42 NW10 Abstracts

IP1 two layers fluid in a scaling regime chosen to capture these Novel Electro-magnetic Phenomena: One-way- observations, in which the internal mode is treated as a waveguides and Wireless Power Transfer long wavelength nonlinear internal wave, while the surface mode is smaller and taken in a modulational regime. This We describe two unusual mechanisms for the transport of talk is based on joint work with Walter Craig and Philippe electromagnetic energy: ONE-WAY WAVEGUIDES: An Guyenne. ordinary waveguide transports waves going both forwards and backwards. A one-way waveguide supports only waves Catherine Sulem propagating in a single direction. Obstacles and disorder University of Toronto can no longer reflect waves, which exhibit 100% transmis- [email protected] sion even across seemingly impassible perfectly-conducting barriers. We explain how such phenomena, analogous to quantum-Hall edge states, can arise from gyromagnetic IP4 materials, and present our experimental demonstration. Extreme Elastohydrodynamics NONRADIATIVE WIRELESS POWER TRANSFER: We describe our work on a technique for wireless power trans- The borderlands between elasticity and hydrodynamics fer using strongly-coupled long-lived electro-magnetic res- lead naturally to a number of moving boundary prob- onances. Instead of irradiating the environment with elec- lems in elastohydrodynamics. I will discuss some phenom- tromagnetic waves, the source fills the space around it with ena in this rich area involving extreme geometries using a “non-radiative’ near-field. This technique could poten- a combination of experimental, analytical and numerical tially be used for middle-range power transfer, such as approaches: the flapping of a slender flag in a breeze and within a room, or a factory pavilion. its generalizations, the generation and propagation of elas- tohydrodynamic solitary waves, the dynamics of gait and Marin Soljacic speed selection in fish swimming far from and close to a MIT wall, and the mechanics and control of song production in [email protected] birds. L. Mahadevan IP2 Harvard University Spin Torque with Point Contacts: Generating Non- [email protected] linear Waves with Magnetic Media in a Localized Way IP5 Spin torque offers new means to locally excite nonlinear Coherent Dynamics in Large-scale Cortical Net- dynamics in magnets. Possible applications include oscil- works lators, disk drive heads, and non-volatile memory. Theo- retical analysis is made challenging by the inherent diffi- Spatially structured oscillations and waves arise both in culties of stiff differential equations with non-local terms. vivo and in vitro, and may be observed experimentally us- Originally formulated in 1999, analysis of the nonlinear ing multi-electrode arrays or voltage-sensitive dye optical Landau-Lifshitz-Slonczewski equation has proceeded either imaging. Such organizing activity in the brain has been by perturbative amplitude expansion or brute force micro- purported to play a role in sensory perception, the execu- magnetic simulations. I will survey the current state-of- tion of motor commands, working memory, early cortical the-art, with and emphasis on efforts to accelerate simula- development and certain brain pathologies such as epilepsy. tion algorithms for infinite domains, and analysis of a novel In this talk we describe various forms of coherent brain dy- magnetic excitation: the magnon ’droplet.’ namics including travelling pulses, spiral waves, spatially periodic patterns, and neuronal avalanches. We then show Tom Silva how to analyze such phenomena using population-based The National Institute of Standards and Technology continuum neural field theories. [email protected] Paul Bressloff University of Oxford IP3 Mathematical Institute Interaction Between Internal and Surface Waves in bressloff@maths.ox.ac.uk aTwoLayersFluid We consider a situation in which a fluid is composed of IP6 two essentially immiscible layers separated by a sharp in- Numerical Relativity: An Overview of the Field terface such as a thermocline or a pycnocline of differen- