully explained by mapping the system onto a nonlinear pendulum. We report on recent results concerning a Schrödinger Equa- tion with an attractive nonlinearity concentrated at a Christoph Becker point. In particular, we focus on the problem of the asymp- Institut fur Laserphysik totic stability of the Ground States. We treat both cases of Universitat Hamburg a linearized evolution with or without bound states. This [email protected] is a joint work with D. Noja (Milan) and C. Ortoleva (Mi- lan). Jannes Heinze, Jasper Krauser, Nick Flaeschner, Soeren Riccardo Adami Goetze, Alexander Itin, Ludwig Mathey, Klaus Sengstlock University of Milano Institut fur Laserphysik Universitat Hamburg [email protected] [email protected], [email protected] hamburg.de, nfl[email protected], [email protected], [email protected] MS3 hamburg.de, [email protected], [email protected] Scattering for Nonlinear Schrdinger Equations under Partial Harmonic Confinement

We consider the nonlinear Schrödinger equation under a MS2 partial quadratic confinement. We show that the global Pattern Formation in Solid State Condensates dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates. In my talk I will discuss the phenomena observed in, and We infer the existence of wave operators. Asymptotic com- properties of, microcavity excitonpolariton condensates. pleteness stems from suitable Morawetz estimates, which in These are condensates of mixed light and matter, consist- turn follow from the latest approach, applied to a marginal ing of superpositions of photons in semiconductor micro- of the position density. This is a joint work with Paolo An- cavities and excitons in quantum wells. Because of the im- tonelli and Jorge Drumond Silva. perfect confinement of the photon component, excitonpo- laritons have a finite lifetime, and have to be continuously Remi Carles re-populated. Therefore, excitonpolariton condensates lie CNRS & Universite Montpellier 2 somewhere between equilibrium BoseEinstein condensates [email protected] and lasers. I discuss the coherence properties of excitonpolariton condensates predicted theoretically and studied experimentally, and the wide variety of spatial structures MS3 including quantised vortices, trapped states, states of a Some Collision Problems Related to Nonlinear quantum harmonic oscillator, ring condensates and other Schrdinger Equations coherent structures. These patterns and their dynamics can be reproduced through the solution of the complex In this talk I will discuss a more qualitative lack of stability Ginzburg-Landau or the complex Swift-Hohenberg equa- of several structures of solitary wave type. I will focus this tions. I will discuss to which extent these macroscopic description on the nonlinear Schrdinger model, although quantum states can be understood as linear or nonlinear the proposed account of the dynamics can be adapted to phenomena and what criteria can be applied to make such several other models with unstable solitons. a separation. Claudio Munoz Natalia G. Berloff University Paris Sud-Orsay University of Cambridge [email protected] N.G.Berloff@damtp.cam.ac.uk

MS3 MS2 On the Long-Time Behavior of NLS in the Partially Quantum Hall Eﬀect and Transport Phenomena in Periodic Setting Lossy Photonic Lattices We present results on the Nonlinear Schroedinger equation We discuss properties of photonic cavities aligned in a form posed in the partially periodic setting Rd × T and we look of a lattice. Because of the inherent loss of photons, the at the long-time behavior of the solutions. In particular we steady-state conﬁguration is achieved as a result of contin- shall study the H 1 scattering in the regime of nonlinearities uous gain and loss, which can theoretically be obtained which are both L2-supercritical and energy subcritical. We 74 NW14 Abstracts

also present results related to the modiﬁed scattering for of transition from energy entrapment to energy transport the cubic NLS in the partially periodic setting R × Td. in the system of N coupled, granular chains. Two distinct mechanisms leading to the breakdown of energy localiza- Nicola Visciglia tion on the ﬁrst and the second chains have been revealed University of Pisa and analyzed in the study. Using the regular multi-scale [email protected] asymptotic analysis along with the non-smooth temporal transformation method (NSTT) we formulate the analytic criteria for the formation of the inter-chain energy trans- MS4 port. Results of the analytical approximation are in a very An Asymptotic Model for Small Amplitude Solu- good correspondence with the results of numerical simulations to Newton’s Cradle tions of the full model.

We study the dynamics of a one-dimensional lattice of Yuli Starosvetsky nonlinearly coupled oscillators. The class of problems ad- Technion, Israel Institute of Technology dressed include Newton’s cradle with Hertzian contact in- [email protected] teractions. The Cauchy problem is studied yielding lower bounds for the existence time. We then derive an asymptotic model for small amplitude solutions over large times. MS4 This allows to prove the existence of breather-like solutions Acoustic Wave Filtering and Breather Formation to the initial model. We also estimate the maximal disper- in Coupled Granular Chains Embedded in Material sion of localized initial data. Matrix

Brigitte Bidegaray-Fesquet We present an experimental study of primary pulse trans- CNRS and Grenoble University, France mission in coupled ordered granular chains embedded in [email protected] elastic matrix under harmonic excitation. We find that depending on the magnitude and frequency of the applied Eric Dumas excitation these strongly nonlinear dynamical systems may Institut Fourier, Université Grenoble 1 and CNRS either support propagating pulses (in pass bands), or may [email protected] attenuate higher frequency harmonic components (in stop bands). Hence, these media act as tunable nonlinear acous- Guillaume James tic filters. In the transition between pass and stop bands Laboratoire Jean Kuntzmann, Université de Grenoble these systems support breathers, in the form of weakly and CNRS modulated oscillating waves. These results are confirmed [email protected] theoretically and experimentally. Alexander Vakakis, M. Arif Hasan, Sinhu Cho, D. MS4 Michael McFarland, Waltraud M. Kriven Justification of Leading Order Quasicontinuum Ap- University of Illinois proximations of Strongly Nonlinear Lattices [email protected], [email protected], [email protected], [email protected], After giving a brief introduction to the topic of granular [email protected] crystals, I will consider the leading order quasicontinuum limits of a one-dimensional granular chain governed by the MS5 Hertz contact law under precompression. The approximate model, a so-called p-system, which is derived in this limit Interfaces in a Random Environment is justified by establishing asymptotic bounds for the error with the help of energy estimates. The continuum model We review results on the qualitative behaviour of parabolic predicts the development of shock waves, which we inves- evolution equations with random coefficients which model tigate with the aid of numerical simulations. the evolution of an interface which moves in a random environment. The typical examples are randomly forced Christopher Chong mean curvature flow (with a forcing which depends on University of Massachusetts, Amherst space onlu) and related semi-linear models (random ob- [email protected] stacle model). Nicolas Dirr Panayotis Kevrekidis Cardiff School of Mathematics Department of Mathematics and Statistics Cardiff University University of Massachusetts dirrnp@cardiff.ac.uk [email protected]

Guido Schneider MS5 Unviersity of Stuttgart Not Available at Time of Publication Germany [email protected] Not Available at Time of Publication

Gabriel J. Lord MS4 Heriot-Watt University Transitions from Energy Entrapment to Energy [email protected] Transport in the System of N Coupled Granular Chains MS5 In the present study we analyze the governing mechanisms Noise Induced State Transitions, Intermit- NW14 Abstracts 75

tency and Universality in the Noisy Kuramoto- measurement of three quantities: the clock period in the Sivashinsky Equation posterior PSM, somite length and the length of the PSM. A travelling wavefront, which slows oscillations along the We analyze the effect of pure additive noise on the long- AP axis, is an emergent feature of the model. time dynamics of the noisy Kuramoto-Sivashinsky (KS) equation in a regime close to the instability onset. We Ruth E. Baker show that when the noise is highly degenerate, in the sense University of Oxford that it acts only on the first stable mode, the solution of the Centre for Mathematical Biology KS equation undergoes several transitions between differ- [email protected] ent states, including a critical on-off intermittent state that is eventually stabilized as the noise strength is increased. Such noise-induced transitions can be completely charac- MS6 terized through critical exponents, obtaining that both the Ca2+/calmodulin-Dependent Protein Kinase KS and the noisy Burgers equation belong to the same uni- Waves in Heterogeneous Spiny Dendrites versality class. The results of our numerical investigations are explained rigorously using multiscale techniques. Not Available at Time of Publication

Greg Pavliotis Paul C. Bressloﬀ Imperial College London University of Utah and University of Oxford, UK [email protected] Department of Mathematics bressloﬀ@math.utah.edu

MS5 Stability of Traveling Waves in Bistable Reaction- MS6 Diﬀusion Equations Actin Nucleation Waves in Motile Cells

Stability of traveling waves in stochastic bistable reaction- Not Available at Time of Publication diffusion equations with both additive and multiplicative noise is proven, using a variational approach based on func- William R. Holmes tional inequalities. Explicit estimates on the rate of stabil- University of California, Irvine ity are derived that can be shown to be optimal in the Department of Mathematics special case of stochastic Nagumo equations. Our anal- [email protected] ysis allows to derive a stochastic differential equation for the motion of the wave front and a mathematical rigorous MS7 decomposition of the full dynamics into the phase-shifted traveling wave and non-Gaussian fluctuations. Jet-Particle Methods for Incompressible Fluids

Wilhelm Stannat A Lagrangian model of ideal incompressible fluid flow is TU Darmstadt introduced in which a kth order deformation is attached [email protected] to each particle. The data associated to the deformation gradient is formally represented as an internal degree of freedom within each particle. In particular, it is a gauge MS6 associated to the kth order jet-group of a point, and we re- Actin Polymerization and Subcellular Mechanics fer to the particle as a k-jet particle. Through an infinite- Drive Nonlocal Excitable Waves in Living Cells dimensional reduction by symmetry, solutions may be obtained to the fluid equation by solving a finite dimensional Not Available at Time of Publication ODE. The resulting ODE is then solvable using standard integration schemes. Interesting behavior occurs when we Jun Allard consider particle collisions. We find that collisions of a k- University of California, Irvine jet particles and an -jet particles yield (in infinite time) a Department of Mathematics (k + )-jet particle. This can be interpreted as a model of [email protected] the energy cascade of an ideal incompressible fluid.

Colin Cotter MS6 Imperial College Clock and Wavefront Model of Somitogenesis [email protected].

The currently accepted interpretation of the clock and Darryl D. Holm, Henry Jacobs wavefront model of somitogenesis is that a posteriorly mov- Imperial College London ing molecular gradient sequentially slows the rate of clock [email protected], [email protected] oscillations, resulting in a spatial readout of temporal oscillations. However, while molecular components of the clocks and wavefronts have now been identified in the pre- MS7 somitic mesoderm (PSM), there is not yet conclusive ev- Structure-preserving Discretization of Fluid Me- idence demonstrating that the observed molecular wave- chanics fronts act to slow clock oscillations. Here we present an alternative formulation of the clock and wavefront model In this talk we will geometrically derive discrete equations in which oscillator coupling, already known to play a key of motion for fluid dynamics from first principles. Our ap- role in oscillator synchronisation, plays a fundamentally proach uses a finite-dimensional Lie group to discretize the important role in the slowing of oscillations along the an- group of volume-preserving diffeomorphisms, and the dis- teriorposterior (AP) axis. Our model has three parame- crete Euler equations are derived from a variational princi- ters that can be determined, in any given species, by the ple with non-holonomic constraints. The resulting discrete 76 NW14 Abstracts

equations of motion induce a structure-preserving time in- Courant Institute of Mathematical Sciences tegrator with good long-term energy behavior, for which an New York University exact discrete Kelvin circulation theorem holds. Our meth- [email protected] ods can be used to derive structure-preserving integrators for many systems, such as equations of magnetohydrody- namics, complex fluids etc. MS8 Large Deviations for Stochastic Partial Differential Dmitry Pavlov Equations with Applications in Planetary Atmo- Imperial College London sphere Dynamics [email protected] We will discuss several recent result for the computation of large deviation rate function for stochastic differential MS7 equations, including the stochastic two-dimensional Euler Geometric Theory of Garden Hose Dynamics and Navier-Stokes equations, and quasi-geostrophic models that describe planetary atmospheres. Both explicit results Instability of flexible tubes conducting fluid, or ”garden for the quasi-potential expression and numerical compu- hose instability”, is a phenomenon both familiar from ev- tations will be discussed. From those computations, we eryday life and important for applications, and therefore predict non-equilibrium phase transitions and bistable sit- has been actively studied. In spite of its long history, many uations. unanswered questions remain, in particular, how to address a dynamically changing cross-section. We construct a Freddy Bouchet fully three dimensional, exact geometric theory of this phe- Laboratoire de Physique nomenon by coupling the dynamics of the elastic tube (left- ENS de Lyon invariant) with the motion of the fluid (right-invariant) [email protected] with additional volume constraint. We also discuss the boundary conditions through an appropriate, Lagrange- d’Alembert’s like modification of the critical action prin- MS8 ciple. We show that the change of the cross-section affects Center Manifolds and Taylor Disperson the stability properties, derive a class of exact, fully nonlinear solutions of traveling wave type and (time permitting), Taylor dispersion, first described by G. Taylor in 1953, oc- discuss the results of experiments elucidating some impor- curs when the diffusion of a solute in a long channel or tant aspects of the dynamics. pipe is enhanced by the background flow, in that the solute asymptotically approaches a form that solves a differ- Vakhtang Putkaradze ent diffusion equation (with larger diffusion coefficient, in University of Alberta an appropriate moving frame). By introducing scaling vari- [email protected] ables, I’ll show how one can obtain this asymptotic form via a center manifold reduction. This is joint work with Margaret Beck and C.E. Wayne. MS7 Using Multi-Moment Vortex Methods to Study Osman Chaudhary Merger Boston University Department of Mathematics and Statistics We use a low order model to understand how two co- [email protected] rotating vortices transition from a quasi-steady distance from each other to the rapid convective merger that occurs after diffusion causes the vortex core size to exceed some MS8 critical fraction of the separation distance. This model was Pseudospectrum for Oseen Vortices Operators derived from the recently developed Multi-Moment Vortex Oseen vortices are self-similar solutions to the vorticity Method and provides several physical insights as well as 2 pins down precisely what causes the very initial onset of equation in R . T.Gallay and C.E.Wayne proved in 2005 convective merger. that these solutions are stable for any value of the circulation Reynolds number. The linearization of the sys- David T. Uminsky tem around an Oseen vortex naturally gives rise to a non- University of San Francisco self-adjoint operator. We shall discuss spectral and pseu- Department of Mathematics dospectral properties of the linearized operator in the fast [email protected] rotation limit. In particular, we give some (optimal) resol- vent estimates along the imaginary axis.

MS8 Wen Deng Inviscid Damping and the Asymptotic Stability of Mathematical Sciences Planar Shear Flows in the 2D Euler Equations Lund University [email protected] We prove the asymptotic stability of shear flows close to the planar, periodic Couette flow in the 2D incompressible Euler equations. Specifically: given an initial perturbation MS9 of the Couette flow small in a suitable regularity class, Mixed Boundary Value Problems for Stokes Flows: the velocity converges strongly in L2 to a shear flow as New Methods and Applications t-¿infinity. The vorticity is mixed to small scales and in general enstrophy is lost in the weak limit. Joint work Motivated by microfluidics applications where it is re- with Nader Masmoudi. quired to manipulate viscous fluids at small scales we present some novel mathematical approaches to a variety Jacob Bedrossian of mixed boundary value problems for biharmonic fields NW14 Abstracts 77

arising therein. The methods, which are based on a formu- [email protected] lation originally presented by Crowdy & Fokas [Proc. Roy. Soc. A, 460, (2004)], are general and have much wider applicability in other areas. MS10 Not Available at Time of Publication Darren G. Crowdy Imperial College London Not Available at Time of Publication [email protected] Jean-Marc Chomaz LADHYX MS9 Ecole Polytechnique [email protected] The Nonlinear Schrdinger Equation with Periodic Boundary Data MS10 I will present some new results related to the initial- Dissection of Boussinesq Non-linear Interactions boundary value problem for the nonlinear Schrdinger equa- Using Intermediate Models tion on the half-line with an asymptotically periodic boundary condition. Nonlinear coupling among wave modes and vortical modes is dissected in order to probe the question: Can we dis- tinguish the wave-vortical interactions largely responsible Jonatan Lenells for formation versus evolution of coherent, balanced struc- Department of Mathematics tures? It is well known that the quasi-geostrophic (QG) Baylor University equations can be derived from the Boussinesq system in jonatan [email protected] a non-perturbative way by ignoring wave interactions and considering vortical modes only. One qualitative difference between those two models is the lack of skewness in the MS9 QG dynamics. In this talk, non-perturbative intermediate models that include more and more classes of non-linear Well-Posedness and Spectral Representation of interactions will be used to identify their role in different Linear Initial-Boundary Value Problems qualitative properties of the Boussinesq system. Numerical results will be shown to describe the effect of each class in We study initial-boundary value problems for linear the transfer of energy between vortical modes and waves, constant-coefficient evolution equations on a finite 1-space, transfer of energy (or the lack of it) between scales, forma- 1-time domain. Classical separation of variables and tion of vortices and skewness. Fourier transform methods fail for all problems except those of second order or those with very special boundary Gerardo Hernandez-Duenas conditions whereas the method of Fokas solves any such Department of Mathematics well-posed problem. We describe the well-posedness crite- University of Wisconsin, Madison ria and provide a functional-analytic view of the failure of [email protected] classical methods and the success of Fokas’ method. Leslie Smith Mathematics and Engineering Physics David Smith University of Wisconsin, Madison University of Crete [email protected] [email protected] Samuel Stechmann University of Wisconsin - Madison MS9 [email protected] A Retrospective Inverse Problem for the Wave Equation MS10 A Coupled Model of the Interactions Between In- A typical retrospective inverse problem is one where we ertial Waves and Turbulence in the Ocean attempt to recover the full solution to the heat equation, say on the whole line, given time series data at a specific We derive a new model of the interactions between near- location i.e. Dirichlet data. In this talk, I shall consider inertial waves (NIWs) and balanced motion in the ocean this problem for the wave equation: given time-series data using Generalised Lagrangian Mean theory and Whitham at fixed spatial locations, can one recover the full solution averaging. In its simplest form, the model couples the well- to the problem posed on the whole line? By using simple known Young-Ben Jelloul model of NIWs with a quasi- arguments based on Fokas’ method we shall obtain neces- geostrophic model of the balanced motion. The model sary conditions to recover the solution. Further employ- is Hamiltonian and conserves both energy and wave ac- ing the global relations for the associated boundary-value tion. Analytic arguments and numerical simulations of the problems we obtain a system of equations to solve for the model shed light on mechanisms of NIW-mean flow inter- initial condition everywhere. actions in the ocean.

Vishal Vasan Jacques Vanneste,Jin-HanXie Department of Mathematics School of Mathematics The Pennsylvania State University University of Edinburgh 78 NW14 Abstracts

[email protected], [email protected] ues present an ”avoided crossing” : at one given point, the gap between them reduces as the semi-classical parameter becomes smaller. We show that when an initial coherent MS10 state polarized along an eigenvector of the potential prop- Centrifugal, Barotropic and Baroclinic Instabili- agates through the avoided crossing point, there are tran- ties of Isolated Ageostrophic Anticyclones in the sitions between the modes at leading order. In the regime Two-layer Rotating Shallow Water Model and their we consider, we observe a nonlinear propagation far from Nonlinear Saturation the crossing region while the transition probability can be computed with the linear Landau-Zener formula. Instabilities of the isolated anticyclonic vortices in the 2- layer rotating shallow water model are studied at Rossby Lysianne Hari numbers up to 2, with the main goal to understand the University Cergy-Pontoise interplay between the classical centrifugal instability and [email protected] other possible ageostrophic instabilities. We find that different types of instabilities with low azimuthal wavenumbers exist, and may compete. In a wide range of parameters MS11 an asymmetric version of the standard centrifugal instabil- Dispersive Estimates for Non-Autonomous Hamil- ity has larger growth rate than this latter. The dependence tonians and Applications to Nonlinear Asymptotic of the instabilities on the parameters of the flow: Rossby Stability of Solitary Waves and Burger numbers, vertical shear, and the ratios of the layers’ thicknesses and densities is investigated. The zones The pioneering work of Soffer and Weinstein in the ’90 of dominance of each instability are determined in the pa- introduced the method of controlling the nonlinearity by rameter space. Nonlinear saturation of these instabilities dispersive estimates of the linearized operator at a frozen is then studied with the help of a high-resolution finite- time. It has now become a classical method to study volume numerical scheme, by using the unstable modes asymptotic stability of nonlinear waves and has shown that identified from the linear stability analysis as initial con- the actual dynamics shadows different solitary waves be- ditions. Differences in nonlinear development of the com- fore collapsing to one. What is less known is that the peting centrifugal and ageostrophic barotropic instabilities method can be significantly improved if one obtains and are exposed. uses dispersive estimates for the time dependent linearized dynamics, i.e. the linearization is at the solitary wave Vladimir Zeitlin,NoéLahaye shadowed at that moment and changes as the dynamics ENS/University of Paris 6, France moves from one solitary wave to another. I will discuss [email protected], [email protected] how this new perspective has allowed us to tackle, for the first time, subcritical nonlinearities in Schrodinger equations and to significantly reduce smoothness assumptions MS11 in nonlinear Dirac equations. This is joint work with A. Selection Principles for Semiclassical Flows over Zarnescu (U. Sussex, England), O. Mizrak (Mersin U., Conical Singularities: Asymptotic and Computa- Turkey), R. Skulkhu (Mahidol U., Thailand) and N. Bous- tional Investigation said (U. Franche-Comte, France). Eduard Kirr When a pure state interacts strongly with conical singular- Department of Mathematics ities, there are several possible classical evolutions. Based University of Illinois at Urbana-Champaign on recent rigorous results, we propose a selection princi- [email protected] ple for the regularization of semiclassical asymptotics for Schroedinger equations with Lipschitz potentials. Employ- ing a solver with a posteriori error control, we generate rig- MS11 orous upper bounds for the error in our asymptotic approx- Dispersion and Long Time Evolution of Coherent imation. For 1-dimensional problems without interference, States we obtain compelling agreement between the regularized asymptotics and the full solution. This allows the formu- We consider the the semiclassical limit of solutions of the lation of a conjecture for the validity of the regularized time dependent Schrödinger equation with potential, and asymptotics. Both the selection principle, and the compu- with initial conditions given by coherent states. The evo- tation with a posteriori error control to check it, can be lution of the shape of a coherent state depends strongly on extended to the 1D Schroedinger-Poisson equation, where the classical dynamics which emerges in the semiclassical the non-linearity is also generated by a conical singularity. limit, and we use this dynamics to give precise estimates Existing work in the same spirit, and implications of our on how a coherent state disperses for large times and even- approach in that problem are briefly discussed. tually becomes a Lagrangian state. This can then be used to transform a nonlinear Schrödinger equation into a new Agisilaos Athanassoulis form in which the nonlinearity decays in time. University of Leicester [email protected] Roman Schubert University of Bristol [email protected] MS11 Nonlinear Propagation of Coherent States Through Avoided Energy Level Crossing MS12 Granular Crystals at the Microscale We study the propagation of a coherent state for a one- dimensional system of two coupled Schrdinger equations in The nonlinear dynamics of a two-dimensional granular the semi-classical limit. Couplings are induced by a cubic crystal composed of 1 μm silica microspheres assembled nonlinearity and a matrix-valued potential, whose eigenval- on an elastic substrate are studied. The dynamics of the NW14 Abstracts 79

system are shown to differ significantly from macroscale Phononic Crystals granular media due to effects such as interparticle and particle-substrate adhesion. High amplitude surface acous- We present experimental and numerical studies on the tic waves are excited and measured in the coupled system modulational effect of nonlinear waves in woodpile via the laser-induced transient grating technique. The pho- phononic crystals composed of slender, cylindrical ele- toacoustic measurements are compared with our analytical ments. We find that various patterns of wave modulation model and numerical simulations. and attenuation can be achieved in woodpile architectures associated with their natural bending modes. The numer- Nicholas Boechler ical results based on finite element and discrete element Massachusetts Institute of Technology methods corroborate experimental results well. Woodpile Department of Mechanical Engineering phononic crystals can offer new testbed for manipulating [email protected] nonlinear wave propagation. Jinkyu Yang, Eunho Kim MS12 University of Washington [email protected], [email protected] Solitary Waves in a 1D Chain of Repelling Magnets

We study experimentally, numerically and theoretically the MS13 dynamics of a 1D discrete nonlinear lattice composed of Determining Wave Stability Via the Maslov Index repelling magnets. We demonstrate that such lattice sup- ports solitary waves with profile and propagation speed de- The Maslov index is a symplecto-geometric invariant that pending on the amplitude. The system belongs to the kind counts signed intersections of Lagrangian subspaces. It was of nonlinear lattices studied in [Friesecke and Matthies, recently shown that the Maslov index can be used to com- 2 Physica D, 171(2002) 211-220] and exhibits sech profile pute Morse indices of selfadjoint, elliptic boundary value in the low energy regime and atomic scale localization in problems on star-shaped domains. We extend these results the high energy regime. Such systems may find potential to bounded domains with arbitrary geometry, and discuss applications in the design of novel devices for shock absorp- some applications to the stability of nonlinear waves. tion, energy localization and focusing. Furthermore, due to the similarity of the magnetic potential with the poten- Graham Cox tials governing atomic forces, the system could be used for University of North Carolina a better understanding of important problems in physics Chapel Hill, USA and chemistry. [email protected]

Miguel Moleron Chris Jones ETH University of North Carolina-Chapel Hill [email protected] [email protected]

Andrea Leonard Jeremy L. Marzuola Cell Biomechanics Department of Mathematics University of Washington University of North Carolina, Chapel Hill [email protected] [email protected]

Chiara Daraio ETH Z¨urich, Switzerland and California Institute of Tech MS13 [email protected] The Krein Matrix for Quadratic Eigenvalue Prob- lems

MS12 In order to understand spectral stability for Hamiltonian systems, there are two types of eigenvalues that need to Disordered Granular Chains be found: those with positive real part, and those purely imaginary ones with negative Krein signature. The former I investigate the propagation and scattering of highly non- can be easily identified visually; however, the latter can- linear waves in disordered granular chains composed of not. Knowing the signature of an eigenvalue is necessary spheres that interact via Hertzian contact. Experiments in order to determine which eigenvalues can contribute to and numerical simulations both reveal the existence of two a Hamiltonian-Hopf bifurcation. The Krein matrix is a different mechanisms of wave propagation. In low-disorder meromorphic matrix which can be used to not only find chains, one observes a decaying solitary pulse; beyond a eigenvalues, but it can be used to graphically determine critical level of disorder, the decay is faster and the wave the Krein signature of purely imaginary eigenvalues. We transmission becomes insensitive to the level of disorder. will discuss its construction, and how it can be used in I will also compare results from different families of disor- conjunction with the Hamiltonian-Krein instability index dered arrangements. theory in order to study Hamiltonian eigenvalue problems. Mason A. Porter University of Oxford Todd Kapitula Oxford Centre for Industrial and Applied Mathematics Calvin College [email protected] Dept of Math & Statistics [email protected]

MS12 Shamuel Auyeung, Eric Yu Modulation of Nonlinear Waves in Woodpile Calvin College 80 NW14 Abstracts

[email protected], [email protected] take the Swift-Hohenberg equation, which is the simplest nonlinear model with a finite critical wavenumber, and use it to study dynamic pattern formation and evolution on MS13 time-dependent spatial domains. In particular, we discuss On the Maslov Index for Periodic and for Multidi- the effects of a time-dependent domain on the stability of mensional Problems spatially homogeneous and spatially periodic base states, and explore the effects of convection and dilution on the In this talk we discuss some recent results on connections Eckhaus instability of periodic states. We show that this between the Maslov and the Morse indices for differential instability is repeatedly triggered as the domain grows and operators. The Morse index is a spectral quantity defined that it leads to phase slips that are responsible for the in- as the number of negative eigenvalues counting multiplici- sertion of additional wavelengths required to maintain a ties while the Maslov index is a geometric characteristic de- preferred wavenumber. This behavior is captured by an fined as the signed number of intersections of a path in the evolution equation for the pattern wavenumber that can space of Lagrangian planes with the train of a given plane. be derived from a generalized complex Ginzburg-Landau The problem of relating these two quantities is rooted in equation describing the nonlinear evolution of the pattern Sturm’s Theory and has a long history going back to the amplitude on time-dependent domains in the presence of classical work by Arnold, Bott, Duistermaat, Smale, and convection and dilution. The results are compared with to a more recent paper by Deng and Jones. Two situa- those on fixed domains. tions will be addressed: First, the case when the differential operator is a Schroedinger operator equipped with Rouslan Krechetnikov theta-periodic boundary conditions, and second, when the Department of Mechanical Engineering Schroedinger operators are acting on a family of multi- University of California at Santa Barbara dimensional domains obtained by shrinking a star-shaped [email protected] domain to a point and are equipped with either Dirichlet or quite general Robbin boundary conditions. This is a joint Edgar Knobloch work with G. Cox, C. Jones, R. Marangell, A. Sukhtayev, University of California at Berkeley and S. Sukhtaiev. Dept of Physics [email protected] Yuri Latushkin Department of Mathematics University of Missouri-Columbia MS14 [email protected] Recent Advances in Pattern Formation on Grow- ing and Evolving Surfaces: Theory, Numerics and Applications MS13 Computing the Maslov Index In this talk I will present theoretical results on the stability analysis of reaction-diffusion systems on evolving We address the problem of computing the Maslov index domains. There are two fundamental biological differ- for large linear symplectic systems on the real line. The ences between the Turing conditions on fixed and grow- Maslov index measures the signed intersections (with a ing domains, namely: (i) we need not enforce cross nor given reference plane) of a path of Lagrangian planes. The pure kinetic conditions and (ii) the restriction to activator- natural chart parameterization for the Grassmannian of inhibitor kinetics to induce pattern formation on a growing Lagrangian planes is the space of real symmetric matrices. biological system is no longer a requirement. Our theoreti- Linear system evolution induces a Riccati evolution in the cal findings are confirmed and reinforced by numerical sim- chart. For large order systems this is a practical approach ulations for the special cases of isotropic linear, exponential as the computational complexity is quadratic in the order. and logistic growth profiles. In particular we illustrate an The Riccati solutions, however, also exhibit singularites example of a reaction-diffusion system which cannot ex- (which are traversed by changing charts). Our new results hibit a diffusively-driven instability on a fixed domain but involve characterizing these Riccati singularities and two is unstable in the presence of slow growth. Generalisations trace formulae for the Maslov index. We demonstrate the to reaction-diffusion systems with cross-diffusion as well as effectiveness of these approaches by applying them to a results on evolving surfaces will be presented. large eigenvalue problem. We also discuss the extension of the Maslov index to the infinite dimensional case. Anotida Madzvamuse University of Sussex Simon Malham Dept of Mathematics Heriot-Watt University [email protected] [email protected]

Margaret Beck MS14 Boston University and Heriot-Watt University Eﬀect of Dissolution-Driven Convection on the [email protected] Partial Mixing Between CO2 and Hydrocarbons Or Reactive Solutions

MS14 Reducing greenhouse gas emissions to mitigate climate Spatial Patterns on Time-Dependent Domains: change is one of the most timely and important challenges Convective and Dilution Effects in environmental and energy-related issues. Subsurface carbon sequestration has emerged as a promising solution We explore the key differences in the near-critical behav- to the problem of rising atmospheric carbon dioxide (CO2) ior between extended systems on time-fixed and time- levels. However, the question remains as to how the effi- dependent spatial domains, with particular emphasis on ciency and safety of such a sequestration process depend on the effects of convection and dilution. As a paradigm, we the physical and chemical characteristics of the storage site. NW14 Abstracts 81

This question is emblematic of the need to better under- University Paris Sud-Orsay stand the dynamics of CO2 in subsurface formations where [email protected] it is known to dissolve into the pre-existing fluid (hydrocarbons in depleted oil fields and brine in saline aquifers). This dissolution, known to improve the safety of the se- MS15 questration by reducing the risks of leaks of CO2 to the OntheDressingMethodforDiscreteIntegrable atmosphere, leads to an increase of the density of the liq- Systems uid phase in the gravity field, thereby leading to natural convection. We aim to describe the mass transport between This talk will discuss the discrete analogue of the dressing the non-equilibrium gas and liquid phases of a binary mix- method for continuous and discrete integrable systems. We ture. Our model accounts for the motion of the interface, will show how, starting from appropriate matrix Riemann- diffusion and convective mass transport due to compress- Hilbert problems, one can derive a number of completely ibility, non ideality, and natural convection. We show that integrable nonlinear differential-difference systems of equa- natural convection influences the mixing time drastically. tions, together with their associated Lax pairs. On the basis of a linear stability analysis of a simplified model, we also show that a chemical reaction of CO2 with Qiao Wang a dissolved chemical species can either enhance or decrease State University of New York at Buffalo the amplitude of the convective dissolution compared to qiaowang@buffalo.edu the non reactive case. On this basis, we classify the various possible cases and identify the parameters that could in- Gino Biondini fluence the dissolution-driven convection in the subsurface State University of New York at Buffalo formations, and thus impact the safety of carbon seques- Department of Mathematics tration. biondini@buffalo.edu

Laurence Rongy Universit´e libre de Bruxelles MS15 [email protected] Stability of Line Standing Waves Near the Bifur- cation Point for Nonlinear Schrodinger Equation

MS14 We consider the transverse instability for nonlinear Modelling and Simulation of Biological Pattern Schrodinger equation. One dimensional nonlinear Formation on Evolving Surfaces Schrödinger equation has a stable soliton. Here, we regard this soliton as a line soliton of two dimensional nonlinear We investigate models for biological pattern formation via Schrodinger equation on the periodic boundary condition reaction-diffusion systems posed on evolving surfaces. The in the transverse direction with the period 2πL. Rousset nonlinear reaction kinetics inherent in the models and the and Tzvetkov showed that there exists the critical period evolution of the surface mean that analytical solutions are 2πL∗ such that the line soliton is stable for LL∗. In this talk, we will sary. We discuss numerical simulation of the model equa- present the transverse instability in the degenerate case tions by finite elements and focus on applications to cell L = L∗ and the relation between the stability and the bi- motility, chemotaxis and skin pigment pattern formation. furcation.

Chandrashekar Venkataraman Yohei Yamazaki University of Warwick Kyoto University Kyoto, Japan [email protected] [email protected]

MS15 MS16 Stability of mKdV Breathers in the Energy Space Phase Mixing and Hydrodynamic Stability

In this talk I will show some recent results about the H1 Phase mixing is thought to play an important role in the stability of breather solutions of mKdV. I will also present stability of certain coherent structures in inviscid fluid dy- Bäcklund transformations for the mKdV and I will explain namics and collisionless kinetic theory. However, math- how we use them as a technical tool to get stability at the ematically rigorous nonlinear stability results have been level of H 1 regularity. somewhat elusive. We will discuss some recent advances on this topic including a simplified proof of nonlinear Landau Miguel Alejo damping in the Vlasov equations joint with Nader Mas- IMPA, Brasil moudi and Clement Mouhot and the asymptotic stability [email protected] of shear flows in 2D ideal fluids joint with Nader Masmoudi. Jacob Bedrossian MS15 Courant Institute of Mathematical Sciences On the Interaction of Nonlinear Schrödinger Soli- New York University tary Waves [email protected]

The aim of this talk is go give a brief account of some interaction problems involving nonlinear solitary waves for MS16 the Schrödinger equation, where the nonlinearity is either Reaction-Diffusion Equations with Spatially Dis- integrable or non integrable. We will give a qualitative tributed Hysteresis description of the involved dynamics. We consider continuous and discrete reaction-diffusion Claudio Munoz equations with hysteresis which is given at every spatial 82 NW14 Abstracts

point. Such equations arise when one describes hysteretic [email protected] interaction between several diﬀusive and nondiﬀusive sub- stances. In the talk, we will discuss mechanisms of appearing global and local spatio-temporal patterns due to MS17 hysteresis as well as their interconnection. This is a joint Initial-Boundary-Value Problems for Nonlinear work with Sergey Tikhomirov. Wave Equations

Pavel Gurevich The lecture will focus upon both the motivation and some Freie Universitat Berlin of the recent analysis of boundary-value problems for non- [email protected] linear, dispersive wave equations. Jerry Bona MS16 University of Illinois at Chicago [email protected] Waves Through Lattices with Impurities

We consider scalar lattice differential equations posed on MS17 square lattices in two space dimensions. Under certain nat- The Effect of Boundary Conditions on Linear and ural conditions we show that wave-like solutions exist when Nonlinear Waves obstacles (characterized by “holes’) are present in the lattice. Our work generalizes to the discrete spatial setting In this talk, I will discuss the effect of boundary conditions the results obtained by Berestycki, Hamel and Matano for on the solvability of PDEs that have formally an integrable the propagation of waves around obstacles in continuous structure, in the sense of possessing a Lax pair. Many spatial domains. of these PDEs arise in wave propagation phenomena, and boundary value problems for these models are very impor- Hermen Jan Hupkes tant in applications. I will discuss the extent to which University of Leiden general approaches that are successful for solving the ini- Mathematical Institute tial value problem extend to the solution of boundary value [email protected] problem. I will survey the solution of specific examples of integrable PDE, linear and nonlinear. The linear theory Aaron Hoffman is joint work with David Smith. For the nonlinear case, I Franklin W. Olin College of Engineering will discuss boundary conditions that yield boundary value aaron.hoff[email protected] problems that are fully integrable, in particular recent joint results with Thanasis Fokas and Jonatan Lenells on the Erik Van Vleck solution of boundary value problems for the elliptic sine- Department of Mathematics Gordon equation. University of Kansas [email protected] Beatrice Pelloni University of Reading, UK [email protected] MS16 Using Graph Limits for Studying Dynamics of MS18 Large Networks A Dynamical Scheme for Prescribing Mesoscale Eddy Diffusivity in the Ocean The theory of graph limits uses analytical methods for describing structural properties of large graphs. We discuss The resolution of the oceanic component of climate models some applications of this theory to constructing and justi- will soon enter a regime where quasigeostrophic dynam- fying continuum limits and to studying stability of spatial ics dominate. This talk will introduce a numerical scheme patterns of nonlocally coupled dynamical systems. for dynamically adjusting the eddy diffusivity and viscosity of these models in a way that is appropriate for quasi- Georgi S. Medvedev geostrophic motion at the gridscale. This scheme is in- Drexel University spired by subgrid modeling techniques that are common in [email protected] 3D Large Eddy Simulation (e.g. Smagorinsky), but which are not appropriate for climate-scale flows.

MS17 Scott Bachman A New Approach for Linear Elliptic Boundary DAMPT Value Problems University of Cambridge [email protected] We will discuss the functional analytic foundations that un- derline Fokas’ uniﬁed approach to elliptic boundary value Baylor Fox-Kemper problems. It will be demonstrated that the global relation Brown University completely characterizes the Dirichlet-Neumann map for [email protected] a large class of elliptic boundary value problems. A detailed treatment of linear elliptic boundary value problems on convex polygons will be presented and we will discuss MS18 the numerical implementation of the uniﬁed method in this On Resurging a Bore-Soliton-Spash case. We were able to create an anomalously high rogue wave Anthony Ashton splash in a water wave channel with a convergence involv- University of Cambridge ing a bore and solitons, and we could control it. Unex- NW14 Abstracts 83

pectedly, this man-made extreme or rogue wave is related the Gibbs measure as initial data. By viewing such ini- to rogue waves in multidirectional seas, solutions of the tial data as a diﬀusion process in x, we extend this result KP-equation, tsunami run-up into coastal valleys, and cer- to (sub-)quintic NLS. Moreover, we identify the limiting tain wave-energy devices, as we will show. Some steps to- Gibbs measure on the real line and show its invariance. wards the mathematical and numerical modeling of this bore-soliton-splash will be discussed, including accurate Tadahiro Oh conservative discretizations of water waves, and a single- University of Edinburgh phase mixture theory for air- and water in an approximate [email protected] model for intermittent wave breaking.

Onno Bokhove MS19 School of Mathematics University of Leeds Two-soliton Solutions to a Focusing Cubic Half- [email protected] wave Equation on R In this talk we discuss work in progress regarding a nonlo- MS18 cal focusing cubic half-wave equation. Evolution problems A Strange Gas? Revisiting the Point Vortex Model with nonlocal dispersion naturally arise in physics as mod- to Understand the Condensation Process in Two- els for wave turbulence and gravitational collapse. The Dimensional and Quasi-Geostrophic Turbulence goal of the present work is to construct asymptotic global- in-time two-soliton solutions and to discuss their behaviour The statistical mechanics of a system of point vortices in- in terms of growth of high Sobolev norms. The talk is based teracting in a bounded two-dimensional domain is revis- on joint work with P. Gérard (Orsay, France), E. Lenzmann ited. It is shown that the familiar sinh-Poisson theory is (Basel, Switzerland), and P. Raphaël (Nice, France). incomplete, and describes the behaviour only for inverse temperatures below a critical value (β<βc), when the Oana Pocovnicu ‘gas’ of point vortices has ‘condensed’ generating a mean Institute for Advanced Studies, Princeton circulation. For β>βc a theory of fluctuations, recently [email protected] generalised by the authors, is shown to describe the ‘uncon- densed’ state. Behaviour near the critical value is investigated using statistical sampling and DNS. The implications MS19 for the behavior of unforced two-dimensional Navier-Stokes Going Beyond the Threshold: Scattering vs. Blow- turbulence, magnetized plasmas, and superfluids are dis- up Dichotomy in the Focusing NLS Equation cussed, and it is shown that the theory extends easily to related problem of quasi-geostrophic vorticity dynamics. We study the focusing nonlinear Schrodinger equation N 2 Gavin Esler in R ,intheL -supercritical regime with finite energy UCL and finite variance initial data and investigate solutions [email protected] above the energy (or mass-energy) threshold. We extend the known scattering versus blow-up dichotomy above the mass-energy threshold for finite variance solutions in the MS18 energy-subcritical and energy-critical regimes, obtaining Cloud-Edge Dynamics and Mysterious Holes in the scattering and blow-up criteria for solutions with arbitrary Sky large mass and energy. This is a joint work with T. Duy- ckaerts. A holepunch cloud is a curious atmospheric phenomenon where a disturbance in a thin cloud layer, as can be caused Svetlana Roudenko by an ascending aircraft, leaves behind a growing circular George Washington University hole of clear air. Observed since the dawn of aviation, only [email protected] very recently has the holepunch feature been simulated in a full-physics numerical weather model. Although the initiation process has been clearly attributed to ice crystal MS19 formation, we explain that the continued expansion of the Dispersive Blow-Up in Schrdinger Type Equations hole is a travelling front between two phases of moist air — unsaturated and weakly-stratified (clear) intruding into saturated and moist-neutral (cloudy). Furthermore our We review results from a recent joint paper together with fluid model, a non-hyperbolic conservation law system, il- J. Bona, J. C. Saut, and G. Ponce on the possibility of lustrates an unusual example of a travelling discontinuity finite-time, dispersive blow up for nonlinear equations of that satisfies the Rankine-Hugoniot conditions, yet exists Schroedinger type. This mathematical phenomena is one of despite the absence of underlying characteristics. the conceivable explanations for oceanic and optical rogue waves. We extend the existing results in the literature David Muraki in several aspects. In one direction, the theory is broad- [email protected] ened to include the Davey-Stewartson and Gross-Pitaevskii [email protected] equations. In another, dispersive blow up is shown to obtain for nonlinear Schroedinger equations in spatial dimensions larger than one and for more general power-law MS19 nonlinearities. As a by-product of our analysis, a sharp Invariant Gibbs Measures for NLS on the Real Line global smoothing estimate for the integral term appearing in Duhamels formula is obtained. We discuss invariant Gibbs measures for NLS on the real line. Previously, Bourgain ’00 considered this problem and Christof Sparber proved uniqueness of solutions to (sub-)cubic NLS with University of Illinois at Chicago 84 NW14 Abstracts

[email protected] [email protected]

MS20 MS20 Kerr Optical Frequency Combs: from Fundamental Modelocking Quantum Cascade Lasers Using Theory to Engineering Applications Quantum Coherent Saturable Absorption

We present the recent developments on the topic of Kerr Experimental eﬀorts to create short, stable mid-infrared optical frequency comb generation using whispering gallery pulses using quantum cascade lasers (QCLs) have not mode resonators. These combs have are expected to pro- been successful to date. Time-evolving, strong spatial vide optical signals with exceptional amplitude and phase hole-burning due to very fast gain recovery hinders mode- stability. We discuss recent theoretical developments which locking. We show that two-section quantum cascade lasers are compared with numerical simulations and experimental can be designed in which one of the sections acts as a measurements. The technological interest of these combs quantum coherent absorbing medium, so that spatial hole- for various microwave photonics applications is also re- burning is suppressed and ultra-short pulses on the order viewed. of 100 fs can be obtained.

Yanne Chembo Muhammad Talukder Optics department, FEMTO-ST Institute, Besancon, Department of Electrical and Electronic Engineering France Bangladesh University of Engineering and Technology [email protected] [email protected]

MS20 Curtis R. Menyuk University of Maryland Baltimore County From Microresonator Combs to Solitons [email protected] Microresonator based Kerr combs open the way for novel compact photonic devices. This promise is supported by predicted and recently experimentally demonstrated con- MS21 trolled transition from chaotic to phase locked combs asso- Stability and Computation of Interacting Nonlin- ciated with temporal solitons. We studied experimentally ear Waves and in systematic numerical simulation the correlation between mode structure and comb dynamics. We found that We consider solutions of nonlinear reaction diffusion equa- solitons may appear in presence of mode coupling induced tions composed of several localized waves which interact avoided crossings, only if affected modes are sufficiently far strongly or weakly. We propose an extension of the freez- from the pump frequency. ing method which generates multiple coordinate frames. In these frames single waves stabilize independently while Michael Gorodetsky still keeping their nonlinear interaction. Asymptotic sta- Faculty of Physics bility of the decomposition system is shown in one space di- M.V.Lomonosov Moscow State University mension and numerical experiments are provided for both [email protected] weakly and strongly interacting waves in dimensions one and higher.

MS20 Wolf-Juergen Beyn Pulse Energy Enhancement of Mode Locked Fiber Bielefeld University Laser by Cascading Nonlinear Polarization Rota- [email protected] tion

The periodic response curve of mode locked fiber laser with MS21 nonlinear polarization rotation limits the single pulse en- Stability of Fronts in Spatially Inhomogeneous ergy in the laser cavity by triggering multi-pulse lasing. We Wave Equations propose to engineer the effective response curve of the fiber cavity by cascading multiple stages of nonlinear polariza- Model equations describing waves in anisotropic media tion rotation schemes in fiber cavity which can enhance the or media with imperfections usually have inhomogeneous single pulse energy in the cavity by improving the thresh- terms. This talk considers the effects of (non-local) inho- old of multi-pulsing with the combination of the multiple mogeneities on fronts and solitary waves in nonlinear wave response curves. equations. Inhomogeneities break the translational symmetry and travelling waves are no longer natural solutions. Feng Li Instead, the travelling waves tend to interact with the inho- Dept. of Electronic and Information Engineering mogeneity and get trapped, reflected, or slowed down. The Hong Kong Polytechnic University underlying Hamiltonian structure of the wave equation al- [email protected] lows for a rich family of stationary front solutions and the energy densities inside the inhomogeneities provide s nat- J. Nathan Kutz ural parametrization for these solutions. In this talk, we University of Washington will show that changes of stability can only occur at criti- Dept of Applied Mathematics cal points of the length of the inhomogeneity as a function [email protected] of these energy densities and we give a necessary and sufficient criterion for the change of stability. We will illustrate P. K. Alex Wai the results with an example related to a Josephson junc- Hong Kong Polytechnic University tion system with a finite length inhomogeneity associated Dept. of Electronic and Information Engineering with variations in the Josephson tunneling critical current. NW14 Abstracts 85

Department of Applied Mathematics and Theoretical Physics Gianne Derks [email protected] University of Surrey Department of Mathematics [email protected] MS22

Christopher Knight Contagion Shocks in One Dimension Department of Mathematics University of Surrey We consider an agent-based model of emotional contagion [email protected] coupled with motion in one dimension that has recently been studied in the computer science community. The Arjen Doelman model involves movement with speed proportional to a Mathematisch Instituut “fear’ variable that undergoes a temporal consensus av- [email protected] eraging with other nearby agents. We study the effect of Riemann initial data for this problem, leading to shock dynamics that are studied both within the agent-based Hadi Susanto model as well as in a continuum limit. We examine the The University of Nottingham model under distinguished limits as the characteristic con- United Kingdom tagion interaction distance and the interaction timescale [email protected] both approach zero. Here, we observe a threshold for the interaction distance vs. interaction timescale that produces MS21 qualitatively different behavior for the system - in one case particle paths do not cross and there is a natural Eule- Spectral Analysis for Transition Front Solutions in rian limit involving nonlocal interactions and in the other Multidimensional Cahn-Hilliard Systems case particle paths can cross and one may consider only a We consider the spectrum associated with the linear oper- kinetic model in the continuum limit. ator obtained when a Cahn-Hilliard system on Rn is linearized about a planar transition front solution. In the Martin Short case of single Cahn-Hilliard equations on Rn,it’sknown Georgia Tech that under general physical conditions the leading eigen- [email protected] value moves into the negative real half plane at a rate |ξ|3, where ξ is the Fourier transform variable corresponding Andrea L. Bertozzi with components transverse to the wave. Moveover, it has UCLA Department of Mathematics recently been verified that for single equations this spectral [email protected] behavior implies nonlinear stability. In this talk, I’ll discuss recent results in which it is shown that this cubic rate Li Wang, Jesus Rosado law continues to hold for a broad range of multidimensional UCLA Mathematics Cahn-Hilliard systems. [email protected], [email protected] Peter Howard Department of Mathematics MS22 Texas A&M University [email protected] Agent-Based and Continuum Models for the For- mation of Stripes in Zebrafish

MS21 Zebrafish (Danio rerio), a small fish with black and yellow A Geometric Approach to Counting Eigenvalues: stripes, has the ability to regenerate its stripes in response The Evans Function in Some Cellular Transport to growth or artificial disturbance. We simulate past laser Models ablation experiments using discrete (agent-based) and continuum (nonlocal conservation law) models. Both models This talk will focus on some geometric techniques in Evans consider two cell types diffusing and reacting based on rules function computations. The aim is to show how such meth- of short range attraction and long range repulsion. Our ods can 1) facilitate spectral calculations, and 2) describe results suggest that the radius of interaction is a key de- dynamical properties via said spectral calculations. The terminant of stripe formation. application in mind is for systems of travelling fronts in cellular transport models, specifically those describing tu- Alexandria Volkening, Bjorn Sandstede mour invasion and wound healing. Brown University Robert Marangell alexandria [email protected], The University of Sydney bjorn [email protected] [email protected]

MS22 MS22 Co-Dimension One Self-Assembly Front Propagation in Bacterial Suspensions

Not Available at Time of Publication Not Available at Time of Publication

Ray E. Goldstein James von Brecht University of Cambridge University of California, Los Angeles 86 NW14 Abstracts

[email protected] type systems as well as various classifications of 2 + 1- dimensional differential-difference equations. The talk is based on joint work with E. Ferapontov, A. Moro, B. Huard MS23 and I. Roustemoglou. Geometric Wave Equations with Multi-component Solitons Vladimir Novikov Loughborough University, UK I will survey some recent work deriving integrable geomet- [email protected] ric wave equations that are generalizations of Schrodinger map equations and their KdV analogs, as well as a geometric map version of integrable Camassa-Holm type MS23 equations. Through a generalized Hasimoto transforma- Well Posedness and Breakdown for Cauchy Prob- tion, these geometric map equations are related to multi- lems of Integrable Equations component integrable systems which have soliton or peakon solutions. We will discuss well posedness and breakdown results for Stephen Anco some integrable equations that can be derived as Euler- Department of Mathematics Arnold equations. Brock University [email protected] Feride Tiglay Ohio State University, USA [email protected] MS23 TheAnalysisofaClassofIntegrableEvolution Equations MS24 A Spectral Theory of Linear Operators on a We shall consider a class of nonlinearevolution equa- Gelfand Triplet and Its Application to the Dynam- tions that are integrable and discuss their analytic prop- ics of Coupled Oscillators erties,including the existence of peakon traveling wave solutions, well-posedness, ill-posedness and the stability The Kuramoto model is a system of ordinary differential of their data-to-solution map. This class includes the equations for describing synchronization phenomena de- Camassa-Holm equation, the Degasperis-Procesiequation, fined as a coupled phase oscillators. In this talk, an infinite the Novikovequation,and the Fokas-Olver-Rosenau-Qiao dimensional Kuramoto model is considered. Kuramoto’s equation.The integrabilityof these equations has beenstud- conjecture on a bifurcation diagram of the system will be ied in a unified way by V. Novikov. proved with the aid of a new spectral theory of linear operators based on Gelfand triplets. Alex Himonas University of Notre Dame, USA Hayato Chiba [email protected] Institute of Mathematics for Industry Kyushu University MS23 [email protected] Integrable Equations in 3D: Deformations of Dis- persionless Limits MS24 Classification of integrable systems remains as a topic of Pacemakers in a Large Array of Oscillators with active research from the beginning of soliton theory. Nu- Nonlocal Coupling merous classification results are obtained in 1 + 1 dimensions by means of the symmetry approach. Although the We model pacemaker effects in a 1 dimensional array of os- symmetry approach is also applicable to 2 + 1-dimensional cillators with nonlocal coupling via an algebraically local- systems, one encounters additional difficulties due to the ized heterogeneity. We assume the oscillators obey simple appearance of nonlocal variables. There are several tech- phase dynamics and that the array is large enough so that niques to tackle the problem (e.g. the perturbative symme- it can be approximated by a continuous nonlocal evolution try approach). In the perturbative symmetry approach one equation. We concentrate on the case of heterogeneities starts with a linear equation having a degenerate disper- with negative average and show that steady solutions to the sion law and reconstructs the allowed nonlinearity. In this nonlocal problem exist. In particular, we show that these talk we present a novel perturbative approach to the clas- heterogeneities act as a wave source, sending out waves in sification problem. Based on the method hydrodynamic the far field. This effect is not possible in 3 dimensional reductions, we first classify integrable quasilinear systems systems, such as the complex Ginzburg-Landau equation, which may potentially occur as dispersionless limits of inte- where the wavenumber of weak sources decays at infinity. grable 2 + 1-dimensional soliton equations. Subsequently To obtain our results we use a series of isomorphisms to we construct dispersive deformations preserving integrabil- relate the nonlocal problem to the viscous eikonal equa- ity deforming the hyrdrodynamic reductions by dispersive tion. The linearization about the constant solution results deformations and requiring that all hydrodynamic reduc- in an operator, L, which is not Fredholm in regular Sobolev tions of the dispersionless limit will be inherited by the spaces. We show that when viewed in the setting of Kon- corresponding dispersive counterpart. The method also al- dratiev spaces the operator, L, is Fredholm. These spaces lows to effectively reconstruct Lax representations of the can be described as Sobolev spaces with algebraic weights deformed systems. We present various classification results that increase in degree with each derivative. obtained in the frame of the new approach, e.g. the classification of scalar 2+1-dimensional equations generaliz- Gabriela Jaramillo ing KP, BKP/CKP, the classification of Davey-Stewartson University of Minnesota NW14 Abstracts 87

[email protected] based on Riemann-Hilbert problems, for the Degasperis- Procesi and Ostrovsky-Vakhnenko equations on the line. The formulation of the initial value problem in terms of MS24 associated (3×3) matrix Riemann-Hilbert problems allows From Particle to Kinetic and Hydrodynamic Mod- us to get the principal term of the long time asymptotics of els of Flocking its solution, and also to find soliton solutions. Work with D. Shepelsky. In nature, one can find many species who’s combined interaction give rise to large scale coherent structures without Anne Boutet de Monvel any force or leader guiding the interaction. The typical Universite’ Paris 6 examples of such dynamics are flocking of birds, schools of France fish, or insect swarms. In this talk, we will consider some of [email protected] the models proposed to capture such interaction. We shall start at the level of particles and from there derive kinetic and hydrodynamic models. The main novelty of the talk MS25 is that all passages will be made using rigorous arguments. The Unified Transform and the Riemann-Hilbert From particles to kinetic models, convergence is measured Formalism in Wasserstein distance, while from kinetic to hydrodynamic models, convergence is measured using relative en- The Unified Transform provides a novel method for tropy. We shall also consider flocking in an incompressible analysing boundary value problems for linear and for in- fluid governed by the Navier-Stokes equations. (Collabo- tegrable nonlinear PDEs. The relationship of this method rators: Jose A. Carrillo, Young-Pil Choi, Antoine Mellet, with the Riemann-Hilbert formalism for both scalar and and Konstantina Trivisa) matrix problems will be elucidated.

Trygve K. Karper Athanassios S. Fokas Norwegian University of Science and Technology University of Cambridge [email protected] [email protected]

MS24 MS25 A Tale of Two Distributions: from Few to Many Long-Time Asymptotics for the Toda Shock Prob- Vortices In Quasi-Two-Dimensional Bose-Einstein lem Condensates Consider the doubly infinite Toda lattice with steplike ini- Motivated by the recent successes of particle models in cap- tial data and non-overlapping background spectra. We ap- turing the precession and interactions of vortex structures ply the method of nonlinear steepest descent to compute in quasi-two-dimensional Bose-Einstein condensates, we re- the leading term of the long-time asymptotics of the Toda visit the relevant systems of ordinary differential equations. shock problem in all principal regions of the space-time half We consider the number of vortices as a parameter and ex- plane under the assumption that no discrete spectrum is plore the prototypical configurations (“ground states’) that present. arise in the case of few or many vortices. In the case of few vortices, we modify the classical result of Havelock [Phil. Johanna Michor,GeraldTeschl Mag. 11, 617 (1931)] illustrating that vortex polygons in University of Wien (Austria) the form of a ring are unstable for N ≥ 7. Addition- [email protected], [email protected] ally, we reconcile this modification with the recent identification of symmetry breaking bifurcations for the cases Iryna Egorova of N =2,...,5. We then examine the opposite limit of ILTPE Kharkov large N and illustrate how a coarse-graining, continuum [email protected] approach enables the accurate identification of the radial distribution of vortices in that limit. MS25 Theodore Kolokolnikov A Coupling Problem for Entire Functions and Its Dept. of Mathematics and Statistics Application to the Long-Time Asymptotics of In- Dalhousie University tegrable Wave Equations [email protected] We propose a novel technique for analyzing the long- Panayotis Kevrekidis time asymptotics of integrable wave equations in the case Department of Mathematics and Statistics where the underlying isospectral problem has purely dis- University of Massachusetts crete spectrum. To this end we introduce a natural cou- [email protected] pling problem for entire functions which serves as a re- placement for the usual Riemann-Hilbert approach, which Ricardo Carretero does not work for these kind of problems. As a prototyp- SDSU ical example we investigate the long-time asymptotics of [email protected] the dispersionless Camassa-Holm equation improving the currently known results.

MS25 Gerald Teschl A Riemann-Hilbert Approach for the Degasperis- University of Wien (Austria) Procesi and Ostrovsky-Vakhnenko Equations [email protected]

I will present an inverse scattering transform approach, Jonathan Eckhardt 88 NW14 Abstracts

Cardiff University have exponentially blowing up and/or decaying solutions as [email protected]ff.ac.uk well. In these cases the concept of an energy propagation speed looses meaning, but direction of propagation does not. Equations to determine the direction of propagation MS26 can then be implemented by considering the flux of energy. Effective Boundary Conditions: An Application to In this first talk (out of two) we show how this can be the Atmosphere implemented to develop radiation boundary conditions for semi-open dispersive systems. We also show that there is In this talk we present an application of the theory in a connection between this concept and the branch cuts of the previous talk to the dry atmosphere. While the at- the dispersion relation in the complex plane. mosphere does not have a definite top, it can be modeled as finite because the buoyancy frequency has a jump at the Rodolfo R. Rosales tropopause. This, effectively, acts as a “leaky” lid on the Massachusetts Inst of Tech lower atmosphere. The leakage of gravity waves from the Department of Mathematics troposphere to the stratosphere is a significant physical ef- [email protected] fect that cannot be ignored — as happens when a rigid lid is added at the tropopause. In this talk we show how to Lyubov Chumakova develop effective boundary conditions at the tropopause, University of Edinburgh using the theory in the previous talk and transform meth- [email protected] ods. Esteban G. Tabak Lyubov Chumakova Courant Institute University of Edinburgh New York Universityb [email protected] [email protected]

Rodolfo R. Rosales Massachusetts Inst of Tech MS26 Department of Mathematics Diapycnal Mixing and Overturning Circulations [email protected] This talk explores a framework to quantify the large-scale Esteban G. Tabak effects of fluid mixing without resolving the associated Courant Institute small scale motion. The equations of motion for hydro- New York Universityb static flows adopt the form of a hyperbolic system of non- [email protected] linear equations, which typically yield breaking waves. In order to model the shock waves that ensue, one needs to involve integral conserved quantities, such as mass and mo- MS26 mentum. Yet in a system composed of layers that may Particle Trajectories Beneath Stokes’ Waves mix, first physical principles do not provide a set of conserved quantities large enough to completely determine the We study particle trajectories beneath a Stokes’ wave in flow. Our proposal is to replace the conventional conser- the following regimes: (a) in the presence of a current, (b) vations laws of each layer’s mass and momentum, invalid in the presence of a submerged structure, and (c) in the after shocks form, by others, such as energy, in a way that presence of constant vorticity, where the particle phase- provides a natural description of the mixing process. This space and related pressure distribution are described in closure is then applied to modeling overturning circula- detail. tions, such as the Hadley cells of tropical convection.

Paul A. Milewski Esteban G. Tabak Dept. of Mathematical Sciences Courant Institute University of Bath New York Universityb [email protected] [email protected]

Andre Nachbin MS27 IMPA,Rio,Brazil [email protected] Solitary Wave Dynamics in Plasmonic Binary Ar- rays

Roberto Ribeiro-Junior Novel photonic devices provide the possibility of being IMPA, Brazil more efficient, smaller and richer in their functionality. At [email protected] the same time they are an excellent platform in the ongo- ing general study of nonlinear light propagation dynamics. Our work represents efforts in the study of photonic binary MS26 systems that are discrete in nature. Results will be pre- Effective Boundary Conditions for Semi-Open Dis- sented on the existence and properties of localized modes persive Systems and on the natural extension of the model to a continuum approximation leading to systems of coupled PDEs. this is In the classical linear dispersive wave theory it is shown a joint effort with Prof. C. De Angelis, Drs. A. Auditore i(kx−ωt) that sinusoidal waves (i.e., ∝ e ) carry energy with and M. Conforti from U. Brescia in Italy the group speed cg = dω/dk. This concept is limited to the case where both the frequency ω(k) and the wavenumber Alejandro Aceves k must be real. On the other hand, semi-open dispersive Southern Methodist University systems allow more than just sinusoidal solutions: they can [email protected] NW14 Abstracts 89

Costantino De Angelis [email protected] University of Brescia, Brescia, Italy [email protected] MS28 Aldo Auditore University of Brescia Azimuthal Dissipative Structures in Whispering- [email protected] Gallery Mode Resonators

Matteo Conforti Diﬀerent regimes of Kerr optical frequency combs can be Universita di Brescia, Italy generated in ultra-high Q resonators pumped by a contin- [email protected] uous wave laser. We use a 1-dimension Lugiato-Lefever equation to theoretically describe their dynamics. In particular, we show that these combs correspond to diﬀerent MS27 spatio-temporal structures such as Turing patterns, bright and dark solitons. We discuss the parameter ranges for the Stochasticity and Coherent Structures in Ultra- generation of such combs, and relate them to our experi- Long Mode-Locked Lasers mental results obtained in crystalline resonators.

Ultra-long mode-locked lasers are known to be strongly Aurelien Coillet influenced by nonlinear interactions in long cavities that Optics department results in noise-like stochastic pulses. Here, by using an ad- FEMTO-ST Institute, Besancon, France vanced technique of real-time measurements of both tem- [email protected] poral and spatial (over round-trips) intensity evolution, we reveal an existence of wide range of generation regimes. Different kinds of coherent structures including dark and grey solitons and rogue-like bright coherent structures are MS28 observed as well as interaction between them are revealed. Injection Locking of Frequency Combs in Mode Locked Lasers Dmitry Churkin Aston University [email protected] Injection locking of mode locked laser to an external pulse train entails the entrainment of two frequencies, the pulse to pulse timing and overall phase shift, synchronizing the source and target frequency combs. For a given injec- MS27 tion strength, we demonstrate the domain of stable syn- Movement of Gap Solitons across Deep Gratings in chronization in the plane of comb spacing and offset mis- the Periodic NLS matches. The shape of the synchronization domain depends on the pulse shape and chirp, and it is bounded by a Moving pulses in nonlinear periodic media can be applied, line of saddle-node and higher co-dimension bifurcations. e.g., as information carriers in photonic crystals. We consider the 1D periodic nonlinear Schroedinger equation as Omri Gat a prototype model. For asymptotically small contrasts of Hebrew University of Jerusalem, Department of Physics the periodic structure the equation has moving pulses ap- Jeruslam, ISRAEL proximated by gap solitons of the coupled mode equations [email protected] [Aceves, Wabnitz, Phys. Lett. A, 1989]. These gap solitons ∈ − have a wide range of velocities v ( cg,cg ). We show that David Kielpinski analogous moving pulses exist also in large contrast struc- Australian Attosecond Science Facility tures described by perturbed finite band potentials due to Griffith University, Brisbane, Australia the opening of spectral gaps from a transversal intersection [email protected] of band functions.

Tomas Dohnal MS28 TU Dortmund Department of Mathematics Dynamics of Frequency Comb Generation and De- [email protected] sign of Planar Microring Resonators

We study the generation of optical frequency combs from MS27 microresonators and show that a truncated dynamical Periodic and Relative Periodic Solutions in a Mul- model can capture the essential nonlinear dynamics of both tiple Waveguide System stable and chaotic patterns. We also present an analytical approach for obtaining linear and nonlinear design param- We describe the propagation of light in multi-waveguide eters of planar microring resonators. Closed form approxi- systems modeled by the nonlinear Schrodinger and discrete mations for the eigenmode/eigenfrequency problem are de- nonlinear Schrodinger equations using normal form meth- rived for resonators possessing a large radius to width ratio ods and other ideas from Hamiltonian systems. We report and examples are presented for silicon microrings. on new families of relative periodic orbits that coexist with the familiar nonlinear normal modes in these systems. Tobias Hansson, Daniele Modotto, Stefan Wabnitz University of Brescia Roy Goodman [email protected], New Jersey Institute of Technology [email protected], ste- 90 NW14 Abstracts

[email protected] of asymptotic regimes, including near the onset of hydrodynamic instability and in the inviscid limit. Our analysis relies on Evans function calculations, geometric singular MS28 perturbation theory, Whitham theory, and spectral per- Tunable Photonic Oscillators turbation theory.

Limit cycle oscillators are used to model a broad range of Mat Johnson periodic nonlinear phenomena. Using the optically injected University of Kansas semiconductor oscillator as key Paradigm architecture we [email protected] will demonstrate that at specific islands in the optical de- tuning and injection level map, the Period one limit fre- Pascal Noble quency is simultaneously insensitive to multiple perturba- University of Toulouse tion sources. In our system these include the temperature [email protected] fluctuations experienced by the master and slave lasers as well as fluctuations in the bias current applied to the slave Miguel Rodrigues laser. Tuning of the oscillation frequency then depends University of Lyon only on the injected optical field amplitude. Experimen- [email protected] tal measurements are in good quantitative agreement with numerical modeling and analysis based on reduced generalized Adler phase equations. These special operating re- Kevin Zumbrun gions should prove valuable for developing ultrastable non- Indiana University linear oscillators, such as sharp linewidth, frequency tun- USA able photonic microwave oscillators. Finally the concept of [email protected] an Isochron originally developed in mathematical biology will be reviewed and placed on context for efficient design of Blake Barker stable frequency sources via systems of coupled limit cycles Indiana University oscillators. Time permitting a few new theoretical results [email protected] on injecting monothematic signals into quantum cascade laser gain media will be outlined. MS29 Vassilios Kovanis Stability of Traveling Waves on Vortex Filaments Bradley Department of Electrical and Computer Engineering We develop a framework for studying the linear stability Virginia Tech solutions of the Vortex Filament Equation (VFE), based [email protected] on the correspondence between the VFE and the NLS provided by the Hasimoto map. This framework is applied to vortex filaments associated with periodic NLS solutions. MS29 We focus on a class of torus knots VFE solutions and show Transverse Dynamics of Periodic Gravity-Capillary that they are stable only in the unknotted case. We also Water Waves establish the spectral stability of soliton solutions.

We study the transverse dynamics of two-dimensional Stephane Lafortune, Annalisa M. Calini gravity-capillary periodic water waves in the case of crit- College of Charleston ical surface tension. We show that, as solutions of the Department of Mathematics full water-wave equations, the periodic traveling waves [email protected], [email protected] are linearly unstable under three-dimensional perturbations which are periodic in the direction transverse to the direction of propagation. Then we discuss the nonlinear Scotty Keith bifurcation problem near these transversely unstable two- College of Charleston dimensional periodic waves. We show that a one-parameter [email protected] family of three-dimensional doubly periodic waves is generated in a dimension-breaking bifurcation. MS29 Mariana Haragus Orbital Stability of Periodic Waves and Black Soli- Universite de Franche-Comte tons in the Cubic Defocusing NLS Equation [email protected] Standing periodic waves of the cubic defocusing NLS equa- Erik Wahlen tion are considered. Using tools from integrability of the Lund University cubic NLS equation, these waves have been shown to be Sweden linearly stable by computing the Floquet–Bloch spectrum [email protected] explicitly. We combine the first four conserved quantities of the cubic NLS equation to give a direct proof that the waves are also orbitally stable with respect to the subhar- 2 MS29 monic periodic perturbations in the Hper topology. We also Stability of Viscous Roll-Waves develop a new proof of the orbital stability of the black solitons in H 2(R). This is a joint work with Thierry Gallay Roll-waves are a well observed hydrodynamic instability (University of Grenoble). occurring in inclined thin film flow, mathematically described as periodic traveling wave solutions of the St. Dmitry Pelinovsky Venant system. In this talk, I will discuss recent progress McMaster University concerning the stability of viscous roll-waves in a variety Department of Mathematics NW14 Abstracts 91

[email protected] [email protected]

MS30 MS30 Frustrated Nematic Order in Spherical Geometries Shear-induced Instability of a Smectic A Liquid Crystal Not Available at Time of Publication

A smectic A liquid crystal is usually described as be- Vincenzo Vitelli ing composed of rod-like molecules arranged in reasonably Associate Professor,Soft Condensed Matter Theory Group well-defined equidistance layers. Recent models describing Instituut-Lorentz, Universiteit Leiden the microscopic organisation of smectic A allow the aver- [email protected] age molecular orientation to evolve away from the layer normal direction, see for example I.W. Stewart, Contin. Mech. Thermodyn. 18:343-360 (2007). In this presenta- MS31 tion we will summarise this continuum theory and apply it Thermal Counterflow in a Periodic Channel with to investigate the undulation instability of the smectic A Solid Boundaries phase in the presence of an imposed linear shear flow. We perform numerical simulations of finite temperature Jonathan Dawes quantum turbulence produced through thermal counter- 4 University of Bath flow in superfluid He, using the vortex filament model. [email protected] We investigate the effects of solid boundaries along one of the cartesian directions, whilst assuming a laminar nor- Iain W. Stewart mal fluid with a Poiseuille profile, whilst varying the tem- University of Strathclyde perature and the normal fluid velocity. We analyze the [email protected] distribution of quantized vortices as a function of the wall- normal direction and find that the vortex line density is concentrated close to the solid boundaries. Furthermore, we find that the vortex line density profile is independent MS30 of the counterflow velocity, and offer an explanation as to Imry-Ma-Larkin Clusters in Random Nematics why the peak of vortex line density tends towards the solid boundaries with increasing mutual friction. Finally we of- Not Available at Time of Publication fer evidence that upon the transition to a turbulent normal fluid, there is a dramatic increase in the homogeneity of the tangle, which could be used as an indirect measure of the Samo Kralj transition to turbulence in the normal fluid component in Faculty of Natural Sciences and Mathematics experiments. University of Maribor [email protected] Andrew Baggaley University of Glasgow [email protected] MS30 Harnessing Topological Defects via Liquid Crystal Jason Laurie Microfluidics Weizmann Institute of Science [email protected] Harnessing interactions between flow, and molecular order- ing in complex anisotropic fluids like liquid crystals pro- MS31 vides a novel pathway to engineer flow and topology tem- plates. The interactions emerge due the anisotropic cou- Vortex Reconnections and Implications for Inverse pling between flow and molecular orientation. In microflu- Energy Transfer in Turbulent Superfluid Helium idic parlance, the nature of surface anchoring, channel as- In superfluid helium vorticity takes the form of discrete pect ratios, and strength of the flow provide a rich assort- vortex filaments of fixed circulation and atomic thickness. ment of accessible experimental parameters. The emerging We present numerical evidence of three-dimensional inverse field of Topological Microfluidics promises physics and po- energy transfer from small length scales to large length tential applications beyond the conventional isotropic case. scales in turbulence generated by a flow of vortex rings. We argue that the effect arises from the anisotropy of the flow, which favors vortex reconnections of vortex loops of the Anupam Sengupta same polarity, and that it has been indirectly observed in Max Planck Institute for Dynamics and Self-Organization the laboratory. The effect opens questions about analogies Göttingen, Germany with related processes in ordinary turbulence. [email protected] Carlo F. Barenghi Stephan Herminghaus Applied Mathematics Group Max Planck Institute for Dynamics and Self-Organization Newcastle University [email protected] [email protected]

Christian Bahr Andrew Baggaley Max Planck Institute for Dynamics and Self-Organization University of Glasgow G¨ottingen, Germany [email protected] 92 NW14 Abstracts

Yuri Sergeev larger length scales. In a second experiment, small vortex Newcastle University rings with large mean free path were frequently generated [email protected] when much larger vortex rings are injected into vortex tan- gles of known density. This only occurred at temperatures below 0.7 K where small wavelength Kelvin waves occur MS31 due to the lack of damping provided by mutual friction. Breathers on Quantized Superfluid Vortices All our observations can be explained using simple models based on vortex reconnections and self-reconnections and We consider the propagation of breathers along quantised provide new insight into the quantum regime of superfluid superfluid vortices. Using the correspondence between the turbulence. local induction approximation (LIA) and the nonlinear Schrödinger equation, we identify a set of initial condi- Paul Walmsley, P.M. Tompsett, D.E. Zmeev, A.I. Golov tions corresponding to breather solutions of vortex motion University of Manchester governed by the LIA. These initial conditions, which give [email protected], rise to a long-wavelength modulational instability, result [email protected], in the emergence of large amplitude perturbations that are [email protected], an- localised in both space and time. The emergent structures [email protected] on the vortex filament are analogous to loop solitons. Al- though the breather solutions we study are exact solutions of the LIA equations, we demonstrate through full numer- MS32 ical simulations that their key emergent attributes carry A Spectral Approach to Dbar Problems over to vortex dynamics governed by the Biot-Savart law and to quantized vortices described by the Gross-Pitaevskii We present a fully spectral approach to the numerical solu- equation. In these cases, the breather excitations either de- tion of dbar problems for smooth rapidly decreasing initial cayorinsomecasescanleadtoself-reconnections,amech- data. As an example we will discuss the Davey-Stewartson anism that can play an important role within the cross-over equations. range of scales in superfluid turbulence. Moreover, the observation of breather solutions on vortices in a field model Christian Klein suggests that these solutions are expected to arise in a wide Institut de Mathématiques de Bourgogne range of other physical contexts from classical vortices to 9 avenue Alain Savary, BP 47870 21078 DIJON Cedex cosmological strings. [email protected]

Hayder Salman University of East Anglia MS32 [email protected] The Semiclassical Sine-Gordon and Rational Solu- tions of Painlev´e-II

MS31 We formulate and study a class of initial-value problem Local and Nonlocal Effects in Quantum Turbulence for the sine-Gordon equation in the semiclassical limit. The initial data parametrizes a curve in the phase por- Quantum turbulence consists of an apparently random tan- trait of the simple pendulum, and near points where the gle of quantised vortex lines which move under the influ- curve crosses the separatrix a double-scaling limit reveals a ence of each others’ velocity field. We ask the natural universal wave pattern constructed of superluminal kinks question of the relative importance of local and nonlocal located in the space-time along the real graphs of all of the effects on the turbulent velocity at a point, which is de- rational solutions of the inhomogeneous Painlevé-II equa- termined by the instantaneous distribution of vorticity via tion. The kinks collide at the real poles, and there the solu- the Biot-Savart law. The answer is related to the existence tion is locally described in terms of certain double-kink ex- of (or the lack of) coherent vortex structures in the tur- act solutions of sine-Gordon. This study naturally leads to bulence tangle: the more random the vortex lines are, the the question of the large-degree asymptotics of the rational more cancellation of far-field effects there will be. We shall solutions of Painlevé-II themselves. In the time remaining present numerical results which address and answer this we will describe recent results in this direction, including question. a formula for the boundary of the pole-free region, strong asymptotics valid also near poles, a weak limit formula, Lucy Sherwin and planar and linear densities of complex and real poles. Newcastle University This is joint work with Robert Buckingham (Cincinnati). [email protected] Peter D. Miller University of Michigan, Ann Arbor MS31 [email protected] Reconnections of Vortex Rings in Superfluid He- lium MS32 Quantized vortex rings in superfluid helium provide an Numerical Inverse Scattering for the BenjaminOno ideal model system for investigating the interactions and Equation collisions occurring within a cloud of unidirectional vortex rings with almost identical radii. We present evidence for We investigate the numerical implementation of the for- small vortex rings emitted upon vortex reconnections in ward and inverse scattering transform for the Ben- superfluid 4He at very low temperatures. In one exper- jaminOno equation. Unlike previous numerical inverse iment, pairs of charged vortex rings collided resulting in scattering problems, the transform involves a nonlocal creation of both smaller and larger vortex rings, highlight- spectral problem, requiring new numerical techniques to ing how energy can be redistributed to both smaller and be developed. Joint work with Peter Miller and Tom Trog- NW14 Abstracts 93

don. [email protected]

Sheehan Olver The University of Sydney MS33 [email protected] Pure Gravity Generalised Solitary Waves

Steep waves travelling at a constant velocity at the surface MS32 of an incompressible and inviscid fluid of infinite depth are Oscillatory Integrals and Integrable Systems considered. The flow is assumed to be irrotational and gravity is taken into account. The effect of surface tension High oscillation is an important conservative phenomenon is neglected. It is shown that in addition to the classical in the theory of integrable systems. For integrable PDEs, periodic Stokes waves, there are also non periodic waves. two competing forces are in balance. First, the flow regu- These waves are generalised solitary waves in the sense that larizes and prevents the formation of discontinuities. Sec- they are characterised by a periodic train of waves in the ond, an infinite number of quantities are conserved. In far field. Numerical evidence for the existence of new types the necessary absence of dissipation, oscillations must be ofperiodicwavesisalsopresented. responsible for regularization. These oscillations are seen to be a direct consequence of oscillatory Fourier symbols. Jean-Marc Vanden-Broeck Furthermore, properties of these symbols are used for de- University College London tailed asymptotic and numerical studies of many problems. [email protected] In this talk, I will discuss recent asymptotic and numerical progress on integrable PDEs that exploits the structure of oscillatory integrals. MS34 On the Existence of Breathers in Periodic Media: Thomas Trogdon An Approach Via Inverse Spectral Theory Courant Institute NYU Breathers are considered a rare phenomenon for constant [email protected] coefficient nonlinear wave equations. Recently, a nonlinear wave equation with spatially periodic step potentials MS33 has been found to support breathers by using a combination of spatial dynamics, center manifold reduction and bi- The Effect of Rotation on Shoaling Internal Soli- furcation theory. Via inverse spectral theory for weighted tary Waves Sturm-Liouville equations, we characterize a surprisingly large class of potentials that allow breathers. The research In the weakly nonlinear long wave regime, internal oceanic is motivated by the quest of using photonic crystals as op- solitary waves are often modeled by the Korteweg-de Vries tical storage. equation, which is well-known to support solitary waves. However, when the effect of background rotation is taken Martina Chirilus-Bruckner into account, the resulting relevant nonlinear wave equa- University of Sydney tion is the Ostrovsky equation, which does not support an [email protected] exact solitary wave solution. Instead an initial solitary-like disturbance decays into radiating oscillatory waves with the long-time outcome being the generation of a nonlinear C. E. Wayne wave packet, whose carrier wavenumber is determined by Boston University an extremum in the group velocity. When variable bottom [email protected] topography is also taken into account, although this process may still take place, some new features emerge such MS34 as the formation of secondary undular bores. Reflection of an Incoming NLS Soliton by an At- Roger Grimshaw tractive Delta Potential University of Loughborough, UK [email protected] We consider the dynamics of a single soliton launched from spatial infinity with velocity v toward a delta-potential defect, evolving according to the 1D cubic NLS equation 1 2 | |2 − ∈ R MS33 i∂tu + 2 ∂xu + u u qδu =0,whereq is the coupling Breaking of Interfacial Waves Induced by Back- constant and δ is the delta function at x =0.Thecaseof ground Rotation |q|∼|v|1 was previously studied and quantum splitting of the incoming soliton was observed and quantified. The amplitudes of vertical modes in weakly-nonlinear, In the case of |q|1, the soliton was shown to remain weakly rotating, horizontally-extended stratified flows are intact and obey classical motion. In this paper, we study governed by the reduced Ostrovsky (RO) equation where the case of intermediate interaction strength |q|∼1, and dispersion arises from a non-local integral term. This equa- slow incoming velocity |v|1. We show that this is nei- tion is integrable provided a certain curvature constraint ther a classical nor quantum regime – a small eigenstate is satisfied. We demonstrate, through theoretical analysis emerges on top of the delta potential with phase structure and numerical simulations, that when this curvature con- incompatible with the incoming soliton, resulting in reflec- straint is not satisfied at the initial time, then wave break- tion of the incoming soliton. Interestingly, this occurs even ing inevitably occurs. Similar results are obtained for the for attractive potentials q ∼−1. modified (cubic) RO. Justin Holmer Ted Johnson Department of Mathematics, Brown University, UCL Box 1917, 151 Thayer Street, Providence, RI 02912, USA London, UK [email protected] 94 NW14 Abstracts

Quanhui Lin (b) the non-diagonal analogue of the Lamb shift for clas- Brown University sical photons will be presented. Supporting experimental [email protected] observations of these novel optical phenomena confirm the unique characteristics of the evanescent vertical coupling in an integrated photonic platform. MS34 Analysis of Dispersion Managed Solitary Waves Mher Ghulinyan Fondazione Bruno Kessler We study pulses in optical glass fiber cables which have [email protected] a periodically varying dispersion along the cable. This so- called dispersion management leads to interesting effects in those cables. Mathematically, these pulses are described by MS35 a non-local version of the non-linear Schrödinger equation Terabit Communications Using Optical Frequency which was derived first by Gabitov and Turitsyn (1996), Combs and and later by Ablowitz and Biondini (1998). The nonlocal nature of the non-linearity makes the rigorous analysis Terabit/s optical interconnects rely on advanced hard. In the talk, we will focus on existence of solitary wavelength-division multiplexing (WDM) schemes, pulses in these cables and show that they are very well- where information is transmitted by a multitude opti- localized. cal channels of different wavelengths. In this context, frequency combs are a particularly attractive option Dirk Hundertmark for generating the associated optical carriers. In this Karlsruhe Institute of Technology paper, we give an overview on our recent demonstrations [email protected] of terabit/s communications using optical frequency combs. Our experiments exploit different comb generation schemes relying, e.g., on electro-optic modulation and on MS34 Kerr-nonlinear interaction in high-Q microcavities. Ground States of a Nonlinear Curl-Curl Problem Christian Koos In this talk I will report on recent joint work with Thomas Karlsruhe Institute of Technology Bartsch, Tomas Dohnal and Michael Plum. We are inter- [email protected] ested in ground states for the nonlinear curl-curl equation

−1 3 3 3 ∇×∇×U +V (x)U =Γ(x)|U|p U in IR ,U: IR → IR . MS35 Noise of Frequency Combs Based on Yb Fiber A basic requirement is to find scenarios, where 0 does not Lasers with Self-Similar Pulse Evolution belong to the spectrum of the operator Measurements of the free running carrier envelope offset L = ∇×∇×+V (x). frequency linewidth of a similariton based optical frequency comb show that the frequency noise of the laser is indepen- Under suitable assumptions on V,Γ we construct ground dent of net cavity dispersion. The experimental results are states both for the defocusing case (Γ ≤ 0) and the focusing consistent with simulations of the quantum-limited tim- case (Γ ≥ 0). The main tools are variational methods and ing noise. The resultant possibilities for low-noise, high- the use of symmetries. performance fiber sources will be discussed.

Thomas Bartsch William Renninger University of Giessen Yale [email protected] [email protected]

Tomas Dohnal MS35 KIT, Department of Mathematics Stability of Modelocked Lasers With Slow Sat- [email protected] urable Absorbers

Michael Plum Lasers that are modelocked with semiconductor saturable Karlsruhe University (KIT) absorbers are of great practical interest because they can be Germany more environmentally stable than other modelocked lasers. [email protected] However, the slow saturable absorber opens a gain window that can lead to instability in the wake of modelocked soli- Wolfgang Reichel ton pulse. In this paper, we characterize the spectrum of KIT, Department of Mathematics this instability and its evolution. [email protected] Shaokang Wang UMBC MS35 [email protected] Novel Phenomena in WGM Resonators with Ver- tical Evanescent Coupling to Bus Waveguides Curtis R. Menyuk University of Maryland Baltimore County We will report on novel optical phenomena which manifest [email protected] in a vertically coupled Whispering-gallery resonator and a dielectric waveguide. In particular, theoretical predictions of (a) the oscillatory vertical coupling, characterized by MS36 multiple critical coupling conditions at diﬀerent gaps, and Stability of Solitary Waves in Nonlinear Dirac NW14 Abstracts 95

Equation Department of Mathematics [email protected], [email protected] We study the linear instability of solitary wave solutions φ(x)e−iωt to the nonlinear Dirac equation. That is, we linearize the equation at a solitary wave and examine the MS36 presence of eigenvalues with positive real part. We de- Stability of Traveling Standing Waves for the Klein- scribe recent analytic and numerical results. In particu- Gordon Equation lar, we show that small amplitude solitary waves in the Soler model are linearly unstable in three spatial dimen- For the Klein-Gordon equation, the stability of the travel- sions, but are generically stable in one dimension. We use ing standing waves is considered and the exact ranges of the limiting absorption principle and the Carleman-type the wave speeds and the frequencies needed for stability estimates of Berthier–Georgescu to locate the part of the are derived. This is done in both the whole line case and continuous spectrum of the linearized system, from which the periodic case. point eigenvalues can bifurcate (leading to linear instability). We also show that the border of the linear instability Milena Stanislavova region is described not only by the Vakhitov-Kolokolov con- University of Kansas, Lawrence dition Q(ω) = 0, obtained in the NLS context, but also Department of Mathematics by the energy vanishing condition, E(ω)=0.HereE, Q [email protected] are the energy and the charge of a solitary wave with frequency ω. Some of the results are obtained in collaboration with Nabile Boussaid, Université de Franche-Comté, David MS37 Stuart, University of Cambridge, Stephen Gustafson, Uni- Two Component Condensates: Spin Orbit Cou- versity of British Columbia, Gregory Berkolaiko and Alim pling and Defects Sukhtayev, Texas A&M University. Two component condensates emerge from the physics of Andrew Comech artificial gauge potentials and can describe a single isotope Department of Mathematics in two different hyperfine spin states, two different isotopes Texas A&M University of the same atom or isotopes of two different atoms. I will [email protected] describe both numerical and theoretical results concerning spin orbit coupled two component condensates. I will explain how some segregation cases can be analyzed through MS36 a Gamma limit leading to a phase separation problem of Approximation of Traveling Waves on Finite Inter- de Giorgi type. In the coexistence cases, we try to obtain vals an asymptotic expansion of the energy taking into account the various types of defects. The first term in the expan- Usually, traveling waves are not a meaningfull object for sion is related to the Thomas Fermi limit of the profile and partial differential equations posed on finite intervals. But relies on singular perturbation techniques, while the next by using the method of freezing, proposed by Beyn and ones require a more precise analysis of the defects: vortex Thümmler in 2004, one is able to calculate a suitable co- sheets, vortices or skyrmions moving frame on the fly so that in this new system a stable traveling wave of the original system becomes a stable Amandine Aftalion standing wave in the new system. More precisely, in the University of Paris VI and CNRS method of freezing one rewrites the original problem as [email protected] a partial differential algebraic equation that includes the velocity of a suitable co-moving frame as an additional unknown. For this new system it does make sense to restrict MS37 to a finite interval. To obtain a well-posed problem, one Convergence to an Equilibrium for Wave Maps on then has to impose artificial boundary conditions. We con- aCurvedManifold sider this procedure in the case of semilinear hyperbolic problems and show how the new system on a finite inter- We consider equivariant wave maps from a wormhole-type val approximates the original problem and can be used in spacetime into the three-sphere. This is a toy-model for numerical computations. gaining insight into the dissipation-by-dispersion phenomena, in particular the soliton resolution conjecture. Using Jens Rottmann-Matthes the hyperboloidal formulation of the initial value problem, University of Bielefeld we show that, for a given topological degree of the map, [email protected] all solutions starting from smooth initial data converge to the unique stationary solution as time goes to infinity. The asymptotics of this relaxation process is described in detail. MS36 Concatenated Traveling Waves Piotr Bizon Institute of Physics We consider concatenated traveling wave solutions of Jagiellonian University reaction-diffusion systems. These are solutions that look [email protected] like a sequence of traveling waves with increasing velocity, with the right state of each wave equal to the left state of the next. I will present an approach to the stability theory MS37 of such solutions that does not rely on treating them as a A Gradient Flow Approach to the Keller-Segel Sys- sum of traveling waves. It is based instead on exponential tems dichotomies and Laplace transform. Not Available at Time of Publication Stephen Schecter, Xiao-Biao Lin North Carolina State University Adrien Blanchet 96 NW14 Abstracts

Ceramade, France UFRJ, Rio de Janeiro [email protected] [email protected]

Jean-Claude Saut MS37 Paris-Sud Vortices and Vortex Lattices in the Ginzburg- France Landau and Bcs Theories [email protected] The Ginzburg-Landau theory gives a fairly good macroscopic description of superconductors and the effect of mag- MS38 netic fields on them. This theory is based on the Ginzburg- Local Well-posedness for a Class of Nonlocal Evo- Landau equations, a pair of coupled nonlinear equations for lution Equations of Whitham Type the macroscopic wave function (order parameter) and magnetic potential. (These equations appear also in abelean For a class of pseudodifferential evolution equations of the Higgs model with unknowns called the Higgs and gauge form fields and, in general, serve as a paradigm for the descrip- ut +(n(u)+Lu)x =0, tion of a large class of phenomena in physics, material science and biology.) In this talk I will review recent rigorous we prove local well-posedness for initial data in the Sobolev s results on the key solutions of these equations - the mag- space H , s>3/2. Here L is a linear Fourier multi- netic vortices and magnetic vortex lattices, their existence plier with a real, even and bounded symbol m,andn is ∈ s R and stability. I will also discuss relation of the Ginzburg- a real measurable function with n Hloc( ), s>3/2. Landau theory to the microscopic BCS theories and the The proof, which combines Kato’s approach to quasilinear description of vortices and vortex lattices in the latter. equations with recent results for Nemytskii operators on general function spaces, applies equally well to the Cauchy Israel Michael Sigal problem on the line, and to the initial-value problem with Department of Mathematics periodic boundary conditions. University of Toronto [email protected] Long Pei Department of Mathematical Sciences NTNU, Trondheim, Norway MS38 [email protected] Stability of Periodic Traveling Waves for Nonlocal Dispersive PDE MS38 We discuss recent work on the stability of periodic travel- Three-Dimensional Solitary Water Waves with ing wave solutions to nonlinear PDE of KdV-type, allow- Weak Surface Tension ing for nonlocal phase speeds. Specific examples include fractional KdV with phase speed |ξ|α for some α>1/2, I will discuss a variational existence theory for three- and Whitham’s water wave equation with the exact (unidimensional fully localised solitary water waves with weak directional) phase speed from the Euler equations. We surface tension. The water is modelled as a perfect fluid of demonstrate the existence of an explicit stability index, finite depth, undergoing irrotational flow. The surface ten- computable entirely from the linear phase speed, which sion is assumed to be weak in the sense that 0 <β<1/3, determines the modulational stability of small amplitude where β is the Bond number. A fully localised solitary waves. wave is a travelling wave which decays to the undisturbed state of the water in every horizontal direction. Such so- Mat Johnson lutions are constructed by minimising a certain nonlocal University of Kansas functional on its natural constraint. A key ingredient is a [email protected] variational reduction method, which reduces the problem to a perturbation of the Davey-Stewartson equation. Vera Hur University of Illinois at Urbana-Champaign Boris Buffoni [email protected] EPFL boris.buffoni@epfl.ch

MS38 Mark D. Groves Dispersive Perturbations of Burgers and Hyper- Universit¨at des Saarlandes bolic Equations [email protected]

The aim of this talk is to show how a weakly dispersive Erik Wahlen perturbation of the inviscid Burgers equation improve (en- Lund University large) the space of resolution of the local Cauchy problem. Sweden We will also review several problems arising from weak dis- [email protected] persive perturbations of nonlinear hyperbolic equations or systems. MS39 Felipe Linares Anatomy Induced Drift of Spiral Waves in Human Institute of Pure and Applied Mathematics Atrium BRAZIL [email protected] In biophysically and anatomically realistic model of human atrium, we demonstrate functional eﬀects of atrial anatom- Didier Pilod ical structures on spiral waves spontaneous drift. Spiral NW14 Abstracts 97

waves drift from thicker to thinner regions, along ridge-like study wave propagation and termination by light. structures of pectinate muscules (PM) and cristae termi- nalis, anchor to PM-atrial wall junctions or to some lo- Emilia Entcheva cations with no obvious anatomical features. The insight Stony Brook University can be used to improve low-voltage de-fibrillation proto- [email protected] cols, and predict atrial arrhythmia evolution given a pa- tient specific atrial anatomy. MS39 Irina Biktasheva Spatiotemporal Dynamics of Obstacle-Induced Spi- University of Liverpool ral Wave Initiation [email protected] Inexcitable obstacles in the heart can act as substrates for the initiation of spiral waves. We introduced obsta- Sanjay R. Kharche cles of varying sizes (1-8 mm) into cardiac monolayers with University of Exeter side pacemakers, demonstrating that smaller obstacles (<5 [email protected] mm) are correlated with spiral wave generation. Using a computational model of the experiments, we showed that Gunnar Seemann the location of the obstacle sensitively governed the spa- Karlsruhe Institute of Technology tiotemporal dynamics of the initiated spiral waves. Germany [email protected] Thomas D. Quail McGill University Henggui Zhang [email protected] University of Manchester UK [email protected] MS40 Large-degree Asymptotics of Generalised Hermite Vadim N. Biktashev Polynomials and Poles of Rational Painlevé-IV College of Engineering, Mathematics and Physical Functions Sciences The Painlevé-IV equation admits a family of rational so- University of Exeter lutions, indexed by two integers, that can be expressed [email protected] in terms of certain generalised Hermite polynomials. In the large-degree limit the zeros of these polynomials form MS39 remarkable patterns in the complex plane resembling rect- angles with arbitrary aspect ratios depending on how the Exact Coherent Structures and Dynamics of Car- indexing integers grow. Using Riemann-Hilbert analysis we diac Tissue compute the large-degree asymptotic behavior of the generalised Hermite polynomials and analytically determine High dimensional spatiotemporal chaos underpins the dy- the boundary of the elliptic (i.e. zero/pole) region for the namics associated with life threatening cardiac arrhyth- associated rational Painlevé-IV functions. We also obtain mias. Our research seeks to suppress these arrhythmias asymptotic behavior in terms of Painlevé-I functions at the and restore normal heart rhythms by using low-energy corners of the elliptic region. transfers between unstable solutions underlying the chaotic dynamics of cardiac tissue. We describe the search for un- Robert Buckingham stable solutions of nonlinear reaction diffusion models of University of Cincinnati cardiac tissue and discuss recent efforts to understand lo- [email protected] cal symmetries in terms of pairwise interactions between spiral cores. MS40 Greg A. Byrne, Chris Marcotte Inverse Scattering Transform for the Defocusing Georgia Institute of Technology Nonlinear Schrödinger Equation with Asymmetric [email protected], Nonzero Boundary Conditions [email protected] We develop the inverse scattering transform (IST) for Roman Grigoriev the defocusing nonlinear Schrödinger (NLS) equation with Georgia Institute of Technology asymmetric nonzero boundary conditions q(x, 0) → q± as Center for Nonlinear Science & School of Physics x →±∞,where|q−| = |q+|. The direct problem is formu- [email protected] lated without using a uniformization variable, taking into account the square root discontinuities of the eigenvalues of the asymptotic scattering problems. The inverse problem MS39 is formulated in terms of a discontinuous Riemann-Hilbert Optogenetic Control of Cardiac Electrical Activity: problem (RHP). Experimental Insights Gino Biondini Optogenetics, in the broader sense, refers to the use of State University of New York at Buffalo genetically-encoded molecules serving as optical actuators Department of Mathematics or optical sensors for the active interrogation and imaging biondini@buffalo.edu of biological processes and systems with high specificity and high spatiotemporal resolution. We will report the Emily Fagerstrom application of these genetic tools to cardiac tissue and will State University of New York at Buffalo demonstrate all-optical remote actuation and sensing to emilyrf@buffalo.edu 98 NW14 Abstracts

Barbara Prinari Moist 2-Layer Model University of Colorado, Colorado Springs University of Salento, Lecce (Italy) [email protected] The coupling between moist convective processes and large- scale circulation is at the core of tropical-extratropical atmospheric interactions, which, in turn, are known to MS40 strongly modulate rainfall at low latitudes. For example, within the tropical Eastern Pacific, observations suggest On the Scattering Matrix for AKNS Systems with that poleward transport of moisture and energy is par- Matrix-Valued Potentials ticularly dependent upon intrusion of large-scale transient Rossby waves from the extratropics towards the equator, We discuss spectral properties and analytic properties of particularly during northern winter. Theories for the re- the scattering matrix associated with matrix AKNS sys- lationship between low latitude moist convection and ex- tems under suitable assumptions on the potentials. Special tratropical waves have emphasized the role of upper level attention will be paid to spectral singularities and embed- potential vorticity (PV) intrusions associated with Rossby ded eigenvalues. We present results on the analytic behav- wave breaking. In this talk, two major drawbacks of this ior of the scattering matrix near such points and study how approach are discussed: the limitations of PV inversion at the spectrum changes if the system is perturbed. low latitudes and the neglect of latent heating feedbacks. To address these limitations, an alternative reduced model Martin Klaus is proposed that is based on a barotropic-baroclinic shal- Virginia Tech low water system coupled to a water vapor equation. The [email protected] water vapor equation is coupled to the dynamical variables through a simple parametrization for precipitation, MS40 allowing for a two-way feedback between precipitation and waves. This model is used to investigate modulations of Inverse Scattering Transform for the Focusing NLS equatorial rainfall, in particular, the relative roles of zonal Equation with Fully Asymmetric Boundary Condi- moisture transport by tropical waves and meridional mois- tions ture advection by extratropical waves. By integrating the model using initial conditions consistent with the northern We present the inverse scattering transform (IST) for the 2 winter basic state, we demonstrate the key role of advec- focusing nonlinear Schrödinger equation: iqt = qxx +2|q| q, ∼ tion of moisture towards the equator due to extratropical with non-zero boundary conditions q(x, t) ql/r(t)= Rossby waves. The transient response and tropical rainfall iθl/r(t) →∓∞ Al/re as x in the fully asymmetric case. The sensitivity to precipitation parametrization and extratrop- direct problem is shown to be well-posed for NLS solu- ical wave amplitude and scales are discussed, as well as the − ∈ 1,1 R∓ tions q(x, t) such that q(x, t) ql/r(t) L ( )with model predictions overall agreement with observations and respect to x for all t ≥ 0, for which analyticity properties more complex models. of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations, and as a Riemann- Juliana F. Dias Hilbert problem on a single sheet of the scattering vari- NOAA Earth System Research Laboratory 2 2 [email protected] ables λl/r = k + Al/r,wherek is the usual complex scattering parameter in the IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with the same amplitude as x →±∞, MS41 here both reflection and transmission coefficients have a nontrivial time dependence. Pilot Wave Dynamics in a Rotating Frame: Orbital Quantization and Multimodal Statistics Francesco Demontis Dipartimento di Matematica e Informatica We present the results of a theoretical investigation of Universitá di Cagliari, Italy droplets walking on a rotating vibrating fluid bath. The [email protected] droplet’s trajectory is described in terms of an integro- differential equation that incorporates the influence of its Barbara Prinari propulsive wave force. Predictions for the dependence of University of Colorado, Colorado Springs the orbital radius on the bath’s rotation rate compare fa- University of Salento, Lecce (Italy) vorably with experimental data and capture the progres- [email protected] sion from continuous to quantized orbits as the vibrational acceleration is increased. The orbital quantization is ratio- Cornelis Van der Mee nalized by assessing the stability of the orbital solutions, Dipartimento di Matematica e Informatica and may be understood as resulting directly from the dy- Universitá di Cagliari, Italy namic constraint imposed on the drop by its monochro- [email protected] matic guiding wave. The stability analysis also predicts the existence of wobbling orbital states reported in recent Federica Vitale experiments, and the virtual absence of stable orbits in the Dipartimento di Matematica e Fisica ”Ennio De Giorgi” limit of large vibrational forcing. The droplet’s trajectory Universita’ del Salento, Lecce (Italy) is numerically simulated in this limit, revealing a chaotic [email protected] dynamics whose statistical properties reflect the persistent influence of the unstable orbital solutions.

MS41 Anand Oza Tropical-Extratropical Wave Interactions in a Math. Dept., MIT. NW14 Abstracts 99

[email protected] whose shape can be quite explicitly decsribed.

Riccardo Adami MS41 University of Milano [email protected] Nonlinear Wave Interactions in Global Nonhydro- static Models Enrico Serra, Paolo Tilli In previous works we have analyzed the nonlinear wave in- Politecnico di Torino teractions in a shallow global non-hydrostatic model. Here [email protected], [email protected] this analysis has been extended for the case where the full Coriolis force terms are considered. The full Corilis MS42 force terms include those proportional to 2ΩcosΦw and 2ΩcosΦu in the zonal and vertical momentum equations, Micro/macroscopic Models for Fluid Flow in Net- respectively, which are disregarded in hydrostatic models works of Elastic Tubes for energetics consistence. For simplicity we have considered the mid-latitude beta-plane approximation. We describe a Boussinesq-type system for modeling the dynamics of pressure-flow in arterial networks, considered Carlos Raupp as a 1d spatial network of elastic tubes. Numerical solutions of the system of PDEs are compared with simplified IAG/University of Sao Paulo microscopic models based on particle-tracking arguments, Brazil and are used to study flow optimization task, depending [email protected] on the geometry and size of the network. Physiologically realistic control mechanisms are tested in the context of Andre Teruya these simplified models. IAG/University of Sao Paulo [email protected] Radu C. Cascaval University of Colorado at Colorado Springs [email protected] MS41 The Influence of Fast Waves and Fluctuations on MS42 the Evolution Three Slow Limits of the Boussinesq Equations Solution Formulas for the Linearized KdV Equa- tion on Graphs We present results from a study of the impact of the non- slow (typically fast) components of a rotating, stratified We investigated linearized KdV equation on metric star flow on its slow dynamics. In order to understand how the graphs with at least three semi-infinite bonds. Under some flow approaches and interacts with the slow dynamics we assumption on the coefficients of the vertex conditions decompose the full solution, where u is a vector of all the uniqueness of solution are proven. It is known that theory α of potentials for linearized KdV equation are well studied. unknowns, as u = u + uαwhere α represents the Ro → 0,Fr→ 0 or the simultaneous limit of both (QG for quasi- Using this theory we reduced the considered problem to the α α equivalent system of linear algebraic equations. It is shown geostrophy), with Pαu = u ,Pαuα =0,andwherePαu represents the projection of the full solution onto the null that the obtained system of equations is uniquely solvable space of the fast operator. Numerical simulations indicate under conditions of the uniqueness theorem. By using of that for the geometry considered (triply periodic) and the derived formulas for the solutions we proved the existence type of forcing applied, the fast waves act as a conduit, theorems in the classes of Schwartz functions decreasing at moving energy onto or off of the slow ’manifold’. infinity and in Sobolev classes. Zarif A. Sobirov Beth Wingate Tashkent Financial Institute University of Exeter [email protected] [email protected] Hannes Uecker Jared Whitehead Department of Mathematics Brigham Young University University of Oldenburg [email protected] [email protected] Terry Haut Maksad Akhmedov Los Alamos National Laboratory Tashkent Financial Institute [email protected] [email protected]

MS42 MS42 Minimizing the NLS Energy on General Graphs The NLS on Fat Graphs and the Metric Graph Limit We treat the problem of finding the ground state for a system driven by the Nonlinear Schroedinger Equation and The description of the propagation of waves through net- evolving on a ramified structure (graph). We prove two works is often based on metric graphs, consisting of 1D results: first, if the graph is made of at least two half lines bonds joined at vertices. However, real waveguides are not meeting in an arbitrary compact graph, then there is non 1D, and networks are not constructed of 1D bonds con- minimizer. Second, if the graph is a star graph with two in- nected at 0D vertices, but rather the bonds and the vertex finite edges and a finite one, then there exists a minimizer, regions have a certain thickness . These structures are 100 NW14 Abstracts

often refered as “fat graphs’. Here we compare numeri- general geometric structures connected with the bifurca- cal solutions of Nonlinear Schrödinger equations (NLS) on tion dynamics in the vicinity of PT phase transition points. fat graphs with solutions of the formally derived NLS on The presentation extends earlier results obtained in collab- the metric graph, in the limit → 0. In particular, we oration with Panayotis Kevrekidis, Boris Malomed and Kai show that an interesting transmission formula for the met- Li in (J. Phys. A 45, 444021 (2012)). ric graph limit lifts to the fat graph with small errors. Uwe Guenther Hannes Uecker Helmholtz-Center Dresden-Rossendorf, Germany Department of Mathematics [email protected] University of Oldenburg [email protected] MS43 Zarif A. Sobirov Tashkent Financial Institute Pt-Symmetric Synthetic Materials [email protected] In this talk we will introduce a new class of optical structures that rely on the recently developed notion of parity- MS43 time (PT) symmetry. These PT synthetic materials can ex- One AC and Two PT Nonlinear Schrödinger Dimers hibit a number of unique characteristics provided that the optical processes of gain, loss, and index guiding are judi- We compare nonlinear dynamics and phase space geome- ciously designed. In particular, we examine the scattering PT try of the standard -symmetric nonlinear Schrödinger properties of PT-symmetric optical Fabry-Perot cavities. dimer to those of its two modifications. One observation is Multiple phase transitions and unidirectional invisibility PT that the addition of the nonlinear coupling softens the - for both polarizations are systematically examined. symmetry breaking transition in the standard dimer: stable periodic and quasiperiodic states persist for any value Konstantinos Makris of the gain-loss coefficient. Another result is that the in- School of Engineering, Optics Laboratory troduction of active coupling (AC) changes the nonlinearity Ecole Polytechnique Federale de Lausanne from the promotor to a suppressor of blowup regimes. [email protected] Igor Barashenkov University of Cape Town Stefan Rotter [email protected] Institute for Theoretical Physics Vienna University of Technology, Austria [email protected] MS43 Coupled PT-Symmetric Systems Demetrios Christodoulides College of Optics and Photonics/CREOL Inspired by recent experiments on a PT-symmetric system University of Central Florida of coupled optical whispering galleries, we study a model [email protected] of two coupled oscillators with loss and gain. There are two PT transitions: PT symmetry is broken for small and very large coupling, but unbroken for intermediate cou- MS44 pling. The classical and the quantized systems transition at the same couplings. An oscillating charged particle back- Evans Function Analysis of Viscous Multidimen- reacting with its own electromagnetic field is also a PT- sional Shock Layers symmetric system. We give an overview of our recent work on the stability of Carl M. Bender multidimensional shock layers in multi-dimensional com- Washington University in St. Louis pressible fluid flow, in particular the Navier-Stokes equa- [email protected] tions. Our work involves a careful combination of asymptotic ODE and Evans function methods. We discuss the substantial challenges observed when computing the Evans MS43 function in Eulerian coordinates at high frequencies. We Linear and Nonlinear Properties of PT-symmetric show how to overcome these problems by transforming into Optical Systems canonical and somewhat general coordinates. The results are surprising. Not Available at Time of Publication

Demetrios Christodoulides Jeffrey Humpherys College of Optics and Photonics/CREOL Brigham Young University University of Central Florida jeff[email protected] [email protected] Greg Lyng Department of Mathematics MS43 University of Wyoming New Aspects of Nonlinear PT-Symmetric Plaque- [email protected] ttes Kevin Zumbrun Nonlinear Schrödinger equations for PT-symmetric plaque- Indiana University tte type setups are used as toy models for investigations of USA NW14 Abstracts 101

[email protected] National Chung Cheng University (Taiwan) [email protected]

MS44 Computing the Reﬁned Stability Condition for MS45 Shock Waves Regularity and Stability of Landau-Lifshitz Flows

The classical (inviscid) stability analysis of shock waves is The Landau-Lifshitz equations of ferromagnetism are ge- based on the Lopatinski˘ı determinant, Δ—a function of ometric PDE for maps into the 2-sphere, which include frequencies whose zeros determine the stability of the un- as special cases the harmonic map heat flow and the derlying shock. A careful analysis of Δ shows that in some Schroedinger flow. While flows of topological degree 1 cases the stable and unstable regions of parameter space which form singularities are known, we present results are separated by an open set of parameters. Zumbrun and showing that symmetric flows of higher degree, even with Serre [Indiana Univ. Math. J., 48 (1999) 937–992] have large energy, remain regular, and flow toward harmonic shown that, by taking account of viscous effects not present maps. in the definition of Δ, it is possible to determine the precise location in the open, neutral set of parameter space at Stephen Gustafson which stability is lost. In particular, they show that the Department of Mathematics transition to instability under suitably localized perturba- University of British Columbia tions is determined by an “effective viscosity’ coefficient [email protected] given in terms of the second derivative of the associated Evans function, the viscous analogue of Δ. Here, we describe the practical computation of this coefficient. MS45 Dynamics of Bec of Fermion Pairs in the Low Den- Gregory Lyng sity Limit of Bcs Theory Department of Mathematics University of Wyoming We show that the time-evolution of the wave function de- [email protected] scribing the macroscopic variations of the pair density in BCS theory can be approximated, in the dilute limit, by a time-dependent Gross-Pitaevskii equation. For the static MS44 case we can as well include direct and exchange energies in Linear and Spectral Stability of Solitary Gravity order to obtain a similar result. Waves Christian Hainzl The talk will discuss linear and spectral stability of two- Mathematisches Institut dimensional solitary waves on water of finite depth. It is University of Tuebingen assumed that the fluid is bounded by a free surface and a [email protected] rigid horizontal bottom. The solitary wave is moving under the gravity (no surface tension). It was known that the fully nonlinear Euler equations have a solitary-wave MS46 solution. In this talk, we will show that the linear opera- Dissipation and Dispersion in Shallow Water tor arising from linearizing the Euler equations around the solitary-wave solution has no spectrum points lying on the In a series of physical experiments of surface water waves, right half of the complex plane, which implies the spectral dissipation and dispersion appear to play important roles in stability of the solitary waves. Moreover, under the norm the evolution of waves of depression. This suggests that the with exponential weight, the solitary waves are linearly sta- (conservative, nondispersive) shallow-water wave equations ble. (This is a joint work with R. Pego) miss some important features of the evolution. In this talk we compare measurements from a series of shallow-water Shu-Ming Sun experiments with numerical simulations of dispersive and Virginia Tech dissipative shallow-water wave models. Department of Mathematics [email protected] John Carter Seattle University Mathematics Department MS44 [email protected] The Evolution of Traveling Waves in a Simple Isothermal Chemical System Harvey Segur We study a reaction-diffusion system for an isothermal Dept. of Applied Mathematics chemical reaction scheme governed by a quadratic autocat- University of Colorado alytic step and a decay step. Note that this system does [email protected] not enjoy the comparison principle. Previous numerical studies and experimental evidences demonstrate that if the David L. George autocatalyst is introduced locally into this autocatalytic re- U.S. Geological Survey action system where the reactant initially distributes uni- [email protected] formly in the whole space, then a pair of waves will be generated and will propagate outwards from the initial reaction zone. We will analytically study this phenomena. MS46 Stability of Solitary Waves for the Doubly Disper- sive Nonlinear Wave Equation Je-Chiang Tsai Department of Mathematics The general class of nonlocal nonlinear wave equations 102 NW14 Abstracts

utt − Luxx = B(g(u))xx involves two sources of dispersion, [email protected] characterized by two pseudo-differential operators L and B. Erbay, Erbay and Erkip (2014) established thresholds for global existence versus blow-up in the case of power- MS46 type nonlinearities. In a recent study of Erbay, Erbay and A Numerical Method for Computing Traveling Erkip (submitted), the existence and stability/instability Waves of Nonlinear Dispersive Equations properties of solitary waves for the above class of nonlocal nonlinear wave equations have been established us- We present an automated Python-powered solver for non- ing the concentration-compactness method of Lions. In linear nonlocal wave equations such as the Whitham equa- this talk, we consider the double-dispersion equation with tion, KdV, Benjamin-Ono and alike. The solver is based power-type nonlinearities, which is a member of the above on the cosine collocation method. A user supplies the pro- class, and discuss recent analytical and numerical results gramme with a nonlinear flux, linear dispersion operator on the stability interval of solitary waves. and some parametres. Given that the input satisfies re- Husnu Ata Erbay quired conditions, a traveling-wave solution is computed Ozyegin University, Istanbul and tested with an evolution integrator. Dynamic simula- Turkey tion results are provided for the Whitham and square-root [email protected] Whitham equations. Daulet Moldabayev Saadet Erbay Department of Mathematics Ozyegin University University of Bergen, Norway [email protected] [email protected] Albert Erkip Sabanci University MS47 [email protected] Measuring Scroll Filament Rigidity

MS46 Spherical glass beads can pin three-dimensional scroll Pseudodifferential Operators on a Half-line and the waves in the Belousov-Zhabotinsky reaction. Three pin- Riemann-Hilbert Problem ning sites cause initially circular scroll filaments to approach equilateral triangles. The resulting stationary Consider the initial-boundary value problem on a half-line shapes show convex deviations that increase with anchor for the evolution equation radii. We describe the shapes of the filament segments analytically using an asymptotic theory, that takes into ∂tu + F (D) u = f(x, t),t>0,x>0, account filament tension and rigidity. Fitting the theoretical shapes to experimental data, with knowledge of fila- m1 m2 m3 k βk αk ment tension, allows experimental measurement of filament where F (D)= ck∂x + bk∂x + ak |∂x| is a linear 1 1 1 rigidity. combination of the usual derivative ∂x, the Caputo deriva- β α tive ∂x and the module-fractional derivative |∂x| on half- Vadim N. Biktashev α line, αk,βk ∈ R, k is integer. Here |∂x| is a module- College of Engineering, Mathematics and Physical fractional derivative Sciences University of Exeter α 1−α+[α] [α]+1 |∂x| u = R ∂x u, [email protected] where Elias Nakouzi, Zulma Jimenez +∞ − Rα sign(x y)u(y) Florida State University u = 1−α dy 0 |x − y| [email protected] ¡[email protected], is the modified Riesz potential. The fractional Caputo [email protected] ¡[email protected] β derivative ∂x is defined as Oliver Steinbock β 1−β+[β] [β]+1 Department of Chemistry and Biochemistry ∂x u = I ∂x u, Florida State University where [email protected] x Iβ u(y) u = 1−β dy 0 (x − y) MS47 is Riemann-Liouville fractional derivative. We prove the existence and the uniquness of the solution. We propose a Effective Dynamics of Twisted and Curved Scroll new method to construct solutions to the initial-boundary Waves Using Virtual Filaments value problem. These results could be applied to the study of a wide class of nonlinear nonlocal equations on half-line Scroll waves are three-dimensional excitation patterns that by using the techniques of nonlinear analysis. We give some rotate around a central filament curve. Starting from the examples: nonlinear Schrodinger equation, Benjamin-Ono reaction-diffusion equations, I will show that the detailed, equation, intermediate long wave equation, and nonlinear instantaneous equations of filament motion heavily depend Klein-Gordon equation. on the scroll wave’s rotation phase. However, by filtering out epicycle motion we obtain simpler dynamics for a pre- Elena Kaikina viously unseen companion of the classical filament, which Instituto de Matematicas we call the ‘virtual filament’. [See H. Dierckx and H. Ver- National Autonomous University of Mexico schelde, Phys. Rev. E 88 0629072013 or arXiv:1311.2782.] NW14 Abstracts 103

with the results from numerical reaction-diﬀusion simulations. Hans Dierckx Department of Physics and Astronomy Vladimir Zykov Ghent University Max Planck Institute for Dynamics and Self-Organization [email protected] [email protected]

Eberhard Bodenschatz MS47 Max Planck Institute for Dynamics & Self-Organization Dynamics of Spiral Cores: the Eﬀect of Boundaries eberhard.bodenschatz.ds.mpg.de and Heterogeneities

While reaction-diffusion PDEs used to model cardiac tis- MS48 sue respect continuous translational and rotational sym- Surface Waves in Graded Index Meta-materials metries, the cardiac tissue and the spatially discretized numerical models based on these PDEs do not. The dynamics We have studied the properties of electromagnetic surface are modified due to both boundary conditions and hetero- waves in graded index meta-materials. It is known that geneities associated, for instance, with the cellular struc- in transient layers, where index of refraction changes sign, ture of the tissue or the computational mesh. This talk will these surface waves are associated with field enhancement discuss the effect of such heterogeneities on the dynamics which leads to the anomalous absorption. We have stud- of stable and unstable spiral wave solutions. ied the nonlinear parametric processes due to such field enhancement in case of non collinear incident fields. Roman Grigoriev Georgia Institute of Technology Ildar R. Gabitov Center for Nonlinear Science & School of Physics Department of Mathematics, University of Arizona [email protected] [email protected]

Christopher Marcotte Andrei Maimistov Georgia Institute of Technology Moscow Engineering Physics Institute [email protected] [email protected]

Greg Byrne MS48 George Mason University [email protected] Models for Laser-Driven Generation of THz Radi- ation

MS47 We discuss the generation of THz radiation by two-color ultrashort optical pulses interacting in a nonlinear medium. Spiral Pinballs Photocurrents driven by tunnel ionization in gases are shown to be responsible for the emission of broadband THz Macroscopic systems exhibiting both wave-like and pulses with field amplitude above the GV/m level. Asso- particle-like properties have received much attention in re- ciated spectra, which can be evaluated semi-analytically, cent years. One case is excitable media, for which spiral result from constructive interferences of discrete plasma waves have core regions that behave like particles. Under radiation bursts occurring at the field maxima. Technolog- resonant stimulation, spiral cores drift in straight lines and ical solutions for optimizing THz emission and links with may reflect from boundaries in interesting ways. In this plasma physics are addressed. talk, the ‘pinball’ dynamics of these reflections are studied, via numerical simulations and asymptotic theory. Pedro Gonzalez de Alaiza Departement de Physique Theorique et Appliquee Jacob Langham CEA, DAM, DIF, 91297 Arpajon, France Mathematics Institute [email protected] University of Warwick [email protected] Luc Berge Departement de Physique Theorique et Appliquee Dwight Barkley CEA/DAM Ile de France University of Warwick [email protected] Mathematics Institute [email protected] MS48 Channel Capacity and Nonlinear Fourier Trans- MS47 form in Coherent Fibre-Optic Communications Kinematic Theory of Spirals and Wave Segments I will overview recent progress in studies of the Shannon Spiral waves and wave segments represent typical exam- capacity of nonlinear fibre channels and new opportunities ples of self-organized patterns observed in excitable me- offered by resurrection of the idea of eigenvalue commu- dia. We demonstrate that at least two dimensionless medi- nications in the new context of non-soliton coherent com- ums parameters should be measured to predict the spatio- munications. Application of the well-known nonlinear ana- temporal characteristics of these patterns. The findings logue of the Fourier transform to digital signal processing should be applicable to a wide class of media due to the in coherent optical transmission leads to radically novel generality of the free boundary approach used. The predic- approaches to coding, transmission and processing of in- tions of this approach are in good quantitative agreement formation using specific nonlinear properties of the optical 104 NW14 Abstracts

ﬁbre communication channel. University of Bremen [email protected] Sergei Turitsyn Aston University, Birmingham Frits Veerman UK Oxford University [email protected] [email protected]

MS48 MS49 Nondecaying Solutions of KdV Equation Coherent Structures in a Population Model for Mussel-Algae Interaction We develop the algebraic procedure that makes possible to construct the exact bounded but not vanishing at infinity We consider a model that describes the interaction of mus- solutions of integrable nonlinear systems. Multizone solu- sel biomass with algae in the water layer overlying the mus- tions of the KdV equation are used as a test bed for the sel bed. The model consists of a system of two coupled method. pdes where both the diffusion and the advection matrices in are singular. We use the Geometric Singular Perturba- Vladimir E. Zakharov tion Theory to capture nonlinear mechanisms of pattern Department of Mathematics, University of Arizona, and wave formation in this system. Tucson P.N. Lebedev Physical Institute RAS, Moscow, Russia Anna Ghazaryan [email protected] Department of Mathematics Miami University Dmitry Zakharov [email protected] Courant Institute of Mathematical Sciences New York University Vahagn Manukian [email protected] Miami University Hamilton [email protected] MS49 Modulation Equations for Interacting Localized MS49 Structures Pattern Formation in a Class of Landau-Lifschitz- Gilbert-Slonczewski Equations for Spintronic De- Employing modulation equations for the approximate devices scription of nonlinear waves is a widespread technique which allows to reduce the analysis of complicated systems The Landau-Lifshitz-Gilbert-Slonczewski equation de- to universal models such as the KdV or the NLS equation. scribes magnetization dynamics in the presence of an ap- Recently, the interaction of localized structures was ana- plied ?eld and a spin polarized current. In the case of lyzed by deriving a set of modulation equations giving a axial symmetry and with focus on one space dimension, separate description of internal and interaction dynamics. we investigate the emergence of space- time patterns in We briefly review the procedure for PDEs and display the the form of coherent structures, whose pro?les asymptote full analysis for a polyatomic FPU system. to wavetrains. In particular, we give a complete existence and stability analysis of wavetrains and prove existence of Martina Chirilus-Bruckner various coherent structures, including soliton- and domain University of Sydney wall-type solutions. Decisive for the solution structure is [email protected] the size of anisotropy compared with the di?erence of ?eld intensity and current intensity normalized by the damping. MS49 This is joint work with Christof Melcher (RWTH). Spatially Periodic Patterns, Busse Balloons, and Jens Rademacher the Hopf Dance University of Bremen [email protected] The Busse balloon is the region in (wavenumber, parameter)-space for which stable spatially periodic patterns exist. In the class of singularly perturbed reaction- MS50 diffusion systems, the Busse balloon typically has a non- Dimension-Breaking Phenomena for Solitary smooth edge where it borders on a family of homoclinic Gravity-Capillary Water Waves patterns. Near this edge, the boundary of the Busse balloon has a fine-structure of intertwining curves that repre- The water-wave problem has small-amplitude line solitary- sent various kinds of Hopf bifucations – as will be explained wave solutions which to leading order are described by the anddiscussedinthistalk. Korteweg-deVries equation (for strong surface tension) or nonlinear Schrödinger equation (for weak surface tension). Arjen Doelman We present an existence theory for three-dimensional pe- Mathematisch Instituut riodically modulated solitary-wave solutions to the water- [email protected] wave problem which have a solitary-wave profile in the direction of propagation and are periodic in the trans- Björn De Rijk verse direction. They emanate from the line solitary waves Leiden University in a dimension-breaking bifurcation and are described to [email protected] leading order by the Kadomtsev-Petviashvili equation (for strong surface tension) or Davey-Stewartson equation (for Jens Rademacher weak surface tension). The term dimension-breaking phe- NW14 Abstracts 105

nomenon describes the spontaneous emergence of a spa- Waves tially inhomogeneous solution of a partial differential equation from a solution which is homogeneous in one or more I will present results on the computation and stability of spatial dimensions. periodic surface gravity-capillary waves that are in a near- resonant regime. In the zero amplitude limit, the param- Mark D. Groves eters defining these solutions almost satisfy the resonance Universität des Saarlandes condition that leads to Wilton ripples. This manifests it- [email protected] self as a small divisor problem in the Stokes expansion for these solutions. We compute such solutions and investigate Shu-Ming Sun their stability using Hills method. Virginia Tech Department of Mathematics Olga Trichtchenko [email protected] Department of Applied Mathematics University of Washington Erik Wahlen [email protected] Lund University Sweden Bernard Deconinck [email protected] University of Washington [email protected]

MS50 Lasers and Ripples MS51 CPT-symmetric Spin-orbit-coupled Condensate Three dimensional capillary-gravity waves behave very much like strong laser pulses propagating through a Kerr We introduce a meanfield model of a spin-orbit-coupled medium. We describe the reasons for this similarity and atomic system accounting for pumping and removal of how the evolution in the two cases differ past the collapse atoms from the two states, i.e. giving origin to an open time. spin-orbit-coupled Bose–Einstein condensate. The system possess charge-parity-time (CPT) symmetry and obeys Paul A. Milewski such properties as control of the stability by the external Dept. of Mathematical Sciences trap, singular CPT symmetry phase breaking, the exis- University of Bath tence of stable nonlinear modes with nonzero currents in [email protected] repulsive and attractive condensates, multiple re-entering the CPT-symmetric phase, splitting of nonlinear modes Zhan Wang into sets of solitons (wave-packets) moving with different University College London velocities and in different directions in the condensate re- [email protected] leased from the trap.

Yaroslav Kartashov MS50 Institut de Ci`encies Fotniques The Inﬂuence of Surface Tension Upon Trapped Barcelona, Spain Waves and Hydraulic Falls [email protected]

Steady two-dimensional free-surface flows past submerged Vladimir V. Konotop obstructions on the bottom of a channel are presented. Department of Physics Both the effects of gravity and surface tension are consid- University of Lisbon ered. Critical flow solutions with subcritical flow upstream [email protected] and supercritical flow downstream are sought using fully nonlinear boundary integral equation techniques based on Dmitry Zezyulin the Cauchy integral formula. When a second submerged Centro de Fsica Teórica e Computacional obstruction is included further upstream in the flow config- Universidade de Lisboa uration trapped wave solutions are found for small values [email protected] of the Bond number, for some values of the Froude number. Other types of trapped waves are found for stronger tension when the second obstruction is placed downstream MS51 of the hydraulic fall generated by the first obstacle. Asymmetric Transport in Non-Linear Systems with Parity-Time Symmetry Emilian I. Parau University of East Anglia We will introduce the notion of Parity-Time (PT) symme- [email protected] try and present its implications in classical wave propagation. Using integrated optics and electronics as playfields Charlotte Page we show how one can construct new circuitry designs that School of Computing Sciences allow for asymmetric transport due to interplay of the novel University of East Anglia, Norwich, UK properties of PT-symmetry and non-linearity or gyrotropic [email protected] elements.

Tsampikos Kottos MS50 Wesleyan University Stability of Near-Resonant Gravity-Capillary Dept. of Physics 106 NW14 Abstracts

[email protected] and consider its properties.

Claire Gilson MS51 School of Mathematics & Statistics, University of Glasgow, UK Nonlinear Dynamics in PT-Symmetric Lattices [email protected] We consider nonlinear dynamics in a finite parity-time- symmetric chain of the discrete nonlinear Schrodinger MS52 (dNLS) type. For arbitrary values of the gain and loss parameter, we prove that the solutions of the dNLS equa- A Four-component Camassa-Holm Type Hierarchy tion do not blow up in a finite time but nevertheless, there We consider a 3×3 spectral problem which generates four- exist trajectories starting with large initial data that grow component CH type systems. The bi-Hamiltonian struc- exponentially fast for larger times with a rate that is rigor- ture and infinitely many conserved quantities are con- ously identified. In the range of the gain and loss parame- structed for the associated hierarchy. Some possible re- ter, where the zero equilibrium state is neutrally stable, we ductions are also studied. prove that the trajectories starting with small initial data remain bounded for all times. We also discover a new inte- Qingping Liu grable configuration of a PT-symmetric dimer and classify Department of Mathematics, existence and stability of large-amplitude stationary PT- China University of Mining and Technoly (Beijing) symmetric nonlinear states. The results to be reported in [email protected] this talk were obtained in collaboration with P. Kevrekidis, V. Konotop, and D. Zezyulin. MS52 Dmitry Pelinovsky On Jost Solutions for the Discrete and Ultradis- McMaster University crete KdV Equation Department of Mathematics [email protected] Explicit expressions for the Jost solutions of the discrete Schrödinger equation for compact support solution of the discrete KdV are described. These expressions have ultra- MS51 discrete limits and applications of these results to solving Interactions of Bright and Dark Solitons with Lo- arbitrary initial value problems for the ultra discrete KdV calized PT -Symmetric Potentials equation using the inverse scattering proposed by Willox et al (J. Phys. A: Math. Theor. 43 (2010) 482003) are We study collisions of moving nonlinear-Schrödinger soli- considered. tons with a PT -symmetric dipole embedded into the one- dimensional self-focusing or defocusing medium. Analyti- Jonathan Nimmo cal approximations are developed for both bright and dark School of Mathematics & Statistics solitons. In the former case, an essential aspect of the ap- University of Glasgow, UK proximation is that it must take into regard the intrinsic [email protected] chirp of the soliton, thus going beyond the bounds of the simplest quasi-particle description of the soliton’s dynamics. The analytical results are verified by comparison with MS52 numerical simulations. Collisions result in partial reflection Classification of Tau-symmetric Hamiltonian Evo- and transmission of incident solitons. Critical velocities lutionary PDEs separating these outcomes are found by means of numerical simulations, and in the approximate analytical form. We consider the classification of deformations of the dis- Exact solutions for the dark soliton pinned by the complex persionless KdV hierarchy which possess a Hamiltonian PT -symmetric dipole are produced too. structure and satisfy the so called tau-symmetry property. We conjecture that such deformations are characterized Boris Malomed by an infinite series of constant parameters, and provide Tel Aviv University, Israel evidences to support this conjecture. We also consider [email protected] the classification problem of tau-symmetric deformations of Hamiltonian integrable hierarchies of hydrodynamic type. Hadi Susanto The University of Nottingham Youjin Zhang United Kingdom Department of Mathematics, [email protected] Tsinghua University, China [email protected] MS52 Quasi-Determinants and Non Commutative Equa- MS53 tions with Pfaffian Type Solutions Skyrmions as Models of Nuclei

For some time, quasi-determinants of wronskian and gram- Skyrmions in 3-dimensions are compact soliton solutions in mian type have been known to provide solutions to some a nonlinear theory of pions. They have interesting shapes, non-commutative equations, the most famous of which is and several have Platonic symmetry. There is a conserved the non-commutative KP equation. In this talk we shall topological charge B which is identified with baryon num- use some of the ideas developed for the non-commutative ber. Skyrmions are proposed to model protons, neutrons KP equation to look at a possible construction for a non- and larger nuclei. I will discuss progress in constructing commutative pfaffian which we shall call a ‘quasi-pfaffian’ Skyrmion solutions up to baryon number 108 and beyond, NW14 Abstracts 107

and ﬁnding the properties of their quantised states, includ- University of Bath ing their spin. [email protected]

Nicholas Manton Dept. of Applied Mathematics and Theoretical Physics MS54 University of Cambridge Discrete KP Equation with Self-Consistent Sources [email protected] We show that the discrete Kadomtsev–Petviashvili (KP) equation with sources, obtained recently by the ”source MS53 generalization” method, can be incorporated into the Vorticity Models in Condensed Matter Physics and squared eigenfunctions symmetry extension procedure. Gradient Flows of 1-Homogeneous Functionals Moreover, using the known correspondence between Darboux-type transformations and additional independent We illustrate some recent results on variational and evo- variables, we demonstrate that the equation with sources lution problems concerning a certain class of convex 1- can be derived from Hirota’s discrete KP system of equa- homogeneous functionals for vector-valued maps, related to tions (without sources) in a space of bigger dimension. In vortex density in 3-d superconductors and superfuids, that this way we uncover the origin of the source terms as com- arise as limiting cases of the classical Ginzburg-Landau and ing from multidimensional consistency of the Hirota system Gross-Pitaevskii energies. Minimizers and gradient flows itself. of such functionals may be characterized as solutions of suitable non-local vectorial generalizations of the classical Adam Doliwa obstacle problem. Faculty of Mathematics and Computer Science University of Warmia and Mazury Giandomenico Orlandi [email protected] Department of Computer Science University of Verona Runliang Lin [email protected] Tsinghua University Beijing MS53 [email protected] Long Range Scattering for the Klein-Gordon Equa- tion with Nonhomogeneous Nonlinearities MS54 The asymptotic stability of coherent states , like kinks in Discrete Geometry of Polygons and Hamiltonian one dimension poses new great challenges. This is due Structures to the long range nature of the dispersive equation. This In this talk I will describe how a discrete moving frame talk will focus on one such problem. We study the 1D describing the discrete geometry of polygons in projective Klein-Gordon equation with quadratic and variable coeffi- RPn and other parabolic manifolds describes a Hamilto- cient cubic nonlinearity. This problem exhibits a striking nian structure on the space of discrete curvatures. The resonant interaction between the spatial frequencies of the structure is associated to some differential-difference com- nonlinear coefficients and the temporal oscillations of the pletely integrable systems. In the projective case the solutions. We prove global existence and (in L-infinity) systems are integrable discretizations of generalized KdV scattering as well as a certain kind of strong smoothness equations or AGD flows, systems that posses a realization for the solution at time-like infinity with the help of several as flows of polygons in RPn.ThisisjointworkwithJing new classes of normal-forms transformations. The analysis Ping Wang. also shows the limited smoothness of the solution, in the presence of the resonances. Gloria Mari Beffa University of Wisconsin, Madison Avy Soffer maribeff@math.wisc.edu Rutgers University, Math Dept. soff[email protected] MS54 MS53 Geometric Aspects of Integrable Self-adaptive Moving Mesh Schemes Coherent Motion in Hamiltonian Lattices with Next-to-nearest Neighbour Interactions We will explain how to construct integrable self-adaptive moving mesh schemes of some nonlinear wave equations We consider an infinite chain of particles with nearest such as the short pulse equation and the Camassa-Holm neighbour (NN) and next to nearest neighbours (NNN) be- equation. We also show various properties of self-adaptive ing coupled by nonlinear springs. To mimic the Lennard- moving mesh schemes. Jones potential, the NN and NNN springs act against each other. The existence of supersonic periodic solutions is Kenichi Maruno discussed, as well as the existence of subsonic waves ho- Department of Mathematics moclinic to exponentially small periodic oscillations. The The University of Texas-Pan American contrast to the theory for chains with NN interaction alone [email protected] is discussed.

Johannes Zimmer MS54 Bath Discrete Moving Frames and Discrete Integrable [email protected] Systems Christine Venney Discrete moving frame oﬀers signiﬁcant computational ad- 108 NW14 Abstracts

vantages for our study of discrete integrable systems. We PT − transitions. demonstrate that the discrete analogues of some curvature flows lead naturally to Hamiltonian pairs, which gener- George Tsironis ate integrable differential-difference systems. In this talk, Depart. Physics, Univ. Crete and FORTH we concentrate on the projective space and the integrable Heraklion 71003, Grece equations obtained. This is joined work with Elizabeth [email protected] Mansfield and Gloria Mar Beffa.

Jin-Ping Wang MS55 School of Mathematics, Statistics, and Actuarial Science University of Kent, England Nonlinear Waves in a Homogenized Two-Phase [email protected] Particulate Composite Medium

We consider a two-phase, isotropic, nonlinear, particulate MS55 composite medium whose effective linear permittivity and Solitonic and Vortex Behaviour in Exciting Non- nonlinear susceptibility are determined by the strong per- linear Hyperbolic Metamaterials mittivity fluctuation theory. Soliton propagation in the homogenized composite medium is found to obey the com- Not Available at Time of Publication plex Ginzburg-Landau (CGL) equation. When linear loss is counterbalanced by nonlinear gain, the propagation of A.D. Boardman chirped solitons is permitted. These solitons are classi- Joule Material & Physics Research Centre, fied as Pereira-Stenflo bright solitons or Nozaki-Bekki dark University of Salford, M5 4WT UK solitons. Their propagation regime depends on the volume [email protected] fraction of the component phases and on the correlation length of the spatial fluctuation of the dielectric properties in the medium. In the absence of linear and nonlin- MS55 ear dissipation, the CGL equation reduces to the nonlinear Nonlinear Wave Propagation and Localization in Schrödinger (NLS) equation, and both types of solitons Squid Metamaterials simplify to the usual NLS bright and dark solitons. The exhibited possibility of localization of electromagnetic en- rf SQUIDs are strongly nonlinear oscillators exhibiting ergy may pave the way for interesting applications employ- several fascinating properties. Metamaterials comprising ing such composite media in planar multilayered structures SQUIDs reveal rich electromagnetic behavior, manifested similar to the ones utilizing temporal solitons and beams by tuneability, switching, and multistability. SQUID meta- in nonlinear optical media. materials support flux waves, whose nonlinear propagation generates subharmonic pass-bands. They also support localized excitations in the form of dissipative breathers, in- Nikolaos L. Tsitsas vestigated with respect to their generation conditions and Department of Informatics their evolution. Tetragonal and honeycomb lattices are Aristotle University of Thessaloniki University Campus, considered, and features of nonlinearity peculiar to each 5412 array type are identified and discussed. [email protected]

Nikos Lazarides Akhlesh Lakhtakia Crete Center for Quantum Complexity and Department of Engineering Science and Mechanics, Nanotechnology, Pennsylvania State University, PA 16802-681, USA Department of Physics, University of Crete [email protected] [email protected] Dimitri Frantzeskakis George Tsironis University of Athens Depart. Physics, Univ. Crete and FORTH [email protected] Heraklion 71003, Grece [email protected] MS56

MS55 Modeling Combined UV/IR Filamentation Dynamical Properties of Parity-Time Symmetric Metamaterials Advances in laser technology allows the creation of intense light filaments, in particular in the infrared(IR) and ul- Synthetic systems with matched gain and loss may be re- traviolet(UV) regimes. A prevailing challenge is to deter- alized through parity-time (PT )-symmetric metamaterials mine conditions on the filament that enhance the possibil- consisted of split-ring resonator (SRR) dimers, one with ity of long (kilometer) distance propagation in the atmo- loss and the other with equal amount of gain, coupled sphere. This requires theoretical studies on filament forma- magnetically while nonlinearity and gain are introduced tion and stability that parallel experimental efforts. This through tunnel Esaki diodes. In the absense of nonlinear- work presents our recent results on the existence and stability a PT − phase transition occurs with an accompanying ity properties of multi-colored (UV/IR) filament/filament band modification. The presense of noninearity may in- and filament/vortex states. This work is join with D. Wang duce nonlinearly localized modes in the form of discrete (U. Vermont), A. Sukhinin, J-C Diels and A. Rasoulof (U. breathers with the largest part of the total energy con- New Mexico. Work is supported by a DOD-MURI grant. centrated in two neighboring sites belonging to the same gain/loss dimer. Time-varying gain/loss parameters in- Alejandro Aceves troduce additional control possibilities including reentrant Southern Methodist University NW14 Abstracts 109

[email protected] tions. Our results are confirmed by numerical simulations and agrees with the formula of Herrero and Velázquez for specially constructed solutions. The talk is based on the MS56 jointworkwithS.I.Dejak,D.EgliandP.M.Lushnikov. Some Fundamental Issues in Internal Wave Dy- namics Israel Michael Sigal Department of Mathematics One of the simplest physical setups supporting internal University of Toronto wave motion is that of a stratified incompressible Euler [email protected] fluid in a channel. This talk will discuss asymptotic models capable of describing large amplitude wave propagation in this environment, and in particular of predicting the MS57 occurrence of self-induced shear instability in the waves’ dynamics for continuously stratified fluids. Some curious Dynamics of Vortex Filaments and their Stability properties of the Euler setup revealed by the models will be presented. I will discuss some progress in determining the nonlinear (orbital) stability type of certain closed solutions of the Roberto Camassa Vortex Filament Equation (VFE), such as finite-gap fila- University of North Carolina ments and periodic breathers. The approach is based on [email protected] the correspondence between the VFE and the Nonlinear Schrödinger equation (NLS), and on the representation of a natural sequence of constants of motion in terms of NLS MS56 squared eigenfunctions. Instabilities of Gravity and Capillary Waves on the Surface of the Fluid Annalisa M. Calini College of Charleston We observed numerically and investigated analytically res- Department of Mathematics onant interactions on the discrete greed of wave vectors, [email protected] where resonances are never fulfilled exactly. In this report we shall concentrate on an interesting and useful case of four wave resonances for standing wave. Standing wave is MS57 relatively easy to generate in experiment. We have shown Vortex Dynamics in Bose-Einstein Condensates: that with time such a wave will generate isotropic wave Bifurcations, Chaotic Dynamics and Experimental field, which is important for experiments on wave turbu- Observations lence.

Alexander O. Korotkevich In this talk, we will revisit the topic of N-vortices in Bose- Dept. of Mathematics & Statistics, University of New Einstein condensates (BECs). We will review recent ex- Mexico periments including ones of remarkable symmetry-breaking L.D. Landau Institute for Theoretical Physics RAS phenomena in cases of co-rotating vortices, as well as ex- [email protected] plore steady states and conﬁgurations of counter-rotating vortices. Particle models of such systems will give rise to a considerable wealth of: (a) chaotic dynamics; (b) ex- Alexander Dyachenko otic states and bifurcation phenomena; (c) connections to Landau Institute for Theoretical Physics classical polynomials (e.g. Hermite); (d) coarse graining [email protected] features (for large N). We will brieﬂy discuss (a)-(d) and provide some future perspectives. Vladimir E. Zakharov Department of Mathematics, University of Arizona, Panayotis Kevrekidis Tucson Department of Mathematics and Statistics P.N. Lebedev Physical Institute RAS, Moscow, Russia University of Massachusetts [email protected] [email protected]

MS56 MS57 Blowup Dynamics in the Keller-Segel Model of Chemotaxis Traveling Waves in Holling-Tanner Model with Dif- fusion The Keller-Segel equations model chemotaxis of bio- organisms. It is relatively easy to show that in the critical We study diffusive Holling Tanner model for wide range of dimension 2 and for mass of the initial condition greater parameters from the point of view of existence of traveling than the critical one, the solutions ’blowup’ (or ’collapse’) waves. We geometrically construct qualitatively different in finite time. However, understanding the mechanism of traveling waves that include fronts, periodic wave-trains blowup turned out to be a very subtle problem defying and solutions that asymptotically connect wave-trains to a solution for the long time. This blowup is supposed to de- constant state. scribe the chemotactic aggregation of the organisms and understanding its mechanism would allow to compare the- Vahagn Manukian oretical results with experimental observations. In this talk Miami University Hamilton I discuss recent results on dynamics of solutions of the (re- [email protected] duced) Keller-Segel equations in the critical dimension 2 which include a formal derivation and partial rigorous re- Anna Ghazaryan sults of the blowup dynamics of solutions of these equa- Miami University 110 NW14 Abstracts

[email protected] The Pennsylvania State University [email protected]

MS57 Semi-strong Multipulse Interaction in Reaction MS58 Diﬀusion Systems: The Case of Asymptotically Gravity Capillary Waves and Related Problems Weak Dissipation Nonliinear waves propagating under an elastic membrane We demonstrate the asymptotic stability of semi-strong N- are considered. Results for periodic waves, solitary waves pulse solutions for a class of singularly perturbed reaction- with decaying oscillatory tails and generalised solitary diﬀusion equations for which the essential spectrum is waves are presented. The results are compared to the clas- asymptotically close to the origin. The key step to both sical theory of gravity-capillary waves and new types of the existence and stability is the semi-group analysis of waves are found and discussed. a weakly time-dependent family of non-self adjoint operators. We derive a nonlinear normal hyperbolicity condition Jean-Marc Vanden-Broeck that balances the time dependence of the linear operators University College London against the rate of dissipation of the essential spectrum. [email protected]

Keith Promislow Michigan State University MS58 [email protected] A Quasi-Planar Model for Gravity-Capillary Inter- facial Waves in Deep Water Arjen Doelman We propose a quasi-planar model for gravity-capillary Mathematisch Instituut waves between two semi-infinite, immiscible fluids. Our [email protected] analysis and numerics show that there exists a critical density ratio, below which the interfacial solitary waves and Tom Bellsky the bifurcation diagram are qualitatively similar to those Arizona State University of the free-surface gravity-capillary waves on deep water. [email protected] However, the bifurcation mechanism near the minimum of the phase speed is essentially similar to that of the hydroelastic waves on deep water, when the density ratio is in the MS58 supercritical regime. An Efficient Boundary Integral Method for 3D In- terfacial Flow with Surface Tension: Numerical Re- Zhan Wang sults and Numerical Analysis University College London [email protected] We introduce an efficient boundary integral method for the initial value problem for 3D interfacial flow with surface tension. The method uses a generalized isothermal param- MS59 eterization of the free surface, and requires a fast method Solitons in Quadratically Nonlinear Media with for evaluation of the Birkhoff-Rott integral. We will also PT-Symmetric Potentials discuss a proof of convergence of the method. This is joint work with Yang Liu, Michael Siegel, and Svetlana Tlupova. The system of equations for waves in quadratically nonlinear media with a parity-time(PT) symmetric modulation of refractive indices for fundamental and second harmonics David Ambrose are studied. Exact bright and dark solitary solutions for Drexel University corresponding forms of the PT-symmetric potentials are [email protected] found. Their stability and dynamics are investigated.

Fatkhulla Abdullaev MS58 Centro de Fisica Teorica e Computacional, A Reduction of the Euler Equations to a Single Faculdade de Ciencias, Universidade de Lisboa Time-Dependent Equation [email protected]

In this talk, a new single equation for the time-dependent Bakhram Umarov free-surface of a water-wave under the influence of grav- International Islamic University of Malaysia ity and surface tension is derived. This new equation is Kulliyyah of Science derived without approximation from Euler’s Equations for [email protected] an invicid, irrotational fluid and is valid in two spatial dimensions (one-dimensional wave surface). Using this single equation, various asymptotic models are derived. Classical MS59 reduced models such as KdV are found with ease, and new Nonlinear Quantum Dynamics of a BEC in a PT - asymptotic models are presented. symmetric Double-well Trap Katie Oliveras The dynamics of a Bose-Einstein condensate in a non- Seattle University Hermitian PT -symmetric double-well potential modelling Mathematics Department a coherent in- and outflux of particles is investigated in [email protected] the mean-field limit. The time-dependent Gross-Pitaevskii equation is solved and it is shown that it exhibits stable Vishal Vasan stationary states, however, the dynamical properties are Department of Mathematics very comprehensive. Elliptically stable and hyperbolically NW14 Abstracts 111

unstable stationary solutions are found. The complicated convergence acceleration algorithms related to discrete in- dynamical properties can be related to the subtle interplay tegrable systems. Then the non-autonomous version of the of the nonlinearity and the non-Hermiticity. corresponding integrable systems are derived. The new operator generalizing the usual forward difference operator Holger Cartarius, Daniel Haag, Dennis Dast, Andreas in this method was first proposed by C. Brezinski when Löhle, Jörg Main, Günter Wunner studying generalizations of the epsilon algorithm. Universität Stuttgart [email protected], Yi He [email protected], [email protected] Wuhan Institute of Physics and Mathematics, Chinese stuttgart.de, [email protected], Academy [email protected], [email protected] [email protected] stuttgart.de MS60 MS59 Constructing Probabilistic Particle Cellular Au- Dynamical Theory of Scattering for Complex Po- tomata from Fundamental Diagrams tentials, Inverse Scattering, and Confined Nonlin- earities We propose a probabilistic neighborfood-five particle cellular automata (CA) which includes fifteen deterministic For a given possibly complex scattering potential v(x) in particle CA. The probabilistic particle CA is constructed one dimension, we construct a two-level non-Hermitian from evolution equations of max-min form obtained from Hamiltonian whose S-matrix coincides with the transfer ultradiscrete Cole-Hopf transformation. We also show a matrix of v(x). We use this approach to develop a dynam- fundamental diagram of the probabilistic particle CA ob- ical theory of scattering where the reflection and transmis- tained by numerical simulations. Finally we discuss about sion amplitudes are given as solutions of a set of dynamical future plans. equations and offer an inverse scattering scheme for construction of scattering potentials with desirable properties Junta Matsukidaira at a prescribed wavelength. We will also outline an exten- Ryukoku University, Japan sion of some of our results to situations where a confined [email protected] nonlinearity is present. MS60 Ali Mostafazadeh Koc University, Turkey Soliton Solutions to an Extended Box and Ball Sys- [email protected] tem Equation We consider an extended Box and Ball system equation. MS59 It has two type of ultradiscrete soliton solutions. One is PT a solution of the ultradiscrete KdV type and the other is Partially- -symmetric Potentials with All-real a solution of the ultradiscrete Toda type. We propose a Spectra and Soliton Families solution which includes both of them.

Multi-dimensional complex potentials with partial parity- Hidetomo Nagai PPT time ( ) symmetry are proposed. The usual parity- Tokai University PT time ( ) symmetry requires that the potential is invari- [email protected] ant under complex conjugation and simultaneous reflection in all spatial directions. However, we show that if the potential is only partially PT -symmetric, i.e., it is invariant MS60 under complex conjugation and reflection in a single spa- On Exact Solutions to Lattice Equations tial direction, then it can also possess all-real spectra and continuous families of solitons. These results are estab- We propose dynamical systems defined on algebra of lat- lished analytically and corroborated numerically. Symme- tices, which we call lattice equations. We give exact solu- try breaking of solitons in PPT -symmetric potentials will tions to a class of lattice equations of which complexity of also be discussed. solutions are of polynomial order. Moreover we discuss the relationship between those equations and binary cellular Jianke Yang automata. Department of Mathematics and Statistics University of Vermont Daisuke Takahashi [email protected] Waseda University, Japan [email protected] MS60 Lattice Boussinesq Equation and Convergence Ac- MS61 celeration Algorithms Well-Balanced Positivity Preserving Central- Upwind Scheme for the Shallow Water System In this talk, we will give the molecule solution of an equa- with Friction Terms tion related to the lattice Boussinesq equation with the help of determinantal identities. It is shown that this equation Shallow water models are widely used to describe and study can for certain sequences be used as a numerical conver- free-surface water flow. While in some practical applica- gence acceleration algorithm. Reciprocally, we will derive a tions the bottom friction does not have much influence non-autonomous form of the integrable equation related to on the solutions, there are still many applications, where the lattice Boussinesq equation by a new algebraic method. the bottom friction is important. In particular, the fric- This method starts from constructing generalizations of tion terms will play a significant role when the depth of 112 NW14 Abstracts

the water is very small. In this talk, I will discuss shal- tions efficiently. The MLMC algorithm is suitably modified low water equations with friction terms and their approx- to deal with uncertain (and possibly uncorrelated) data on imation by a well-balanced central-upwind scheme that is each node of the underlying topography grid by the use capable of exactly preserving physically relevant steady of a hierarchical topography representation. Numerical ex- states. The scheme also preserves the positivity of the wa- periments in one and two space dimensions are presented ter depth. We test the designed scheme on a number of to demonstrate the efficiency of the MLMC algorithm. one- and two-dimensional examples that demonstrate robustness and high resolution of the proposed numerical ap- Jonas Sukys proach. The data in the last numerical example correspond ETH, Zurich to the laboratory experiments, designed to mimic the rain [email protected] water drainage in urban areas containing houses. Since the rain water depth is typically several orders of magnitude smaller than the height of the houses, we develop a special MS61 technique, which helps to achieve a remarkable agreement High Order Discontinuous Galerkin Methods for between the numerical and experimental results. the Shallow Water Equations

Alina Chertock Shallow water equations (SWEs) with a non-flat bottom North Carolina State University topography have been widely used to model flows in rivers Department of Mathematics and coastal areas. Since the SWEs admit non-trivial equi- [email protected] librium solutions, extra care need to be paid to approximate the source term numerically. To capture this balance and near-equilibrium solutions, well-balanced meth- MS61 ods have been introduced and performed well in many nu- Well-Balanced Fully Coupled Central-Upwind merical tests. In this presentation, we will talk about re- Scheme for Shallow Water Flows over Erodible Bed cently developed high-order discontinuous Galerkin (DG) finite element methods, which can capture the general mov- Intense sediment transport and rapid bed evolution are ing steady state well, and at the same time are positivity frequently observed under highly-energetic flows, and bed preserving without loss of mass conservation. Some nu- erosion sometimes is of the same magnitude as the flow merical tests are performed to verify the positivity, well- itself. Simultaneous simulation of multiple physical pro- balanced property, high-order accuracy, and good resolu- cesses requires a fully coupled system to achieve an action for smooth and discontinuous solutions. curate hydraulic and morphodynamical prediction. In this Yulong Xing talk, I will present a high-order well-balanced finite-volume Department of Mathematics method for a new fully coupled two-dimensional hyperbolic Univeristy of Tennessee / Oak Ridge National Lab system consisting of the shallow water equations with fric- [email protected] tion terms coupled with the equations modeling the sediment transport and bed evolution. The nonequilibrium sediment transport equation is used to predict the sedi- MS62 ment concentration variation. Since both bed-load, sed- Multi-Component Integrable Wave Equations and iment entrainment and deposition have significant effects Geometric Moving Frames on the bed evolution, an Exner-based equation is adopted together with the Grass bed-load formula and sediment I will survey some recent work giving a geometric approach entrainment and deposition models to calculate the mor- to deriving group-invariant nonlinear wave equations. The phological process. The resulting 5 by 5 hyperbolic system approach is based on applying moving-frame methods to of balance laws is numerically solved using a Godunov-type curve flows in geometric spaces, where the differential co- central-upwind scheme on a triangular grid. A computa- variants of the curve naturally satisfy a multi-component tionally expensive process of finding all of the eigenval- nonlinear wave equation which has an internal symmetry ues of the Jacobian matrices is avoided: The upper/lower group given by the equivalence group of the moving frame. bounds on the largest/smallest local speeds of propaga- In this setting there are natural classes of integrable flows, tion are estimated using the Lagrange theorem. A special which give rise to group-invariant integrable wave equa- discretization of the source term is proposed to guarantee tions with a Lax pair, a bi-Hamiltonian structure, and a the well-balanced property of the designed scheme. The symmetry recursion operator. proposed fully coupled model is verified on a number of numerical experiments. Stephen Anco Department of Mathematics Abdolmajid Mohammadian, Xin Liu Brock University University of Ottawa [email protected] [email protected], li- [email protected] MS62 Some Novel Geometric Realizations of Integrable MS61 Hierarchies MLMC-FVM for Shallow Water Equations with Uncertain Bottom Topography We discuss geometric evolution equations for generic curves in centroaffine 3-space and for Legendrian curves in the The initial data and bottom topography, used as inputs 3-sphere. In the centroaffine case, we construct a double in shallow water models, are prone to uncertainty due to hierarchy of mutually commuting curve flows, whose curva- measurement errors. We model this uncertainty statisti- ture evolution equations realize the two-component Boussi- cally in terms of random shallow water equations. We ex- nesq hierarchy. The passage from the velocity components tend the Multi-Level Monte Carlo (MLMC) algorithm to (with respect to a Frenet-type frame) to invariant evolu- numerically approximate the random shallow water equations gives a Poisson operator for the hierarchy. Within NW14 Abstracts 113

the hierarchy of curve flows there is a sub-family of geo- Centre for Plasma Physics (CPP) metric evolution equations that preserves curves on a cone [email protected] through the origin, and realize the Kaup-Kuperschmidt hierarchy. In the Legendrian case, we can also realize the Kaup-Kuperschmidt equations via a sequence of geometric MS63 flows. Pt-Symmetry and Embedded Modes in the Con- tinuum for Magnetic Metamaterials Thomas Ivey College of Charleston First, we examine the PT-symmetry breaking transition [email protected] for a magnetic metamaterial of a ?nite extent, modeled as an array of coupled split-ring resonators in the equivalent MS62 circuit model approximation. Small-size arrays are solved completely in closed form, while for arrays larger than N = A Finite Dimensional Integrable System Arising in 5 results were computed numerically for several gain/loss Shock Clustering spatial distributions. In all cases, it is found that the pa- In previous work, Menon and Srinivasan studied scalar con- rameter stability window decreases rapidly with the size servation laws with convex flux and initial datum given by of the array, until at N = 20 approximately, it is not pos- spectrally negative Feller processes. For the special case sible to support a stable PT-symmetric phase. A simple in which the initial condition is stationary, it was shown explanation of this behavior is given. Next, we consider the that the generator of the Feller process evolves according problem of building a surface localized mode embedded in to a Lax equation. Subsequently, Menon introduced a fi- the continuous band of a semi-infinite one- dimensional ar- nite dimensional version of this problem, associated with ray of split-ring resonators. We suggest an efficient method the Markov group. In this talk, we will explain how to for creating such surface mode and the local bounded po- solve this finite dimensional system explicitly. tential necessary to support this mode.

Luen-Chau Li Mario Molina Department of Mathematics Departamento de Fisica, Universidad de Chile Pennsylvania State University, USA Santiago, Chile [email protected] [email protected]

MS62 MS64 Integrable Systems and Invariant Geometric Flows High-Frequency Instabilities of Small-Amplitude in Similarity Symplectic Geometry Solutions of Hamiltonian PDEs

In this talk, differential invariants and invariant geometric Generalizing ideas of MacKay, and MacKay and Saffman, a flows in similarity symplectic geometry are studied. Ex- necessary condition for the presence of high-frequency (i.e., plicit formulae of symplectic invariants and Frenet formu- not modulational) instabilities of small-amplitude solutions lae for curves in the similarity symplectic geometry are of Hamiltonian partial differential equations is presented, obtained by the moving frame method. The relationship entirely in terms of the Hamiltonian of the linearized prob- between the of Euclidean symplectic invariants and simi- lem. The entire theory with the exception of a Krein sig- larity symplectic invariants is also identified. As a result, nature calculation can be phrazed in terms of the disper- we show that the matrix mKdV equation and the matrix sion relation of the linear problem. The general theory Burgers equation arise from invariant nonstreching curve changes as the Poisson structure of the Hamiltonian PDE flows in the similarity symplectic geometry. is changed. Two important cases are worked out and different examples are presented. Changzheng Qu Northwest University, China Bernard Deconinck [email protected] University of Washington [email protected] MS63 Quasi-Discrete Solitons in Nonlinear Transmission MS64 Line Metamaterials Multipole and Half-vortex Gap-solitons in Spin- Not Available at Time of Publication orbit Coupled Bose-Einstein Condensates

D.J. Frantzeskakis Using the parity and time reversal symmetries of a two- University of Athens, Zografos, Greece dimensional spin-orbit coupled Bose-Einstein condensate [email protected] in a Zeeman lattice, we found numerically various families of localized solutions, including multipole and half-vortex solitons. The obtained solutions may exist at any direction MS63 of the gauge ﬁeld with respect to the lattice and can be Amplitude Modulation and Envelope Mode For- found either in ﬁnite gaps. The existence of half-vortices mation in Left-Handed Media: a Survey of Recent requires higher symmetry. Stability of these modes makes Results them feasible for experimental observation.

Not Available at Time of Publication Yaroslav Kartashov Institut de Ci`encies Fotniques Ioannis Kourakis Barcelona, Spain Queen’s University Belfast [email protected] 114 NW14 Abstracts

Vladimir V. Konotop ics change as the cavity size decreases. Department of Physics University of Lisbon Lendert Gelens [email protected] Stanford University USA Valery Lobanov [email protected] Institut de Ciencies Fotoniques Barcelona, Spain Pedro Parra-Rivas [email protected] Applied Physics Research Group Vrije Universiteit Brussel, Belgium [email protected] MS64 Wave Turbulence: A Story Far from Over Damia Gomila IFISC-CSIC, Universitat de les Illes Balears Despite some successes, there are still many open questions Palma de Mallorca, Spain as to the validity of the wave turbulence closure and the damia@ifisc.uib-csic.es relevance of Kolmogorov-Zakharov finite flux solutions. In some cases, such as gravity driven water waves, the KZ Fran¸cois Leo spectrum has to be augmented with the Phillips’ spectrum Photonics Research Group, at high wavenumbers. Why? In others, such as the the Ghent University, Belgium MMT equation, the KZ spectrum and indeed the whole [email protected] closure does not obtain at any scale at all. Why? Further we ask: might it be possible to give a priori conditions on Stéphane Coen weakly nonlinear systems so that one is guaranteed that at Department of Physics, University of Auckland, least the natural asymptotic closure holds? I look forward New Zealand to addressing and discussing these challenges. [email protected]

Alan Newell University of Arizona MS65 Department of Mathematics Localised Hexagon Patches on the Surface of a [email protected] Magnetic Fluid

We present experimental results showing the existence of MS64 a patches involving different numbers cellular hexagons on Energy Growth in Switching Hamiltonian Systems the surface of a magnetic fluid under the influence of a of Fermi-Ulam Type vertical magnetic field. These patches are spontaneously generated by jumping into the neighborhood of the unsta- A natural example of a switching Hamiltonian system of ble branch of the domain covering hexagons and are found Fermi-Ulam type (a particle bouncing between two oscillat- to co-exist in the same parameter region. Using numer- ing walls) is considered. A corresponding smooth in time ical continuation techniques, we investigate the existence system would possess invariant KAM tori which would pre- of localized hexagons in the full equations determining the vent energy growth. Numerical simulations suggest that free-surface of magnetic fluid and describing the experi- energy does not grow even in the discontinuous case. We ment. We find that cellular hexagons possess a Maxwell explain this phenomenon using relation with another prob- point where the energy of a single hexagon is equal to the lem considered earlier by Kesten. energy of the flat state providing an energetic explanation for the multitude of various different hexagon patches ob- Vadim Zharnitsky served in experiments. Furthermore, it is found that planar Department of Mathematics hexagon fronts and hexagon patches undergo homoclinic University of Illinois snaking. [email protected] David Lloyd University of Surrey Maxim Arnold [email protected] University of Illinois at Urbana-Champaign Department of Mathematics Reinhard Richter [email protected] University of Bayreuth [email protected] MS65 Dynamics of Cavity Solitons and Optical Frequency MS65 Combs in the Lugiato-Lefever Equation Localized Solutions in Plane Couette Flow with Ro- tation Kerr frequency combs can be modeled in a similar way as temporal cavity solitons (CSs) in nonlinear cavities, using Localized solutions of plane Couette flow (PCF) bifurcate the Lugiato-Lefever equation describing pattern formation from the subcritical 3D periodic state at infinite parameter; in optical systems. Here, we first characterize different forcing the system through rotation changes this. The 3D dynamical regimes of CSs, such as time-periodic oscilla- periodic state now emerges secondarily from a 2D periodic tions and various chaotic dynamics. Secondly, the effect of state, making the localized solution a 3D patch embedded third order dispersion on the stability and snaking struc- in a 2D patterned background. The nontrivial background ture of CSs is studied. Finally, we discuss how the dynam- has some effect, however much of the scenario seen in other NW14 Abstracts 115

systems remains, such as the snakes-and-ladders structure proliferation. A key ingredient in the generation of the already found in non-rotating PCF. labyrinthine patterns formed, is the non-local interaction of the curved domain with its distal segments. Numerical Matthew Salewski tests are used to confirm and illustrate these phenomena. Max Planck Institute for Dynamics and document or other high-level commands. Self-Organization, Ger {E´}cole Polytechnique F{é}d{é}rale de Lausanne, Switze Alan E. Lindsay [email protected] Applied Computational Mathematics and Statistics University of Notre Dame Tobias Schneider [email protected] Max Planck Institute for Dynamics and Self-Organization, [email protected] MS66 Hotspots in a Non-Local Crime Model MS65 Stability Properties of Localized Structures Near We extend the Short et al. burglary hotspot model to allow Snaking for a larger class of criminal movement. Specifically, we allow criminals to travel according to a Lévy flight rather Snaking refers to the existence of localized roll structures in than Brownian motion. This leads to a non-local system spatially extended, reversible 1D partial differential equa- of differential equations. The stability of the homogeneous tions, that exist along a vertical sine-shaped bifurcation state is studied both numerically and through a Turing- curve so that the width of the underlying periodic roll pat- type analysis. The hotspot profiles are then constructed to tern increases along the bifurcation curve. Localized rolls leading order in a singular regime. undergo a series of bifurcations along the branch, and this talk is concerned with analytical and numerical aspects of Scott McCalla their PDE stability spectra. University of California Los Angeles [email protected] Elizabeth J. Makrides Brown University Jonah Breslau Division of Applied Mathematics Pomona College elizabeth [email protected] [email protected]

Bjorn Sandstede Sorathan Chaturapruek Brown University Harvey Mudd College bjorn [email protected] tum [email protected]

Theodore Kolokolnikov MS66 Dalhousie University Modeling Capillary Origami [email protected] To continue the move towards miniaturization in technol- Daniel Yazdi ogy, developing new methods for fabricating micro- and University of California, Los Angeles nanoscale objects has become increasingly important. One [email protected] potential method, called capillary origami, consists of plac- ing a small drop of liquid on a thin, inextensible sheet. In this state the system minimizes its total energy, pulling MS66 the planar sheet upward. Under appropriate conditions the sheet will fully encapsulate the liquid, creating a three- Slowly Varying Control Parameters, Delayed Bi- dimensional structure. In this talk, we discuss recent work furcations, and the Stability of Spikes in Reaction- on modeling a two-dimensional version of this system. diﬀusion Systems

Nicholas D. Brubaker We present three examples of delayed bifurcations for spike University of Arizona solutions of reaction-diffusion systems. The delay effect [email protected] results as the system passes slowly from a stable to unstable regime, and was previously analysed in the context of ODE’s in [P.Mandel and T.Erneux, J.Stat.Phys 48(5-6) MS66 pp.1059-1070, 1987]. Analysis of explicitly solvable nonlo- The Stability and Evolution of Curved Domains cal eigenvalue problems allows for analytic predictions of Arising from One Dimensional Localized Patterns the magnitude of delay. Asymptotic results are compared favorably to numerical computations. In many pattern forming systems, narrow two dimensional domains can arise whose cross sections are roughly one Justin C. Tzou dimensional localized solutions. This talk will present an Dalhousie University investigation of this phenomenon for the variational Swift- [email protected] Hohenberg equation. Stability of straight line solutions is analyzed, leading to criteria for either curve buckling or Michael Ward curve disintegration. A high order matched asymptotic Department of Mathematics expansion reveals a two-term expression for the geometric University of British Columbia motion of curved domains which includes both elastic and [email protected] surface diffusion-type regularizations of curve motion. This leads to novel equilibrium curves and space-filling pattern Theodore Kolokolnikov 116 NW14 Abstracts

Dalhousie University tical transmissions, it also provides an attractive solution [email protected] to generate new temporal and spectral waveforms. We explain in this talk how to take advantage of the progressive temporal and spectral reshapings that occur upon propaga- MS67 tion in a normally dispersive fiber. We base our discussion In-Cavity Transformations for New Nonlinear on several experimental results obtained at telecommuni- Regimes of Pulse Generation in Mode-Locked Fi- cation wavelengths. bre Lasers Kamal Hammani, Julien Fatome We review recent progress in the research on nonlinear University of Bourgogne mechanisms of pulse generation in passively mode-locked fi- [email protected], [email protected] bre lasers. These include parabolic self-similar pulse mode- locking, a mode-locking regime featuring pulses with a tri- Guy Millot angular distribution of the intensity, and spectral compres- University of Bourgogne sion arising from nonlinear pulse propagation. We also Institut Carnot de Bourgogne report on the possibility of achieving various regimes of [email protected] advanced temporal waveform generation in a mode-locked fibre laser by inclusion of a spectral filter into the laser cavity. Sonia Boscolo Aston University, Birmingham, United Kingdom Sonia Boscolo [email protected] Aston University, Birmingham, United Kingdom [email protected] Hervé Rigneault Institut Fresnel Sergei K. Turitsyn [email protected] Aston University Birmingham Stefano Wabnitz [email protected] University of Brescia, Brescia, Italy [email protected] Christophe Finot Universite’ de Bourgogne, Dijon, France Christophe Finot christophe.fi[email protected] Universite’ de Bourgogne, Dijon, France christophe.fi[email protected] MS67 Mechanisms and Definitions of Rogue Waves in MS67 Nonlinear Systems Time Domain Nonlinear Structures and Substruc- Optical rogue waves are rare yet extreme fluctuations in tures for Ultrafast Photonics an optical field. First used to describe an analogy between wave propagation in optical fibre and on deep water, the Applications of the time domain amplitude and phase terminology has since been generalized to many other op- structures and substructures, induced by self- and cross- tical processes. We present here an overview of recent re- phase modulation, phase addition and CARS type pro- search in this field, and although statistical features are cesses, are presented based on our numerical and exper- often considered defining feature of rogue waves, we em- imental studies. The report is focused on the techniques of phasize the underlying physical mechanisms driving the temporal lensing- spectral compression, Fourier conversion- appearance of extreme events. spectrotemporal imaging and frequency tuning, the specificity of Newton rings in a similariton-induced time lens, John Dudley as well as on nonlinear-optic applications of the trains pe- Universite de Franche-Comte, Institut FEMTO-ST, UFR riodically substructured by similaritons superposition. Sciences 16, route de Gray, 25030 Besancon Cedex, FRANCE Levon Mouradian, Aram Zeytunyan, Hrach Toneyan, [email protected] Garegin Yesayan Yerevan State University, Yerevan, Armenia Goery Genty [email protected], [email protected], Tampere University of Technology [email protected], [email protected] Tampere, Finland goery.genty@tut.fi Ruben Zadoyan Technology & Applications Center, Newport Corporation, Frederic Dias USA University College Dublin [email protected] School of Mathematical Sciences [email protected] Frédéric Louradour, Alain Barthélémy XLIM-UMR 6172 Université de Limoges/CNRS, France [email protected], [email protected] MS67 Nonlinear Pulse Shaping in Normally Dispersive Fibres: Experimental Examples MS68 While the combination of Kerr nonlinearity with disper- Complex and Coupled Complex Short Pulse Equa- sion in optical fibers can seriously impair high speed op- tions: Integrability, Discretization and Numerical NW14 Abstracts 117

Simulations syzygies on the generating set of differential invariants are obtained. We propose a complex and a coupled complex short pulse equations which describe the ultra-short pulse propaga- Ruoxia Yao tion in optical-fibers. After showing the integrability School of Computer Science, Shaanxi Normal University of these equations by Lax pairs, we derive the multi- Xi’an, China soliton solutions expressed in pfaffians by Hirotas bilinear [email protected] method. In the second partod the talk, the integrable semi- discretizations are constructed and are successfully used as self-adaptive moving methods for the numerical simula- MS69 tions of these equations. This is a collaborative work with Well-Balanced Positivity Preserving Cell-Vertex K. Maruno and Y. Ohta. Central-Upwind Scheme for Shallow Water Flows

Bao-Feng Feng I will present a novel two-dimensional central-upwind Mathematics Department, the University of Texas-Pan scheme on cell-vertex grids for shallow water equations American with source terms due to bottom topography. This type of 1201 W. University Dr., Ediburg TX 78541-2999 computational cells has the advantage of using more cell [email protected] interfaces which provide more information on the waves propagating in different directions. We prove that the proposed method is well-balanced and guarantees the positiv- MS68 ity of the computed water depth. Our numerical experi- A New Procedure to Approach Integrable Dis- ments confirm stability, well-balanced and positivity pre- cretization serving properties of the proposed method. This scheme can be applied to shallow water models when the bed to- In this talk, we present a new procedure to derive discrete pography is discontinuous and/or highly oscillatory, and on analogues of integrable PDEs in Hirota’s bilinear formal- complicated domains where the use of unstructured grids ism. This new approach is mainly based on introdution is advantageous. of suitable auxiliary equations with additional discrete independent variables such that there is integrable compat- Abdelaziz Beljadid ibility between the integrable system under consideration University of Ottawa and the new introduced equations while keeping the dis- [email protected] crete independent variables converging to original ones. We apply this procedure to several equations, including the Korteweg-de-Vries (KdV) equation, the Kadomtsev- MS69 Petviashvili (KP) equation, the Boussinesq equation, the RVM Finite Volume Methods: Applications to Sawada- Kotera (SK) equation, the Ito equation and etc., Multilayer Shallow Flows and obtain their associated semi-discrete or fully discrete analogues. In the continuum limit, these discrete analogues A new class of incomplete Riemann solvers for the numeri- converge to their corresponding smooth equations. cal approximation of conservative and nonconservative hyperbolic systems is proposed. They are based on viscosity Xing-Biao Hu matrices obtained by rational approximations to the Jaco- AMSS, Chinese Academy of Sciences, bian of the flux evaluated at some average state, and only [email protected] require the knowledge of the maximal characteristic speed. The proposed schemes are applied to nonconservative multilayer shallow water systems, where it has been observed MS68 that intermediate waves can be properly captured for an Rogue Waves for Some Soliton Equations appropriate degree of approximation of the generating rational function used. The numerical tests indicate that the For some soliton equations, the rogue wave solutions are schemes are robust, running stable and accurate with a studied and their algebraic structures are analyzed based satisfactory time step restriction, and the computational on the bilinear technique of soliton theory. cost is more advantageous with respect to schemes using a Yasuhiro Ohta complete spectral decomposition of the Jacobians. Department of Mathematics Jose M. Gallardo Kobe University University of Málaga. Spain [email protected] [email protected]

MS68 Manuel Castro New Results on the Explicit Monge-Taylor Forms University of Malaga for Submanifolds Under Group Actions Using [email protected] Equivariant Moving Frame Method Antonio Marquina Motivated by the widely used equivariant moving frame University of Valencia method and starting from the relations between the in- [email protected] finitesimal generators of transformation groups and the universal recurrence formulae for differential invariants, we elaborate important propositions which indicate the pro- MS69 cedure to construct more higher order fundamental nor- Path-Conservative Central-Upwind Schemes for malized differential invariants such that we can derive the Nonconservative Hyperbolic Systems explicit Monge-Taylor forms extensively for submanifolds under group actions. At the same time, a complete set of I will present a new path-conservative central-upwind 118 NW14 Abstracts

scheme for noncoservative hyperbolic systems of PDEs. can therefore be described by linearizing the NLS equation Such systems arise in a variety of applications and the most around a constant background. Once the perturbations challenging part of their numerical discretization is a robust have grown to O(1), however, the linearization ceases to treatment of nonconservative product terms. Godunov- be valid. On the other hand, the NLS equation is a com- type central-upwind schemes were developed as an effi- pletely integrable infinite-dimensional Hamiltonian system, cient, highly accurate and robust “black-box’ solver for and the initial-value problem is therefore amenable to so- hyperbolic systems of conservation laws. They were suc- lution via the inverse scattering transform (IST). In this cessfully applied to a large number of hyperbolic systems talk we will describe how the recently-developed IST for including such nonconservative systems as two-layer shal- the focusing NLS equation with non-zero boundary condi- low water equations, compressible two-phase flow models, tions can be used to elucidate the nonlinear stage of the Savage-Hutter type system modelling submarine underwa- MI. ter slides. To overcome the difficulties related to the presence of nonconservative product terms, several special tech- Gino Biondini niques were proposed. However, none of these techniques State University of New York at Buffalo was sufficiently robust and thus the applicability of the Department of Mathematics original central-upwind scheme was quite limited. We have biondini@buffalo.edu recently realized that the main drawback of the original approach was the fact that the jump of the nonconserva- Emily Fagerstrom tive product terms across cell interfaces has never been State University of New York at Buffalo taken into account. Rewriting central-upwind schemes emilyrf@buffalo.edu in the form of path-conservative schemes has helped to understand how the nonconservative products should be discretized so that their influence on the numerical solu- MS70 tion is accurately taken into account. The resulting path- Rational Solitons of Wave Resonant-Interaction conservative central-upwind scheme is a new robust tool for Models both conservative and nonconservative hyperbolic systems. The new scheme has been applied to the Saint-Venant sys- Integrable models which describe the resonant interaction tem with discontinuous bottom topography, two-layer shal- of two or more waves in 1 + 1 dimensions are known to low water system, and the two-mode shallow water equa- be of applicative interest in many fields. We consider a tions which was recently derived as a simplified model tat system of three coupled wave equations which includes describes nonlinear dynamics of waves with different ver- as special cases the vector Nonlinear Schroedinger equa- tical profiles. Our numerical results illustrate a superb tions (or Manakov System) and the equations describing performance of the new path-conservative central-upwind the resonant interaction of three waves. The Darboux- scheme, its robustness and ability to achieve very high res- Dressing construction of soliton solutions is applied under olution. the condition that the solutions have rational, or mixed rational-exponential, dependence on coordinates. Our al- Alexander Kurganov gebraic construction relies on nilpotent matrices and their Tulane University Jordan form. We systematically search for all bounded ra- Department of Mathematics tional (mixed rational-exponential) solutions. Rogue waves [email protected] of the three-wave resonant interaction equations belog to this class.

MS69 Sara Lombardo Well-Balanced Ale: a Dynamic Mesh Adaptation Applied Mathematics, CEIS Strategy for Shallow Water Flows Northumbria University [email protected] We discuss an efficient and cost effective strategy toward dynamic unstructured mesh adaptation for shallow water Antonio Degasperis flows. The method combines a mesh movement strategy Dipartimento di Fisica, Universita’ La Sapienza, Roma, based on a solution dependent Lapacian operator with an Italy ALE formulation. The first provides nodal displacements [email protected] dependent on the smoothness of the solution, while the second allows to follow these solution adaptive displacements without the need of any interpolation. We will discuss the MS70 conditions allowing the ALE formulation to remain well On the Spectrum of the Defocusing NLS Equation balanced, and show validation on both smooth and discon- with Non-zero Boundary Conditions tinuous shallow water flows. We revisit the scattering problem for the defocusing nonlin- Mario Ricchiuto ear Schrödinger equation with constant, non-zero bound- INRIA ary conditions at infinity. After reviewing some aspects [email protected] of the general theory, we consider a specific kind of piecewise constant potentials to address and clarify two issues, concerning: (i) the (non)existence of an area theorem re- MS70 lating the presence/absence of discrete eigenvalues to an Integrable Nature of Modulational Instability appropriate measure of the initial condition; and (ii) the existence of a contribution to the asymptotic phase differ- The modulational instability (MI), known as Benjamin- ence of the potential from the continuous spectrum. Feir instability in water waves, is one of the most widespread phenomena in nonlinear science. In many Barbara Prinari cases, the underlying dynamics is governed by the non- University of Colorado, Colorado Springs linear Schrodinger (NLS) equation. The initial stage of MI University of Salento, Lecce (Italy) NW14 Abstracts 119

[email protected] [email protected]

Gino Biondini State University of New York at Buﬀalo MS71 Department of Mathematics On the Travelling Wave Problem for Phase Transi- biondini@buﬀalo.edu tions in the Fermi-Pasta-Ulam Chain

We want to present established and recent results on the MS70 analyse of travelling waves for a lattice model of phase transitions. In particular we consider a bistable model with a Closed Form Solutions of the Hirota Equation piece-wise quadratic interaction potential and a smoothed variant with small spinodal region. Various authors have By using the Inverse Scattering Transform we construct an been able to prove in both cases the existence of families of explicit multisoliton solution formula for the Hirota equa- subsonic travelling waves, so that special interest is focus- tion. The formula obtained allows one to get, as particular ing on relevant selection criteria to identify meaningful so- cases, the N-soliton solution, the breather solution and, lutions. Following ideas from vanishing viscosity approach most relevantly, a new class of solutions called multipole to conservation laws we present ideas to analyse the stabil- soliton solutions. By adapting the Sym-Pohlmeyer recon- isation of waves in the model with dissipation and discuss struction formula to the Hirota equation, we use these ex- how this might help to set up a framework for the interact solutions to study the motion of a vortex filament in an pretation of the various types of wave solutions existing for incompressible Euler fluid with nonzero axial velocity. The the lattice problem. This is joint work with M Herrmann, results are obtained in collaboration with G. Ortenzi (Uni- K Matthies, J Zimmer versity of Milano Bicocca) and F. Demontis (University of Cagliari) Hartmut Schwetlick University of Bath Cornelis Van der Mee Department of Mathematical Sciences Dipartimento di Matematica e Informatica hartmut schwetlick ¡[email protected]¿ Universitá di Cagliari, Italy [email protected] MS71 Breathers in Two-dimensional Fermi-Pasta-Ulam MS71 Lattices Localized Waves in Fully Nonlinear Media Motivated by the observations of Marin et al [Phys Lett A, 281, 21, (2001)], we use asymptotic analysis to approximate We discuss properties of localized waves in a class of Hamil- breathers in a square, triangular and honeycomb Fermi- tonian lattices involving fully nonlinear interactions. This Pasta-Ulam lattices. In scalar two-dimensional systems we problem arises for the study of impact propagation in gran- obtain perturbed 2D NLS equations and conditions on the ular chains, where the Hertz contact force introduces a existence of breathers. We discuss the open problems of fully nonlinear coupling. In this context, new types of breathers’ stability (or decay rate), and interaction prop- strongly nonlinear modulation equations such as the loga- erties. rithmic KdV and discrete p-Schrödinger equations can be introduced. We review open problems concerning localized Jonathan Wattis waves in these systems and the original lattice. School of Mathematical Sciences University of Nottingham Guillaume James [email protected] Laboratoire Jean Kuntzmann, Université de Grenoble and CNRS [email protected] MS72 Branch Cut Singularity of Stokes Wave

MS71 Stokes wave is the fully nonlinear gravity wave propagating with the constant velocity. We consider Stokes wave in New Solutions for Slow Moving Kinks in a Forced the conformal variables which maps the domain occupied Frenkel-Kontorova Chain by fluid into the lower complex half-plane. Then Stokes wavecanbedescribedthroughthepositionandthetypeof In order to study effects of discreteness on dislocation mo- complex singularities in the upper complex half-plane.We tion, we construct new traveling wave solutions of mov- identified that this singularity is the square-root branch ing kink type for a modified, driven, dynamic Frenkel- point. We reformulated Stokes wave equation through the Kontorova model, representing dislocation motion under integral over jump at the branch cut which provides the stress. Formal solutions known so far are inadmissible for efficient way for finding of the explicit form of Stokes wave. velocities below a threshold value. The new solutions fill the gap left by this loss of admissibility. Analytical and numerical evidence is presented for their existence; how- Pavel M. Lushnikov ever, dynamic simulations suggest that they are probably Department of Mathematics and Statistics unstable. We discuss ramifications of this result for slow University of New Mexico dislocation motion. [email protected]

Phoebus Rosakis Applied Mathematics MS72 University of Crete (Greece) Solitons, Self-Induced Transparency, and Quantum 120 NW14 Abstracts

Cascade Lasers in integral neural field models with inhomogeneous synaptic kernels and Heaviside firing rate functions. We consider Self-induced transparency is a phenomenon in which an a simple periodic modulation in the synaptic kernel, apply optical pulse passes through a resonant medium without interface methods to the resulting integral problem and being absorbed because of quantum coherence. This phe- show how to construct localised bumps with a finite activ- nomenon is modeled mathematically using the Maxwell- ity region and two threshold crossings. Localised solutions Bloch equations, and these equations have exact soliton are arranged in a ”snake and ladder” bifurcation diagram. solutions in the limit that the quantum coherence times However, in this context, interface methods allow for the become infinite. In this talk, we show that it is possible to explicit construction of a bifurcation equation for localised use this phenomenon to modelock a quantum cascade laser steady states, so that analytical expressions for snakes and by creating dissipative solitons. ladders can be derived. Similarly, eigenvalue computations can be carried out analytically to determine the stability Curtis R. Menyuk of the solution profiles. In addition, we find regions of University of Maryland Baltimore County parameter space where the trivial homogeneous state co- [email protected] exists with two other stable solutions, an above-threshold periodic state and a cross-threshold periodic state. We Muhammad Talukder show how the multiple stability affects the corresponding BUET, Dhaka, Bangladesh bifurcation diagram and examine models in which the fir- [email protected] ing rate is a steep sigmoid as opposed to the Heaviside step function. This is joint work with Helmut Schmidt (Exeter) MS72 On Multi-Dimensional Compact Patterns Daniele Avitabile School of Mathematical Sciences Solitons, kinks or breathers, are manifestations of weakly University of Nottingham nonlinear excitations in dispersive media like mass-particle [email protected] lattices. In a strongly anharmonic chains, the tails of the solitary patterns rather than exponentially, decay at a doubly-exponential rate which in the continuum limit MS73 collapse into a singular surface with the resulting waves becoming strictly compact,hence their name: compactons. Localized Convection in a Rotating Fluid Layer Using the Z-K, the sub-linear Complex Klein-Gordon and the sub-linear NLS eqs. as examples, we shall demonstrate We study stationary convection in a two-dimensional fluid how N-dimensional compactons emergence and interact. In layer rotating around the vertical and heated from below. general, for compact, and hence non-analytical, structures With stress-free boundary conditions, spatially localized to emerge, the underlying system has to undergo a local states are embedded in a self-generated background shear loss of uniqueness due to, for instance, a degeneracy of zone and lie on a pair of intertwined solution branches ex- the highest order operator or other, singularity inducing, hibiting ”slanted snaking”. Similar solutions with no-slip mechanisms. For a basic introduction see: What is a Com- boundary conditions are computed. They are not embed- pacton, Notices of the AMS, Vol. 52, #7, pp 738, 2005. ded in a background shear and the solution branches exhibit snaking without a slant. Homotopic continuation from free-slip to no-slip boundary conditions is used to Philip Rosenau track the changes in the properties of the solutions and School of Mathematical Sciences the associated bifurcation diagrams. Tel-Aviv University, Israel [email protected] Cedric Beaume UC Berkeley [email protected] MS72 Thresholds and Blow-up Dynamics in the Nonlin- ear Dispersive Equations MS73 Steady and Oscillatory Localised States in Boussi- We consider the focusing nonlinear Schrödinger equation nesq Magnetoconvection with finite energy initial data in the mass-supercritical regime. We discuss the contracting sphere blow-up dynam- Previous studies have shown that it is possible to ics (joint work with Justin Holmer and Galina Perelman), find steady, localised convective cells in two-dimensional then present a new dichotomy for scattering and collapse Boussinesq magnetoconvection. Building on earlier work, solutions with finite variance (joint work with Thomas we demonstrate the existence of oscillatory localised con- Duyckaerts). In both results solutions can have an arbi- 2 vective states in this system. We discuss the properties of trary L -norm (mass). Generalization to other nonlinear these solutions as well as the effects of changing the hor- dispersive equations will be considered. izontal boundary conditions. In particular, we show that Svetlana Roudenko the inclusion of impermeable sidewalls can lead to forma- The George Washington University tion of localised “wall” modes. [email protected] Paul J. Bushby Newcastle University MS73 United Kingdom Localised Solutions in Integral Neural Field Equa- [email protected] tions Matthew Buckley I will discuss the formation of stationary localised solutions Newcastle University NW14 Abstracts 121

[email protected] front.

Matthew Pennybacker MS73 University of Arizona [email protected] Moving Localized Structures in a Doubly Diffusive System MS74 In this talk we will describe the origin and properties of Nanoscale Pattern Formation by Ion Bombardment moving spatially localized structures in natural doubly- of Binary Compounds diffusive convection in a vertically extended cavity. These solutions arise through secondary parity breaking bifurca- When a solid surface is bombarded with a broad ion beam, tions. Both single pulse and multipulse states of this type a plethora of self-assembled nanoscale patterns can emerge, will be described. The numerical results will be related including nanodots arranged in hexagonal arrays of re- to the phenomenon of homoclinic snaking in spatially remarkable regularity. We discuss a theory that explains the versible systems. genesis of the strikingly regular hexagonal arrays of nanodots that can form when binary materials are bombarded. David Lo Jacono In our theory, the coupling between a surface layer of al- Universite de Toulouse tered stoichiometry and the topography of the surface is the France key to the observed pattern formation. We analyze how a [email protected] soft mode related to the mean sputter yield can facilitate defect formation and give rise to less ordered patterns. Alain Bergeon Universite de Toulouse, France Patrick Shipman [email protected] Department of Mathematics Colorado State University Edgar Knobloch [email protected] University of California at Berkeley Dept of Physics [email protected] MS75 Relative Intensity Noise Transfer in Second-Order Raman Amplification with Random Distributed MS74 Feedback Ultra-Long Fibre Lasers

Isometric Immersions and Pattern Formation in The predicted RIN transfer function for second-order dis- Non-Euclidean Elastic Sheets tributed amplification based on different configurations of Random distributed feedback ultralong Raman fiber lasers Swelling thin elastic sheets are archetypical examples of (RDFLs) is found and studied over a broad range of pos- pattern forming systems which are often modeled as the sible lengths and signal powers, showing that the values of minimum an elastic energy which is the sum of a strong the RIN transfer function and the RIN cut-off frequencies stretching energy which penalizes deviations from a growth are dependent upon both parameters. RIN performance is induced geometry and a weak bending energy. A funda- compared to that of a typical cavity ultralong Raman fiber mental question is whether we can deduce the three dimen- laser pumped from the extremes. sional configuration of the sheet given exact knowledge of the imposed geometry. Using a combination of rigorous Javier Nuño analysis and numerical conjectures on I will argue that the Instituto de Optica,´ CSIC periodic patterns observed in nature correspond to low en- [email protected] ergy isometric immersions of the imposed geometry. Juan Diego Ania Castañón John A. Gemmer Brown University Instituto de ptica (CSIC), Madrid john [email protected] [email protected]

Shankar C. Venkataramani MS75 University of Arizona Transverse Disorder-Induced Localizations in Non- Department of Mathematics local Nonlinear Media: Theory and Experiments [email protected] We review our theoretical approach about the way localized states due to disorder are affected by the presence of non- MS74 linearity. We outline the link between Anderson localiza- Pattern-Forming Fronts in Phyllotaxis tion and solitons, and study local and nonlocal responses. We also report on experimental results on two-dimensional Some of the most spectacular patterns in the natural world disorder-induced localization in optical fibres, and the ob- can be found on members of the plant kingdom. Further- servation of action at a distance and collective dynamics more, the regular configurations of organs on plants, col- involving localized states in nonlocal media. lectively called phyllotaxis, exhibit a remarkable predispo- sition for Fibonacci and Fibonacci-like progressions. Start- Claudio Conti ing from a biochemical and mechanical growth model simi- Università Sapienza, Rome, Italy lar to the classic Swift-Hohenberg equation, we discuss the [email protected] ways in which nearly every property of phyllotaxis can be explained as the propagation of a pushed pattern-forming Viola Folli 122 NW14 Abstracts

Dep. Physics Sapienza Rome (IT) [email protected] [email protected]

Marco Leonetti MS76 University Sapienza, Rome, Italy KP Web-solitons from Wave Patterns [email protected] Nonlinear interactions among small amplitude, long wavelength, obliquely propagating waves on the surface of shal- Salman Karbasi, Arash Mafi low water often generate web-like patterns. In this talk, University of Wisconsin-Milwaukee, Milwaukee we discuss how line-soliton solutions of the Kadomtsev- [email protected], mafi@uwm.edu Petviashvili (KP) equation can approximate such web- pattern in shallow water wave. We describe an “inverse problem” which maps a certain set of measurable data from MS75 the solitary waves in the given pattern to the parameters required to construct an exact KP soliton that describes Optical Wave Turbulence: Toward a Unified Non- the non-stationary dynamics of the pattern. We illustrate Equilibrium Thermodynamic Description of Statis- the inverse problem using explicit examples of shallow wa- tical Nonlinear Optics ter wave pattern. Sarbarish Chakravarty We review a unified theoretical formulation of statistical University of Colorado nonlinear optics on the basis of the wave turbulence theory, at Colorado Springs which provides a nonequilibrium thermodynamic descrip- [email protected] tion of incoherent nonlinear waves. On the basis of the generalized NLS equation, we discuss the wave turbulence kinetic equation in analogy with kinetic gas theory, the MS76 long-range Vlasov equation in analogy with Gravitational Invariant Manifold of the KdV Equation dynamics, and the nonequilibrium description of nonin- stantaneous (Raman-like) nonlinearities in analogy with The solution structure of the KdV equation is studied in weak Langmuir turbulence in plasma. light of invariant manifold (IM). According to the definition of IM, the first order IM of the KdV equation is obtained, which turns to describe the traveling wave solution Gang Xu . The relationship is recovered between the second order University of Burgundy group-invariant IM and the group-invariant solutions of the Laboratoire Interdisciplinaire Carnot de Bourgogne KdV. The concept of linear IM is presented to study the [email protected] higher order IM and the two sets of linear IMs concerning soliton and algebraic geometric solutions are investi- Josselin Garnier gated. The direct method for seeking the linear IM of a University Paris Diderot given PDE system is presented and the existence of rich Laboratoire de Probabilités et Modèles Aléatoires linear IMs is a strong indication that the given PDE is in- [email protected] tegrable. We find that the first integrals of the reduced ordinary differential equation can be obtained by the inner Stefano Trillo symmetry of the IMs. It is conjectured that the new (2+1)- University of Ferrara dimensional differential-difference system is an integrable [email protected] system including the well-known KdV reduction.

Antonio Picozzi Yong Chen Universite’ de Bourgogne, Dijon, France Eastern China Normal University [email protected] [email protected] Yuqi Li Eastern China Normal University MS75 China [email protected] Order and Chaos in Fibre Lasers Senyue Lou We analyse the nonlinear stage of modulation instability Ningbo University, China in passively mode locked fiber lasers, leading to either sta- [email protected] ble or chaotic or noise-like emission. We present the diagram of phase transitions among stable temporal patterns MS76 and different regimes of chaotic emission in terms of the key cavity parameters such as dispersion and nonlinear- General Solutions of Arbitraray First Order Au- ity: amplitude or phase turbulence, and spatio-temporal tonomous PDEs intermittency. Whenever the polarizer is removed from the cavity, stabilization of the chaotic emission may occur via Using general symmetry approach, it is proved that ALL spatio-temporal synchronisation induced by the nonlinear solutions of arbitrary autonomous first order partial differ- coupling among two the orthogonal polarization modes. ent equations (PDEs) can be obtained. In other words, a first order autonomous PDE in any dimensions is C- integrable. Especially, for arbitrary two dimensional first Stefan Wabnitz order autonoumous PDEs, the recursion operators, group University of Brescia invariant operators, Lax pairs, higer order general local and NW14 Abstracts 123

nonlocal symmetries and group invariant functions are ex- we derive an asymptotic model for surface-internal wave plicitly given. interactions, where the nonlinear internal waves evolve according to a KdV equation, while the smaller-amplitude Senyue Lou surface waves are described by a linear Schrodinger equa- Ningbo University, China tion. For internal solitons of depression, the Schrodinger [email protected] equation is shown to be in the semi-classical regime and thus admits localized bound states. This leads to the phenomenon of trapped surface modes which propagate as the MS76 signature of the internal waves. Numerical simulations tak- On Nonintegrable Semidiscrete Hirota Equation: ing oceanic parameters into account are performed to il- Gauge Equivalent Structures and Dynamical Prop- lustrate this phenomenon. This is joint work with Walter erties Craig and Catherine Sulem.

In this talk, we investigate nonintegrable semidiscrete Hi- Philippe Guyenne rota equations including nonintegrable semidiscrete Hirota Department of Mathematical Sciences I equation and nonintegrable semidiscrete Hirota II equa- University of Delaware tion. We focus on the topics on gauge equivalent struc- [email protected] tures and dynamical behaviours for the two nonintegrable semidiscrete equations. By using the concept of the prescribed discrete curvature, we show that under the discrete MS77 gauge transformations, nonintegrable semidiscrete Hirota I Dynamics of Localized Solution in Nonlocal NLS and II equations are respectively gauge equivalent to the Equation with Double Well Potential nonintegrable generalized semidiscrete modified Heisen- berg ferromagnet equation and the nonintegrable gener- In this talk, we examine the combined effects of cubic and alized semidiscrete Heisenberg ferromagnet equation. We quintic terms of the long range type in the dynamics of a prove that the two discrete gauge transformations are re- double well potential. Employing a two-mode approxima- versible. We study the dynamical properties for the two tion, we systematically develop two cubic-quintic ordinary nonintegrable semidiscrete Hirota equations. The exact differential equations and assess the contributions of the spatial period solutions of the two nonintegrable semidis- long-range interactions in each of the relevant prefactors, crete Hirota equations are obtained through the construc- gauging how to simplify the ensuing dynamical system. Fi- tions of period orbits of the stationary discrete Hirota equa- nally, we obtain a reduced canonical description for the tions. We discuss the topic that the spatial period prop- conjugate variables of relative population imbalance and erty of the solution to nonintegrable semidiscrete Hirota relative phase between the two wells and proceed to a dy- equation is whether preserved to that of the correspond- namical systems analysis of the resulting pair of ordinary ing gauge equivalent nonintegrable semidiscrete equations differential equations. The relevant bifurcations, the sta- or not under the action of discrete gauge transformation. bility of the branches and their dynamical implications are We also give the numerical simulations for the stationary examined both in the reduced (ODE) and in the full (PDE) discrete Hirota equations. We find that their dynamics are setting. much richer than the ones of stationary discrete nonlinear Schrodinger equations. This is a joint work with Li-yuan Vassilis M. Rothos Ma. Aristotle University of Thessaloniki [email protected] Zuonong Zhu Department of Mathematics, Shanghai Jiaotong University, China MS77 [email protected] The Role of Radiation Loss in the Evolution of El- liptic Solitons in Nonlocal Soft Media

MS77 Optical solitary waves can propagate in a nematic liquid Mobile Localized Solutions for An Electron in Lat- crystal, which exhibits an intensity dependent refractive in- tices with Dispersive and Non-Dispersive Phonons dex change, balancing beam diffraction and self-focusing. The evolution of an elliptical nematicon with orbital an- We consider a one dimensional lattice in which an electron gular momentum is investigated. Modulation theory and can interact both with on-site nondispersive (Einstein) numerical solutions show that shed diffractive radiation is phonons and with longitudinal dispersive acoustic (Debye) critical to this evolution, which occurs in two phases. First, phonons and provide existence conditions for mobile local- angular momentum is shed via spiral waves, reshaping the ized electron excitations in the long wave limit. The role of beam. Secondly, diffractive radiation loss drives the beam both types of phonon modes on localization is also assessed, towards its steady state. together with a discussion of differences existing between the discrete and the continuum approaches. A striking re- Luke Sciberras sult is that, under certain conditions localized states can Universidad Nacional Autonoma de Mexico only be stable if they have a non-zero velocity. [email protected]

Luis Cisneros ESFM, Instituto Polit´ecnico Nacional MS78 [email protected] Waves on the Interface of Two-Layer Liquid System Subject to Longitudinal Vibrations: Stability and Collision of Solitons MS77 Surface Signature of Internal Waves We consider dynamics of the internal interface in two-layer liquid system subject to longitudinal vibrations. We derive Based on a Hamiltonian formulation of a two-layer ocean, the equations of average dynamics of large-scale patterns 124 NW14 Abstracts

on the interface, reveal a one-parametric family of solitons Sergey Shklyaev and show that the persisting system dynamics is merely Institute of Continuous Media Mechanics the kinetics of a set of these solitons. Collisions of solitons Perm, Russia are comprehensively studied. Class of collisions leading to [email protected] explosive layer rupture is revealed; formation of the rupture is both observed numerically and described analytically. Alexander Nepomnyashchy Technion Denis S. Goldobin Israel Institute of Technology Institute for Physics, University of Potsdam [email protected] Potsdam, Germany [email protected] MS78 Kseniya Kovalevskaya Theoretical Physics Department Symmetries and Parametric Instabilities in Vibrat- Perm State University ing Containers k [email protected] Symmetries, either exact or approximate, play an essential role in pattern forming systems, in particular under oscil- MS78 latory excitation, suppressing some of the resonances ex- Coherent Structures Interaction and Self- pected in the generic case and enhancing others. This will Organization in Dissipative Turbulence be illustrated considering parametric excitation of surface waves in finite containers in several distinguished regimes, We study interactions of coherent structures in a falling including single-mode and modulated patterns, horizontal film as a generic prototype of dissipative turbulence. Our and vertical excitation, and various excitation mechanisms analysis is based on a low-dimensional formulation and di- such as vibration of the whole container and immersed rect numerical simulations (DNS). A rich and complex dy- wave makers. namics is observed, with monotonic, underdamped or self- sustained oscillations, finding excellent DNS-theory agree- Jose Manuel Vega ment. We also consider a shear-thinning falling film that Universidad Politecnica de Madrid yields hysteresis on the coherent structures as the Reynolds [email protected] number increases which we quantify with appropriate stability analysis. MS79 Serafim Kalliadasis Department of Chemical Engineering Model Reduction for Waves in Networks Imperial College London [email protected] The propagation of localized waves in nonlinear networks is an ubiquitous problem. Examples are fluxon motion in Marc Pradas arrays of Josephson junctions, pulse propagation in the cir- Department of Mathematical Sciences culatory system. Modeling such problems is difficult and [email protected] it is helpful to simplify the equation and the geometry. We will illustrate these issues with the analysis of the propa- Te-Sheng Lin gation of sine-Gordon waves through Y junctions. Department of Mathematical Sciences Loughborough University Jean-Guy Caputo [email protected] INSA de Rouen France Dmitri Tseluiko [email protected] Department of Mathematical Sciences [email protected] MS79 MS78 Partial Continuum Limits for Exciton Pulses in Large Molecules Modulation of a Heat Flux in a Layer of Binary Mixture Numerical solutions of ODEs related to the discrete NLS We consider nonlinear dynamics of a longwave Marangoni equation often develop propagating pulses with slow spatial convection in a binary liquid layer subject to a modulated variation, which has motivated the search for approxima- heat flux at the bottom. It is shown that nonlinear evolutions by PDEs. In this talk, it is observed that these pat- tion for synchronous and subharmonic modes is governed terns are substantially different from previously-considered by the same set of standard Landau equations with the co- NLS approximations, and several new PDE approxima- efficients calculated numerically. For both modes the tran- tions are presented: systems related to the Airy PDE, and sition to the case of no modulation is nontrivial and the a quite different system describing oscillations near end- corresponding intermediate asymptotics are developed. points. Numerical solutions and linear analysis show good agreement with the long-wave phenomena in the ODEs. Irina Fayzrakhmanova Department of General Physics Brenton J. LeMesurier Perm State Technical University, Perm, Russia College of Charleston [email protected] Department of Mathematics NW14 Abstracts 125

[email protected] [email protected]

Tom J. Bridges MS79 University of Surrey Disorder in One-Dimensional Granular Crystals [email protected] We study the effects of multiple types of disorder — both correlated and uncorrelated — and nonlinearities on Matthew R. Turner the evolution of an initially localized excitation in a one- University of Brighton dimensional granular crystal. In the linear limit, we exam- [email protected] ine various regimes of subdiffusivity, superdiffusivity, and absence of diffusion. These regimes depend on the strength MS80 and type of disorder as well as on the normal modes of the linear spectrum. When we introduce nonlinearity, we ob- Transversality of Solitary Waves and Their Stabil- serve a gradual delocalization. ity Alejandro J. Martinez Variationnal partial differential equations like the Swift- Mathematical Institute Hohenberg equation or the Kawahara equation have an University of Oxford Hamiltonian steady part. This steady part can have an ho- [email protected] moclinic tangle, a phenomenon which implies the existence of an infinity of isolated waves or solitary waves for the PDE. In this talk, we will describe how the Lazutkin invari- MS79 ant, a quantity originally introduced to study the steady Model Reduction in Optics and Traffic Modeling part, can be related to the stability problem.

This talk introduces some recurring ideas and methods in Frederic Chardard model reduction by describing two apparently diverse prob- Universite Jean Monnet/Institut Camille Jordan lems where similar reduction techniques are useful and re- France lated wave phenomena arise. The first is the Zeno effect for [email protected] soliton type pulses in a nonlinear directional coupler with dissipation. The effect consists in increase of the coupler transparency with increase of the dissipative losses in one MS80 of the arms (joint work with F. Abdullaev et al). The sec- An Application of Maslov Index to the Stability ond is traffic modeling, and methods for reducing traffic Analysis for Standing Pulses jams through modulations of parameters, such as stochastic modulation of the safety distance (joint work with Yuri For the FitzHugh-Nagumo equations, a standing pulse so- Gaididei et al). lution is a homoclinic orbit of a second order Hamiltonian system. We employ the Maslov index theory to give certain Mads Sørensen criteria for the stability of standing pulse. Related results Department of Applied Mathematics and Computer are also applicable to more general skew-gradient systems. Science Technical University of Denmark [email protected] Chiao-Nen Chen Department of mathematics National Changhua University of Education, Taiwan, MS80 ROC Sloshing Dynamics with the Hamiltonian Particle- [email protected] mesh Method The Hamiltonian Particle-Mesh (HPM) method of [J. MS80 Frank, G. Gottwald & S. Reich. A Hamiltonian Stability of Periodic Waves in Hamiltonian Pdes particle-mesh method for the rotating shallow-water equations, Meshfree Methods for Partial Differential Equa- Many partial differential equations endowed with a Hamil- tions,Lecture Notes in Computational Science and Engi- tonian structure are known to admit rich families of peri- neering, Springer 26 131142 (2002).] is used to develop odic traveling waves. The stability theory for these waves a symplectic integrator in Eulerian-Lagrangian coordinate is by many respects still in its infancy though. The main systems for the problem of dynamic coupling between purpose of this talk is to point out a few results, for KdV- shallow-water sloshing and horizontal vehicle motion. A like equations and systems, that make the connection be- simple and fast numerical algorithm with excellent energy tween three kind of approaches: spectral, variational, and conservation over long times, based on the Störmer-Verlet modulational. method is implemented. Numerical simulations of the coupled dynamics are presented and compared to the results Miguel Rodrigues of [H. Alemi Ardakani & T.J. Bridges. Dynamic coupling University of Lyon between shallow-water sloshing and horizontal vehicle mo- [email protected] tion, European Journal of Applied Mathematics 21 479517 (2010).] where the coupled fluid-vehicle problem is analysed in the pure Lagrangian Particle-Path (LPP) setting. MS81 Experimental Observation of Unstable Mode 2 In- ternal Waves Hamid Alemi Ardakani Department of Mathematics The structure and stability of mode 2 internal solitary-like University of Surrey waves were investigated. The amplitude of the wave and 126 NW14 Abstracts

the offset of the pycnocline (with regard to the mid-depth Gravity Wave Packets of the water) were varied. In non-zero offset cases, it was found that the critical amplitude required for instability We compare and contrast existing and new results for the was less than in the zero offset counterpart case. It was Stokes drift and associated return flow for surface gravity also found that increasing the offset value led to increases wave packets on an unstratified fluid to new results for the in the asymmetry of the wave. Stokes drift and its “return flow’ for Boussinesq internal gravity wave packets on a linearly stratified fluid. We do Magda Carr so by exploring a perturbation expansion in two small pa- School of Mathematics and Statistics rameters, the steepness and the bandwidth of the packet, University of St Andrews, UK and provide a numerical validation. [email protected] TonvandenBremer,PaulTaylor Peter A Davies University of Oxford Department of Civil Engineering [email protected], University of Dundee, UK [email protected] [email protected] Bruce Sutherland Ruud Hoebers University of Alberta Department of Applied Physics [email protected] Eindhoven University of Technology, NL [email protected] MS82 Stablility of Localized Structure for a Semi-Arid MS81 Climate Model Hydraulic Falls Under a Floating Ice Plate This talk will give a brief overview of dynamical systems Steady two-dimensional nonlinear flexural-gravity hy- and climate. We will also detail recent stability results draulic falls past a submerged obstruction on the channel and multi-pulse interaction laws for N-pulse semi-strong bottom are considered. The fluid is assumed to be ideal solutions to the Gray-Scott model. This model has recently and is covered above by a thin ice plate. Cosserat the- been used to describe vegetative pattern formation in a ory is used to model the ice plate as a thin elastic shell, semi-arid ecosystem. and boundary integral equation techniques are employed to find critical flow solutions. By utilising a second ob- Thomas Bellsky struction, solutions with a train of waves trapped between Arizona State University the obstructions are investigated. [email protected]

Charlotte Page School of Computing Sciences MS82 University of East Anglia, Norwich, UK Model Reduction and Response for Two-Timescale [email protected] Systems Using Fluctuation-Dissipation

Emilian I. Parau Direct numerical simulation of a large multiscale system is University of East Anglia typically cost prohibitive. We present a method to gener- [email protected] ate reduced models for the slow variables of two-timescale ODE systems using an averaging method combined with the fluctuation-dissipation theorem to obtain a linear re- MS81 sponse correction term for the averaged fast dynamics. We Analysis and Computations of the Initial Value apply this method to a two-scale Lorenz ’96 model and Problem for Hydroelastic Waves analyze the statistics and perturbation response of the resulting reduced system. First, we summarize a new existence and uniqueness proof for the initial value problem for hydroelastic waves in 2D Marc Kjerland flow. The proof uses energy methods, following earlier work University of Illinois at Chicago by Ambrose on vortex sheets with surface tension. Sec- [email protected] ond, an efficient, nonstiff boundary integral method for 3D hydroelastic waves is presented. The stiffness is removed by developing a small-scale decomposition, in the spirit of MS82 prior work on 2D vortex sheets with surface tension by Tipping and Warning Signs for Patterns and Prop- Hou, Lowengrub and Shelley. The existence theory and agation Failure in SPDEs numerical method both rely on a formulation of the problem using a generalized isothermal parameterization of the In this talk, I shall report on recent results on early- warn- interface. This is joint work with David Ambrose. ing signs for pattern formation in stochastic partial differential equations. In particular, it will be shown that Michael Siegel classical scaling laws from stochastic ordinary differential New Jersey Institute of Technology equations can be carried over to the SPDE case. This is [email protected] illustrated in the context of the Swift- Hohenberg equation, analytically and numerically. Furthermore, I shall discuss numerical results for warning signs for the stochas- MS81 tic Fisher- KPP equation in the case of noisy invasion front Comparing Stokes Drift for Internal and Surface travelling waves near positive absorption probability events NW14 Abstracts 127

leading to propagation failure. applied electrical signal. We prove theoretically and numerically that by slightly changing the chemical potential, Karna V. Gowda a graphene coupler can switch from the bar to the cross Northwestern University state. [email protected] Costantino De Angelis Christian Kuehn University of Brescia, Brescia, Italy Vienna University of Technology [email protected] [email protected] Aldo Auditore, Andrea Locatelli University of Brescia MS82 [email protected], [email protected] Semi-Strong Desertification Dynamics with a Slowly Changing Parameter Alejandro Aceves Southern Methodist University We introduce a slowly changing parameter in the frame- [email protected] work of semi-strong interaction of pulses in reaction diffusion systems, which appear as models for vegetation in (semi-)arid regions. Geometric singular perturbation MS83 theory yields laws of motion for the pulses and spectral Self-Tuning Nonlinear Optical Systems (in)stability of the configuration. Crucial is understanding the interplay between the inherent slow dynamics of We demonstrate that the integration of data-driven ma- the pulse interactions and the rate of change of the chang- chine learning strategies with adaptive control produce an ing parameter: will ecosystem response be fast enough to efficient and optimal self-tuning algorithm for mode-locked postpone destabilization by a changing parameter? fiber lasers. The adaptive controller is capable of obtaining and maintaining high-energy, single-pulse states in a Eric Siero mode-locked fiber laser while the machine learning char- Leiden University, Mathematical Institute acterizes the cavity itself for rapid state identification and [email protected] improved optimization. The theory developed is demonstrated on a nonlinear polarization rotation (NPR) based Arjen Doelman laser using waveplate and polarizer angles to achieve opti- Mathematisch Instituut mal passive mode-locking despite large disturbances to the [email protected] system. The physically realizable objective function introduced divides the energy output by the fourth moment of Thomas Bellsky the pulse spectrum, thus balancing the total energy with Arizona State University the time duration of the mode-locked solution. The meth- [email protected] ods demonstrated can be implemented broadly to optical systems, or more generally to any self-tuning complex systems. MS83 Thermo-Optical Effects and Mode Instabilities in J. Nathan Kutz Large Mode Area Photonic Crystal Fibers University of Washington Dept of Applied Mathematics In recent years fiber laser systems have shown a rapid evo- [email protected] lution in terms of beam quality and power. High pulse energies and peak power require large effective area and Steven Bruton, Xing Fu new photonic crystal fiber designs are proposed. Thermo- Applied Mathematics optical effects can lead to transverse mode instability University of Washington (TMI), a nonlinear effect that sets in at a threshold power [email protected], [email protected] level. The numerical models of TMI for understanding the origin and mechanism behind, and for future mitigation strategies are discussed. MS83 Ultrafast Spatial and Temporal Soliton Dynamics Annamaria Cucinotta in Gas-Filled Hollow-Core Photonic Crystal Fibres Università degli Studi di Parma, Parma, Italy [email protected] Optical solitons form the core of the many of the most exciting nonlinear phenomena in nonlinear fibre optics for Enrico Coscelli, Federica Poli, Stefano Selleri example, they are the basis of a plethora of supercontin- University of Parma, Parma, Italy uum mechanisms. Next generation photonic-crystal fibres [email protected], [email protected], ste- filled with gas provide new opportunities and regimes to ex- [email protected] plore soliton physics, ranging from the influence of soliton- induced plasma, to the possibility of guiding and generating sub-cycle and vacuum-UV optical fields. A range of MS83 theoretical and experimental results will be discussed. Tuning of Surface Plasmon Polaritons Beat Length in Graphene Directional Couplers John Travers, Francesco Tani, Ka Fai Mak, Philipp Hölzer, Alexey Ermolov, Philip Russell We investigate the tuning of the coupling of surface plas- Max Planck Institute for the Science of Light, Erlangen, mon polaritons between two spatially separated graphene Germany layers. We demonstrate that the coupling coefficient in [email protected], [email protected], such structures can be easily controlled by means of an [email protected], [email protected], 128 NW14 Abstracts

[email protected], [email protected] play a rich variety of modes with similar behaviour.

Franck Plouraboue MS84 CNRS Standing Waves and Heteroclinic Networks in the [email protected] Nonlocal Complex Ginzburg-Landau Equation for Electrochemical Systems Adama Creppy, Olivier Praud Toulouse University, IMFT UMR 5502, 31400 Toulouse The nonlocal complex Ginzburg-Landau equation [V. [email protected], [email protected] Garcia-Morales and K. Krischer, Phys. Rev. Lett. 100, 054101 (2008)] describes the interaction between the elec- Pierre Degond trostatic potential and the oscillatory nonlinear kinetics Imperial College of London, Applied mathematics Dpt of spatially extended electrochemical systems. This equa- [email protected] tion is shown to exhibit heteroclinic netwroks and standing waves (under global coupling). The latter lead to complex Hui Yu interference patterns, with active subharmonic modes. Lin- Universit´e de Toulouse; UPS, INSA, UT1, UTM ; ear stability and symmetry analysis allow detailed features [email protected] of these coherent structures to be elucidated.

Vladimir Garcia-Morales Nan Wang Department of Physics National University of Singapore, Department of Technische Universit¨at M¨unchen Mathematics [email protected] [email protected]

MS84 MS84 Bifurcations in the Langmuir-Blodgett Transfer On Electrical Diﬀuse Layers in Ionic Liquids and Problem Heteroclinic-Type Connections

Spontaneous pattern formation in deposition processes at Studies of room temperature ionic liquids, showed that receding contact lines has become a versatile tool to coat electrical diffuse layers can be thick and exhibit spatially substrates with well controlled micro- and nanostructures. extended non-monotonic (oscillatory) and monotonic de- As a paradigmatic example, the coating of substrates with cays. These unconventional properties are fundamentally periodically structured monolayers has in recent years been different from traditional (dilute) electrolytes and demon- investigated by theoreticians and experimentalists [Köpf et strate the limited mechanistic understanding of highly con- al, New J. Phys. 14 (2012) 02316 and Langmuir 26 (2010) centrated electrolytes. To advance the understanding of 10444, Li et al, Small 8 (2012) 488] alike. Here, we present electrical diffuse layers in ionic liquids we use a semi- recent progress, allowing for the first time to understand phenomenological modified Poisson-Nernst-Planck equa- the intricate bifurcation diagram of the system that ex- tions and regulate weak dilutions. Using spatial dynamics hibits a snaking branch of stationary solutions. Each nose and numerical methods, we analyze distinct diffuse layer of the snake is connected to a branch of time periodic so- characteristics and provide for each type the analytic con- lutions. Using numerical continuation, we detect various ditions and the validity limits in terms of applied voltage, local and global bifurcations and investigate how the solu- domain size, molecular packing, and short range electro- tion structure depends on the system size. These results static correlations. We also discuss the qualitative general- are of wide interest for the theoretical description of pat- ity of the results via global heteroclinic-type connections. tern formation in systems with nontrivial boundary conditions. Arik Yochelis Michael H. Koepf Department of Chemical Engineering Département de Physique Technion - Israel Institute of Technology Ecole Normale supérieure [email protected] [email protected] MS85 Uwe Thiele Controlability of Schroedinger Equation with a Institute of Theoretical Physics Hartree Type Nonlinearity University of Muenster [email protected] We are concerned with the control problem for 2 MS84 iut = −uxx + α(x) u + (|x − y|−|x|)|u(y,t)| dy u, Coherent Structures in Confined Swarming Flows where the Hartree nonlinearity stems from the coupling Swarming flows display new interesting behaviours as- with the 1D Poisson equation, and α(x) ∈ C∞ has linear sociated with the collective dynamics of interacting ac- growth at infinity, including constant electric fields. We tive swimmers. We report new observations of collective shall show that for both small initial and target states, motion of concentrated semen arising inside an annulus and for distributed controls supported outside of a fixed through spontaneous symmetry breaking. The resulting compact interval, the model equation is controllable. We self-sustained dynamics shows various interesting regimes shall also give non controllability results for distributed such as constant rotation, oscillating rotation, and damped controls with compact support oscillations. A similar configuration is analyzed through a linear instability analysis of the Viksek model which dis- Mariano De Leo NW14 Abstracts 129

Universidad Nacional General Sarmiento force action perpendicular to rotation axis, are consid- [email protected] ered. Excitation of interface diﬀerential rotation is observed. With an increase of external forcing, an azimuthal wave, characterized by pronounced two-dimensional crests MS85 parallel to rotation axis, is excited on the interface accom- Exact and Approximate Solutions for Solitary panied with interface diﬀerential rotation velocity increase. Waves in Nematic Liquid Crystals Thresholds of interface stability are studied.

The system described solitary waves in nematic liquid Nikolay Kozlov crystals (NLC) is general, arising in many other areas of Department of General and Experimental Physics physics.. Isolated exact solutions and general variational Perm State Pedagogical University, Perm, Russia approximations are found for solitary waves in NLC’s. [email protected] These approximate solutions are compared with numerical solutions and the relative merits of different ansatze are Anastasiia Kozlova, Darya Shuvalova discussed. Physically, we find a type of bistability in the Laboratory of Vibrational Hydromechanics system and a minimum power for solitary waves to exist. Perm State Humanitarian Pedagogical University, Perm, Russia Michael MacNeil [email protected], [email protected] University of Edinburgh [email protected] MS86 MS85 Thin Film Flows over Spinning Discs: Numerical Simulation of Three-Dimensional Waves Localized Solutions and Traveling Waves in a Nonlocal Parametrically Forced Nonlinear We model the flow of a thin film over a rotating disc us- Schroedinger Equation ing the integral method coupled to the method of weighted residuals to derive a set of evolution equations for the in- The parametrically forced nonlinear Schroedinger equation terface, and the radial and azimuthal flow rates. These is a model for nonlinear dissipative systems, including de- equations account for centrifugal, inertial, and capillary tuned operation of optical parametric oscillators. It has forces and the transition from two-dimensional to three- been shown to support a rich variety of localized solu- dimensional waves as these waves travel towards the disc tions, including standing, periodic and (unstable) travel- periphery. We compare our results with those from full ing waves. We show how coupling this equation to a heat Navier-Stokes simulations. equation produces stable traveling waves whose dynamics can be understood through a simple reduction to collective Omar K. Matar coordinates, and we discuss their connection to solutions Imperial College London of the uncoupled equation. [email protected] Richard O. Moore New Jersey Institute of Technology MS86 [email protected] Compactons Induced by Nonconvex Advection ± 3 − 2 3 MS86 Using the model equation ut (u u )x +(u )xxx =0 we study the impact of a non-convex convection on forma- Continuously Quenched Pattern Dynamics in Dry- tion of compactons. In the ositive version, both traveling ing Liquid Films and stationary compactons are observed, whereas in the The nonlinear dynamics of Bnard-like surface-tension- negative branch, compactons may form only for a bounded driven patterns in thin liquid films evaporating into am- range of velocities. Depending on their relative speed, in- bient air is analyzed experimentally and theoretically. Due teraction of compactons may be close to being elastic or a to the layer thickness decrease, the average size of convec- fission process wherein the collision gives rise to additional tion cells continuously decreases by a fast mitosis process, compactons. which results in highly disordered structures. Their characteristics are studied using a long-wave convection model, Alex Oron which compares qualitatively well with experiments. A Department of Mechanical Engineering simple fitting-parameter-free model is also proposed for the Technion - Israel Institute of Technology, Haifa, Israel threshold thickness below which patterns disappear. [email protected]

Pierre Colinet Philip Rosenau ServicedeChimiePhysiqueE.P. Tel Aviv University Universite Libre de Bruxelles, Brussels, Belgium [email protected] [email protected]

MS87 MS86 Settling and Rising in Density Stratiﬁed Fluids: Wave Instability of a Rotating Liquid-Liquid Inter- Analysis and Experiments face Not Available at Time of Publication Experimental study is performed of rotating system of two immiscible liquids of diﬀerent density. Inertial waves, ex- Roberto Camassa cited on the centrifuged interface under external periodic University of North Carolina 130 NW14 Abstracts

[email protected] [email protected]

MS87 MS88 Transcritical Flow of a Stratiﬁed Fluid Over Topog- Multisymplectic Structure and the Stability of Soli- raphy and Non-classical Dispersive Shock Waves tary Waves

We study transcritical flow of a stratified fluid past a broad A challenge in the stability analysis of solitary waves is to localised topographic obstacle using the recent develop- find general conditions for stability or instability. In this ment of the nonlinear modulation theory for the forced talk it is shown how multi-symplectic structures leads to Gardner equation. We identify some of the wave struc- a general instability condition for a large class of solitary tures generated by the transcritical flow as dispersive coun- waves of Hamiltonian PDEs. It generalizes theory of Pego terparts of non-classical shocks and double waves occurring & Weinstein (1992) and Bridges & Derks (2001), and uses in the regularisation of non-convex hyperbolic conservation new results of Bridges & Chardard (2014) on transversality. laws. The talk is based on joint works with R. Grimshaw, It is shown that the product of the the Lazutkin-Treschev A. Kamchatnov and M. Hoefer. invariant, and dI/dc, the derivative of the momentum with respect to the speed, generates a sufficient condition for Gennady El linear instability of solitary waves. Loughborough University, United Kingdom [email protected] Tom J. Bridges University of Surrey [email protected] MS87 Solute Dynamics Within Settling Marine Snow at Density Discontinuities MS88 Standing Pulse Solution of Fitzhugh-Nagumo Not Available at Time of Publication Equations

Morten Iversen Using a variational formulation, we study standing pulse Marum - Geophysical Sciences solutions of the FitzHugh-Nagumo equations when the ac- Bremen University tivator diffusion coefficient is small compared to that of [email protected] the inhibitor. There is no global minimizer in this case, we therefore look for local minimizer by imposing appropriate MS87 topological constraints on the solution space. Interesting techniques are developed to deal with such classes of topo- Experiments and Theory for Porous Spheres Set- logical constraints. tling in Sharply Stratified Fluids Yung-Sze Choi Marine snow, aggregates composed of organic and inor- University of Connecticut ganic matter, play a major role in the carbon cycle. Most [email protected] of these particles are extremely porous, allowing diffusion of a stratifying agent from the ambient fluid to affect the density and therefore the settling dynamics. We study the MS88 case of a single spherical particle settling in water strati- Dispersive Shear Shallow Water Flows fied by salt, focusing on effects of porosity and diffusion. A parametric study of the settling behaviors and comparisons We derive a new dispersive model of shear shallow water between modeling and experiments will be presented. flows through the Hamilton principle. The model generalizes the well-known Green-Naghdi model. Shilpa Khatri Department of Mathematics Sergey Gavrilyuk University of North Carolina at Chapel Hill University Aix-Marseille, UMR CNRS 6595 IUSTI [email protected] [email protected]

Roberto Camassa University of North Carolina MS88 [email protected] A Hamiltonian Analogue of the Meandering Tran- sition Richard McLaughlin UNC Chapel Hill A Hamiltonian analogue of the meandering transition from [email protected] rotating waves to modulated traveling waves (TWs) in systems with Euclidean symmetry is presented. In dissipative Jennifer Prairie systems this transition is associated with a Hopf bifurca- University of North Carolina at Chapel Hill tion. In the Hamiltonian case, for example in models of [email protected] point vortex dynamics, the conserved quantities of the system are bifurcation parameters. Depending on symmetry, it is shown that either modulated TWs do not occur, Brian White or modulated TWs are the typical scenario near rotating University of North Carolina waves. [email protected] Claudia Wulﬀ Sungduk Yu University of Surrey University of North Carolina at Chapel Hill Dept. of Math and Statistics NW14 Abstracts 131

c.wulff@surrey.ac.uk discuss all waves, from small-amplitude waves to the highest one with a corner flow at the crest. This work inves- tigates singularity structure of this approximate solution, MS89 and compares the high-order Davies approximations with Axisymmetric Solitary Waves on a Ferrofluid Jet some other standard methods including long wave approximation. The propagation of axisymmetric solitary waves on the surface of a cylindrical ferrofluid jet subjected to a mag- Sunao Murashige netic field is investigated. A numerical method is used to Faculty of Systems Information Science compute fully-nonlinear travelling solitary waves and pre- Future University Hakodate dictions of elevation waves and depression waves by Ran- [email protected] nacher & Engel (2006) using a weakly-nonlinear theory are confirmed in the appropriate ranges of the magnetic Bond Wooyoung Choi number. New nonlinear branches of solitary wave solutions Dept of Mathematics are identified. As the Bond number is varied, the solitary New Jersey Institute of Tech wave profiles may approach a limiting configuration with [email protected] a trapped toroidal-shaped bubble, or they may approach a static wave if the wave speed approaches zero. For a sufficiently large axial rod, the limiting profile may exhibit a MS90 cusp. Automatic Recognition and Tagging of Topologi- cally Different Regimes in Dynamical Systems Mark Blyth School of Mathematics We discuss robust methods to characterize and detect University of East Anglia, Norwich, UK sudden shifts, eg. critical transitions, between different [email protected] regimes in stochastic dynamical systems, with examples from classical dynamics as well as real world climatologi- cal data. We develop techniques for the detection of such MS89 critical transitions from sparse observations contaminated On the Interaction Between Surface and Internal by noise. We present a machine learning framework which Waves accurately tags different regimes of a dynamical systems based on topologically persistent features near an attrac- We consider a system of two fluids of different densities tor. In particular, our methodology performs well in the bounded above by a free surface. The propagation of sur- context of periodic orbits and saddle-type bifurcations. face and internal waves generated by an external forcing and their interaction are studied both theoretically and Jesse Berwald experimentally. Institute for Mathematics and its Applicatons University of Minnesota Wooyoung Choi [email protected] Dept of Mathematics New Jersey Institute of Tech [email protected] MS90 Data Assimilation for Quadratic Dissipative Dy- namical Systems MS89 Computing the Pressure in Fully Nonlinear Long- Data assimilation refers to the incorporation of a model of a Wave Models for Surface Water Waves system together with data in the form of noisy observations of that system in order to infer the underlying state and/or The effect of a linear background shear flow on the pres- parameters of the system. In the case in which the data is sure beneath steady long gravity waves at the surface of received online and incorporated sequentially, it is referred a fluid is investigated. Using an asymptotic expansion for to as filtering. We present accuracy and well-posedness the streamfunction, we derive a model equation given in results for two prototypical filters, 3DVAR and EnKF, for terms of the background vorticity, the volume flux, the a broad class of quadratic dissipative dynamical systems, total head and the momentum flux. It is shown that a including Navier-Stokes, Lorenz ’96, and Lorenz ’63. strongly sheared flow leads to nonmonotonicity of the fluid pressure in the sense that the maximum pressure is not lo- Kody Law cated under the wave crest, and the pressure just beneath SRI UQ Center, CEMSE, KAUST the wavecrest can be below atmospheric pressure. [email protected] Henrik Kalisch University of Bergen MS90 [email protected] Mixed Mode Oscillations in Conceptual Climate Models: An In-depth Discussion

MS89 This talk analyzes a fast/slow system with one fast and two Davies Approximation of Levi-Civita’s Surface slow variables. Geometric singular perturbation theory is Condition for Water Waves in the Complex Do- used to demonstrate the existence of a folded node singu- main larity. A parameter regime is found in which the folded node leads to a stable MMO orbit through a generalized Levi-Civita’s surface condition is the free surface condition canard phenomenon. for an irrotational plane ﬂow using the logarithmic hodo- graph variable. This condition allows us to apply Davies’ Andrew Roberts approximation to the problem of steady water waves, and Department of Mathematics 132 NW14 Abstracts

The University of North Carolina at Chapel Hill [email protected] [email protected]

Esther Widiasih MS91 Department of Mathematics Orbital Stability of Solitary Waves of Moderate The University of Arizona Amplitude in Shallow Water [email protected] We study the orbital stability of solitary traveling wave solutions of the following equation for surface water waves Chris Jones of moderate amplitude in the shallow water regime: University of North Carolina-Chapel Hill 2 3 [email protected] ut+ux+6uux−6u ux+12u ux+uxxx−uxxt+14uuxxx+28uxuxx =0.

Martin Wechselberger Our approach is based on a method proposed by Grillakis, University of Sydney Shatah and Strauss in 1987, and relies on a reformulation [email protected] of the evolution equation in Hamiltonian form. We deduce stability of solitary waves by proving the convexity of a scalar function, which is based on two nonlinear functionals that are preserved under the ﬂow. MS90 Nilay Duruk Mutlubas Mixed Mode Oscillations in Conceptual Climate Istanbul Kemerburgaz University Models: A General Perspective [email protected]

Much work has been done on relaxation oscillations and Anna Geyer other simple oscillators in conceptual climate models. The University Autonoma de Barcelona oscillatory patterns frequently found in climate data how- [email protected] ever are often more complicated than what can be described by such mechanisms. In this talk, I will present some recent work incorporating ideas from the study of MS91 mixed mode oscillations into conceptual modeling of cli- Local Well-posedness for a Class of Nonlocal Evo- mate systems. lution Equations of Whitham Type

For a class of pseudodifferential evolution equations of the Esther Widiasih form Department of Mathematics u +(n(u)+Lu) =0, The University of Arizona t x [email protected] we prove local well-posedness for initial data in the Sobolev space H s, s>3/2. Here L is a linear Fourier multiplier with a real, even and bounded symbol m,andn is ∈ s R a real measurable function with n Hloc( ), s>3/2. MS91 The proof, which combines Kato’s approach to quasilinear The Nonlocal Nonlinear Wave Equation with Peri- equations with recent results for Nemytskii operators on odic Boundary Conditions: Analytical and Numer- general function spaces, applies equally well to the solitary ical Results and periodic cases. Mats Ehrnstrom We consider a general class of nonlinear nonlocal wave Department of Mathematical Sciences equation arising in one-dimensional nonlocal elasticity [N. Norwegian University of Science and Technology Duruk, H. A. Erbay, A. Erkip, ”Global existence and blow- [email protected] up for a class of nonlocal nonlinear Cauchy problems arising in elasticity”, Nonlinearity, 23, 107-118, (2010)]. The model involves a convolution operator with a general ker- MS91 nel function whose Fourier transform is nonnegative. We Traveling Surface Waves of Moderate Amplitude in focus on well-posedness and blow-up results for the nonlo- Shallow Water cal wave equation with periodic boundary conditions. We then propose a Fourier pseudo-spectral numerical method. We study traveling wave solutions of an equation for sur- To understand the structural properties of the solutions, face waves of moderate amplitude arising as a shallow water we present some numerical results illustrating the effects approximation of the Euler equations for inviscid, incom- of both the smoothness of the kernel function and the pressible and homogenous fluids. We obtain solitary waves strength of the nonlinear term on the solutions. This work of elevation and depression, including a family of solitary has been supported by the Scientific and Technological Re- waves with compact support, where the amplitude may in- search Council of Turkey (TUBITAK) under the project crease or decrease with respect to the wave speed. Our MFAG-113F114. approach is based on techniques from dynamical systems and relies on a reformulation of the evolution equation as an autonomous Hamiltonian system which facilitates an Handan Borluk explicit expression for bounded orbits in the phase plane Department of Mathematics, Isik Universty to establish existence of the corresponding periodic and [email protected] solitary traveling wave solutions. Gulcin M. Muslu Anna Geyer Department of Mathematics, Istanbul Technical University Autonoma de Barcelona University [email protected] NW14 Abstracts 133

Armengol Gasull phologies Universitat Autnoma de Barcelona Departament de Matemàtiques Abstract: Functionalized polymer membranes have a [email protected] strong affinity for solvent, imbibing it to form charge-lined networks which serve as charge-selective ion conductions in a host of energy conversion applications. We present a MS92 continuum model, based upon a reformulation of the Cahn- Hilliard free energy, which incorporates solvation energy Variational Models and Energy Landscapes Asso- and counter-ion entropy to stabilize a host of network mor- ciated with Self-Assembly phologies. We derive geometric evolution for co-dimension 1 bilayers and co-dimension two pore morphologies and In this talk, I will address two paradigms for self-assembly. show that the system possesses a simple mechanism for The first is via phase-field, energy-driven pattern formation competitive evolution and bifurcation of co-existing net- induced by competing short and long-range interactions. works through the common far-field chemical potential. A nonlocal perturbation (of Coulombic-type) to the well- known Ginzburg-Landau/Cahn-Hilliard free energy gives Keith Promislow rise to a mathematical paradigm with a rich and com- Michigan State University plex energy landscape. I will discuss some recent work [email protected] (with Dave Shirokoff and J.C. Nave at McGill) on developing methods for (i) assessing whether or not a particular Shibin Dai metastable state is a global minimizer and (ii) navigating Department of Mathematics from metastable states to states of lower energy. The sec- New Mexico State University ond paradigm is purely geometric and finite-dimensional: [email protected] Centroidal Voronoi tessellations (CVT) of rigid bodies. I will introduce a new fast algorithm for simulations of CVTs Greg Hayrapetyan of rigid bodies in 2D and 3D and focus on the CVT energy Department of Mathematics landscape. This is joint work with Lisa Larsson and J.C. Carnegie Mellon University Nave at McGill. [email protected] Rustum Choksi Department of Mathematics MS92 McGill University Multiple Water-Vegetation Feedback Loops Lead [email protected] to Complex Vegetation Diversity Through Com- peting Turing Mechanisms

MS92 In my presentation I will use the context of dryland vege- A Generalized Otha-Kawasaki Model with Asym- tation to study a general problem of complex pattern form- metric Long Range Interactions ing systems - multiple pattern-forming instabilities that are driven by distinct mechanisms but share the same spectral Energy-driven pattern formation induced by interactions properties. This study shows that the co-occurrence of two on various scales is common in many physical systems. The Turing instabilities, when the driving mechanisms coun- Ohta-Kawasaki model is a classic nonlocal Cahn-Hilliard teract each other in some region of the parameter space, model which gives rise to pattern formation driven by the results in the growth of a single mode rather than two inter- competition between short and long range interactions be- acting modes. The interplay between the two mechanisms tween phases A and B. The interactions described in the compensates for the simpler dynamics of a single mode by Otha-Kawasaki model are symmetric, i.e., phase A inter- inducing a wider variety of patterns, which implies higher acts with phase B in the same way phase B interacts with biodiversity in dryland ecosystems. phase A. Therefore, the only way to break symmetry is by ShaiKinast,YuvalR.Zelnik, Golan Bel introducing mass asymmetry between phase A and phase Department of Solar Energy and Environmental Physics B, a parameter which is not directly related to long range Ben-Gurion University interactions. In this talk, I will present a Generalized Otha- [email protected], [email protected], Kawasaki Model with asymmetric long range interactions, [email protected] whose derivation is motivated from a model of a binary mixture of charged phases. Of particular interest is the mass symmetric case, as in this case, any asymmetry be- Ehud Meron tween the two phases is directly related to the nature of Ben-Gurion University long-range interaction. I will present a systematic analy- [email protected] sis of the equation. In particular, I will demonstrate that proper tuning of the long range interaction gives rise to MS93 patterns which cannot exist in the Otha-Kawasaki model, and show that it is possible to attain isolated structures. Some Results About the Relativistic NLS Equation We study the existence of ground states of the non Nir Gavish linear√ pseudo-relativistic Schrödinger equation iψt = Department of Mathematics 2 2 2 m − ∂x − m ψ −|ψ| ψ, by using a suitable adapta- Technion ITT tion of the concentration-compactness principle. We prove [email protected] existence of the boosted ground states with profile ϕv ∈ H 1/2 (R). Also, we present results on the regularity of the ground states and on the weak orbital stability of the flow MS92 of the equation. In addition we present an efficient nu- Competitive Geometric Evolution of Network Mor- merical method to compute the profile ϕv of the boosted 134 NW14 Abstracts

ground state solution. [email protected]

Juan Pablo Borgna Universidad Nacional General Sarmiento MS94 [email protected] Mixing in Stratiﬁed Shear Flows: The Central Role of Coherent Structures

MS93 Stratified shear flow instabilities can modulate, and in some Interaction of Dark Nonlocal Solitons circumstances significantly accelerate, the transition to turbulence in stratified fluids, which are very common in the In typical nonlinear media the response to the propagating environment. We demonstrate that the life-cycles of such wave in a particular point is determined by the intensity coherent vortical primary and secondary instabilities play of the wave in the same point. On the other hand, in me- a central role in the commonly observed non-monotonic dia with nonlocal nonlinearity the nonlinear response in dependence of turbulent mixing on overall stratification, a particular location depends on the strength of the wave leading, through a classical physical mechanism first iden- in a certain neighbourhood of this location. Nonlocality tified by Phillips, to the inevitable, generic development of appears to be a generic feature of various nonlinear sys- layers in stratified fluids. tems ranging from optical beams to matter waves. It may result from certain transport processes such as, atom dif- Colm-cille Caulfield fusion, heat transfer or the long range of the inter-particle DAMPT interaction as in the case of nematic liquid crystals. Such a Cambridge University spatially nonlocal nonlinear response is also inherent to co- c.p.caulfi[email protected] herent excitations of Bose-Einstein condensates where it is due to the finite range of the inter-particle interaction po- Ali Mashayek tential. Propagation of waves in nonlocal media has been MIT extensively studied both theoretically and experimentally ali [email protected] in last decade. It turns out that nonlocality may dramat- ically affect propagation of waves, their localization and W. Richard Peltier stability. In particular, it has been shown that while nonlo- Physics cality generally slows down modulational instability it may University of Toronto actually promote it in defocusing media if the nonlocal re- [email protected] sponse function is sufficiently flat. Moreover, nonlocality has been shown to arrest collapse of finite beams and stabilize spatial solitons, including vortex and rotating solitons. MS94 In addition nonlocal response of the medium drastically af- Large Amplitude Solitary Waves and Dispersive fects soliton interaction. In this talk we will discuss propa- Shock Waves in Conduits of Viscous Liquids gation of dark spatial solitons in spatially nonlocal media. I will show how nonlocality affects properties of individ- A dispersive shock wave (DSW) represents the combina- ual dark solitons and their interaction. In particular we tion of solitary and linear dispersive wave phenomena into demonstrate that nonlocality introduces attractive force one coherent structure. DSWs are therefore fundamen- between normally repelling dark solitons and ultimately tal nonlinear structures that can occur in any conservative lead to the formation of their bound states. Our analytical hydrodynamic setting, e.g., superfluids and ”optical flu- and numerical results are confirmed by experiments with ids”. Experimental studies of DSWs in all media have been spatial dark solitons in weakly absorbing liquids. restricted by inherent physical limitations such as multidimensional instabilities, difficulties in capturing dynami- W. Krolikowski cal information, and, eventually, dissipation. These limit Australian National University DSW amplitudes, evolution time, and spatial extent. In [email protected] this talk, a new medium is proposed in which to study DSWs that overcomes all of these difficulties, allowing for the detailed, visual investigation of dispersive hydrody- MS93 namic phenomena. The vertical evolution of the interface Optical Solitons in Nematic Liquid Crystals: Con- between a buoyant, viscous liquid conduit surrounded by tinuous and Discrete Models a miscible, much more viscous fluid exhibits nonlinear self- steepening (wave breaking), dispersion, and no measurable We present some results on optical solitons in nematic liq- dissipation. First, it will be shown experimentally and the- uid crystals, and in waveguide arrays built from liquid crys- oretically that the three Lax categories of KdV two-soliton tals. The corresponding models are continuous and discrete interaction geometries extend into the strongly nonlinear cubic NLS equations with Hartee-type nonlinearities that regime. Then, DSW experiments and novel DSW-soliton involve Bessel potentials. We first review results on en- interaction behaviors will be presented and compared with ergy minimizing solitary wave solutions for the continuous modulation theory. The talk will cover multiple scales, 2-D case. We show the existence of smooth, radial, and from the microscopic (Navier-Stokes), mesoscopic (interfa- monotonic (up to symmetries) sollitons, and we also show cial conduit equation), and macroscopic (Whitham modu- that these solutions can only exist above a power thresh- lation equations), to the truth (experiments). old. This is joint work with T. Marchant, U. Wollongong, Australia. We also report on more recent work on the dis- Mark A. Hoefer crete 1-D problem, where we discuss existence and stability North Carolina State University of various types of breather solutions. [email protected]

Panayotis Panayotaros Nicholas K. Lowman Depto. Matematicas y Mecanica Department of Mathematics IIMAS-UNAM North Carolina State University NW14 Abstracts 135

[email protected] cient. The existence and uniqueness results of periodic and asymptotic travelling waves of the system are presented, Gennady El using dynamical system methods. We compare our analyt- Loughborough University, United Kingdom ical results with numerical simulation. [email protected] Makrina Agaoglou, Makrina Agaoglou Aristotle University of Thessaloniki MS94 University of Essex Buoyant Jets and Vortex Rings in Stratiﬁcation makrina [email protected], makrina [email protected]

We present theoretical, computational, and experimental Vassilis M. Rothos studies of the motion of buoyant fluid through a stratified Aristotle University of Thessaloniki background density field focusing on the evolution of jets [email protected] and vortex rings impinging upon sharp stratification. Both cases depict an interesting critical phenomena in which the Hadi Susanto buoyant fluid may escape or be trapped as the propagation The University of Nottingham distance is varied. For the case of jets, an exact solution United Kingdom is derived for the Morton-Taylor-Turner closure hierarchy [email protected] which yields a simple formula for this critical distance, both with and without a nonlinear ”entropy” condition. These Giorgos P. Veldes formulae will be compared directly to experimental mea- Nationaland Kapodistrian University of Athens surements. Additionally, analysis will be shown demon- [email protected] strating that the sharp two-layer background is the optimal stably stratified mixer. For the case of the buoyant vortex ring, full DNS simulations of the evolving ring impinging PP1 upon sharp stratification will compared directly with experimental measurements of the critical length. The Dual-Weighted Residual Method Applied to a Discontinuous Galerkin Discretization for the Shal- Richard McLaughlin low Water Wave Equations UNC Chapel Hill [email protected] The dual-weighted residual (DWR) method provides an error estimator which can be used as a mesh refinement criterion for adaptive methods.Our aim is to use the DWR MS94 scheme for goal oriented mesh refinement applied to a dis- On the Extreme Runup of Long Surface Waves on continuous Galerkin (DG) discretization of the shallow wa- a Vertical Barrier ter equations. Since the DG method allows discontinuous solutions the dual equations have discontinuous coeffi- The runup of long, strongly nonlinear waves impinging on cients. We solve the dual equations also by the DG method, a vertical barrier can result in a remarkable amplification using a Riemann solver designed to handle discontinuous (beyond 6 times) of the far-field amplitude of incoming coefficients. waves. Such an extreme runup is the result of an evolution process in which long waves experience strong ampli- Susanne Beckers,Jörn Behrens fication under the combined action of nonlinear steepening KlimaCampus, University of Hamburg and dispersion, followed by the formation of undular bores. [email protected], [email protected] In this work we study and discuss the conditions that result optimal for producing vertical runup, as well as some Winnifried Wollner of its consequences, by means of numerical simulations of Univeristy of Hamburg the free-surface Euler equations. In particular, we analyze [email protected] the pressure fluctuations on the wall during strong runup cycles. We show that non-hydrostatic effects can strongly affect the dynamic loads exerted on the wall, and that, as PP1 a consequence, the high-frequency component of the pres- Dipolar Bose-Einstein Condensates sure loads is significantly enhanced with respect to that of the wave spectrum itself. This observation suggests that In this poster we analyze the Gross-Pitaevskii equation also long oceanic waves can act as a source of seismic noise. modeling the dynamics of dipolar Bose-Einstein condensates: Claudio Viotti Department of Mathematical Sciences 1 iψ (x, t)=− Δψ(x, t)+V (x)ψ(x, t) University College Dublin t 2 [email protected] 2 2 +α |ψ(x, t)| ψ(x, t)+β(W ∗|ψ| )ψ(x, t), 3 t ∈ R>0,x∈ R . PP1 2 2 2 Bifurcation of Travelling Waves in Nonlinear Mag- where V (x)=ω1x1 + ω2x2 + ω3x3 states for a double-well netic Metamaterials harmonic trap potential (with ωk being the trap frecuency 2 1−3(x·n) in x direction) and W (x)= 3 is the kernel of the We consider a model of one-dimensional metamaterial k |x| formed by a discrete array of nonlinear split-ring resonators dipolar interaction potential (with n ∈ R3, |n| =1,being where each ring interacts with its nearest neighbors. At the dipole axis). first we study this problem without taking into account the normalized electro-motive force and the loss coeffi- We show some known facts concerning the existence 136 NW14 Abstracts

of solutions and well-posedness and present a time- ables, it is necessary to explicitly keep track of the energy of splitting method for computing the dynamics. the fast sub-system. Therefore, we develop a new stochastic mode reduction process in this case by introducing en- Roberto I. Ben ergy of the fast subsystem as an additional hidden slow Universidad Nacional de General Sarmiento variable. We use several prototype models to illustrate the [email protected] approach.

Ankita Jain PP1 Dept of Applied and Computational Mathematics and Oscillatory Pulses in the Fitzhugh-Nagumo Equa- Statistics tion University of Notre Dame [email protected] It is well known that the FitzHugh-Nagumo system exhibits stable, spatially monotone traveling pulses. Also, there is numerical evidence for the existence of spatially PP1 oscillatory pulses, which would allow for the construction Stochastic Analysis of Turbulent Mixing of multi-pulses. Here, we show the existence of oscillatory pulses rigorously, using geometric blow-up techniques and We study fluid mixing layers which grow out of acceler- singular perturbation theory. ation driven instabilities, including the classical cases of Rayleigh-Taylor instability, driven by a steady acceleration Paul A. Carter and Richtmyer-Meshkov instability, driven by an impulsive Department of Mathematics acceleration. Numerical simulations of the microphysical Brown University equations of fluid mixing are validated through compari- [email protected] son to laboratory experiments. We mathematically analyze properties of fluid mixing and averaged equations. Bjorn Sandstede Brown University Hyeonseong Jin bjorn [email protected] Jeju National University [email protected] PP1 Stability of Spatially Periodic Pulse Solutions in PP1 General Singularly Perturbed Reaction-Diffusion Abundant Soliton Solutions of the General Nonlo- Systems cal Nonlinear Schrodinger System with the Exter- nal Field The spectral stability of spatially periodic pulse solutions is studied in a general class of singularly perturbed reaction- Periodic and quasi-periodic breather multi-solitons solu- diffusion systems that significantly extends the theory tions, dipole-type breather soliton solution, the rogue wave for slowly non-linear equations of Gierer-Meinhardt/Gray- solution and the fission soliton solution of the general non- Scott type. The Evans function is approximated by a prod- local Schrodinger equation are derived by using the similar- uct of an analytic ‘fast’ component Df and a meromorphic ity transformation and manipulating the external potential ‘slow’ component Ds, corresponding to singular limits of function. And the stability of the exact solitary wave so- the stability problem. Stability is determined by Df , Ds lutions with the white noise perturbation is investigated and ‘small’ spectrum around 0. Each of these aspects are numerically.. analyzed in full asymptotic detail. Ji Lin Björn De Rijk Institute of Nonlinear Physics, Leiden University Zhejiang normal university [email protected] [email protected]

Arjen Doelman Xiaoyan Tang Mathematisch Instituut East China Normal University [email protected] [email protected]

Jens Rademacher University of Bremen PP1 [email protected] Elastic Nonlineal Model for a Unidimensional Chain with Clustering Zones Frits Veerman Oxford University We study weakly nonlineal localized oscillations in an quar- [email protected] tic elastic network model. Elastic network models describe sets of particles interacting through pairwise elastic potentials of finite range, We use a small displacement as- PP1 sumption to derive a quartic FPU-type model in which the Stochastic Mode-Reduction in Models with Con- number of interacting neighbours depends on the site. The servative Fast Sub-Systems spatial inhomogeneity of the interaction, specifically the presence of clusters of particles that interact with many We will consider application of the stochastic mode re- neighbours, leads to the existence localized linear modes. duction to multi-scale models with deterministic energy- Additionally we show examples where the properties of the conserving fast sub-system. Since there is energy exchange linear spectrum, and the nonlinear mode interaction coeffi- between the fast conservative sub-system and the slow vari- cients allow us to bring the system into a normal form that NW14 Abstracts 137

has invariant subspaces with additional symmetries and Department of Mathematics periodic orbits that represent spatially localized motions. Leiden University [email protected] Francisco J. Martinez Universidad Nacional Autonoma de Mexico Henk Schuttelaars sairaff@hotmail.com Delft University of Technology Department of Applied Mathematical Analysis [email protected] PP1 Scattering from a Large Cylinder with a Cluster of Eccentrically Embedded Cores PP1 Hydrodynamic Rogue Waves The onset of Anderson localization at millimeter/sub- millimeter wavelengths is experimentally studied by exam- The poster primarily discusses the hierarchy of rogue wave ining a scale model of cylindrical scatterers embedded into structures generated by the Peregrine soliton in envelope a large host cylinder. Analytic methods are developed to equations. This covers how to generate higher order so- compare the laboratory measurements with the theoretical lutions from those known and how these behave in higher predictions. A numerical implementation of these methods order models, such as the Dysthe equation. The poster also allows us to gain physical insight into the onset of Anderson touches on the stability of the Peregrine soliton to different localization in two and three dimensions. kinds of disturbances and how these can be generalised to other rational solutions of the Nonlinear Schrodinger equa- Brittany Mccollom tion. Colorado School of Mines [email protected] Daniel Ratliff University of Surrey Alex Yuffa D.Ratliff@uea.ac.uk U.S. Army Research Laboratory 2800 Powder Mill Road, Adelphi, MD 20783-1197 ayuff[email protected] PP1 Bifurcation of Travelling Waves in Nonlinear Mag- netic Meta-Materials PP1 Oscillons Near Hopf Bifurcations of Planar Reac- We consider a model of one-dimensional meta-material tion Diffusion Equations formed by a discrete array of nonlinear split-ring resonators where each ring interacts with its nearest neighbors. At Oscillons are planar, spatially localized, temporally oscil- first we study this problem without taking into account lating, radially symmetric structures. They have been the normalized electro-motive force and the loss coeffi- observed in several experimental media, including fluids, cient. The existence and uniqueness results of periodic and granular particles, and chemical reactions. Oscillons of- asymptotic travelling waves of the system are presented, ten arise near forced Hopf bifurcations. Such systems are using dynamical system methods. We compare our analyt- modeled mathematically by the forced complex Ginzburg- ical results with numerical simulation. Landau equation (CGL). We present a proof of the existence of oscillons CGL through a geometric blow-up anal- Vassilis M. Rothos ysis. Our analysis is complemented by a numerical contin- Aristotle University of Thessaloniki uation study of oscillons using Matlab and AUTO. [email protected]

Kelly Mcquighan, Bjorn Sandstede Hadi Sussanto Brown University Univeristy of Essex kelly [email protected], [email protected] bjorn [email protected] G.P Veldes Technological Institution of Lamia PP1 [email protected] Stability of Morphodynamical Equilibria in Tidal Basins PP1 Interesting patterns are observed in the tidal basins in Tracking Pattern Evolution Beyond Center Mani- The Wadden Sea. To get a better understanding of these fold Reductions with Singular Perturbations patterns, a morphodynamical model is constructed. This model describes the interaction between water motion, sed- We consider 2-component, singularly perturbed PDE- iment transport and bed evolution. The goal is to find the systems with a known stability spectrum, close to a bi- morphodynamic equilibria, to investigate their sensitivity furcation. A method which reduces them to 2D systems to parameter variations, and to understand the physical of ODEs governing the flow on an exponentially attracting mechanisms resulting in these equilibria. 2D manifold, is presented. Other than center manifold reduction, this method remains robust even if the primary Corine J. Meerman eigenvalue is of the same asymptotic magnitude as other Leiden University eigenvalues. We also show explicit systems for which the Mathematical Institute reduction is 3D or 5D, and contains chaos. [email protected] Lotte Sewalt Vivi Rottschäfer Leiden University 138 NW14 Abstracts

[email protected] Shanti Toenger CNRS Institut FEMTO-ST Arjen Doelman Besancon, France Mathematisch Instituut [email protected] [email protected] Goery Genty Vivi Rottsch¨afer Tampere University of Technology Department of Mathematics Tampere, Finland Leiden University goery.genty@tut.ﬁ [email protected] Frederic Dias Antonios Zagaris University College Dublin Universiteit Twente School of Mathematical Sciences [email protected] [email protected]

PP1 PP1 Reflection and Transmission of Plane Quasi Longi- Evolution Equations for Weakly Nonlinear Internal tudinal Waves at Semiconductor Elastic Solid In- Ocean Waves terface We follow a perturbation approach to derive an evolution This paper deals with the study of reflection and transmis- equation from the equations of momentum for long, weakly sion characteristics of acoustic waves at the interface of a nonlinear internal ocean waves for a two-layer system. At semiconductor half-space and elastic solid. the second order of the momentum equations, the extended Kortewegde Vries equation was derived in the literature. Amit Sharma For this work, we will attempt to rewrite the momentum National Institute of Technology Hamirpur equations at the third order and higher order. [email protected] Eric J. Tovar,DaqiXin The University of Texas-Pan American PP1 [email protected], [email protected] Robust Pulse Generators in An Excitable Medium with Jump-Type Heterogeneity Zhijun Qiao Department of Mathematics We study a spontaneous pulse-generating mechanism oc- [email protected] curred in an excitable medium with jump type heterogeneity. We investigate firstly the conditions for the onset of robust-type PGs, and secondly, we show the organizing PP1 center of their complex ordered sequence of pulse genera- Soliton Interactions in the Sasa-Satsuma Equation tion manners. To explore the global bifurcation structure on Nonzero Backgrounds of heterogeneity-induced ordered patterns (HIOPs) including PGs, we devise numerical frameworks to trace the long- In this poster, we obtain two families of anti-dark soliton term behaviors of PGs as periodic solutions, and we detect solutions of the Sasa-Satsuma equation on nonzero back- the associated terminal homoclinic orbits that are homo- grounds, one of which can the W-shaped solitary struc- clinic to a special type of HIOPs with a hyperbolic saddle. ture. The anti-dark solitons admit both the elastic collisions and resonant interactions, which has been first re- Takashi Teramoto ported in a one-dimensional nonlinear dispersive equation Chitose Institute of Science and Technology with only one field. In general, the solutions can exhibit [email protected] more complicated phenomena which are combined of elastic and resonant interactions.

PP1 Tao Xu On the Statistics of Localized Rogue Wave Struc- China University of Petroleum, Beijing tures in Spontaneous Modulation Instability [email protected]

We use numerical simulations of the stochastic NLSE to Min Li investigate the statistics of emergent intensity peaks in the North China Electric Power University chaotic ﬁeld generated from spontaneous modulation in- [email protected] stability. We show that these emergent structures display time and space properties well-described by analytic NLSE solutions including Akhmediev breather, Peregrine Soliton and Kuznetsov-Ma soliton forms. Examining the statistics of these structures allows us to associate the highest intensity rogue wave peaks as due to higher-order breather superpositions.

John Dudley Universite de Franche-Comte, Institut FEMTO-ST, UFR Sciences 16, route de Gray, 25030 Besancon Cedex, FRANCE [email protected]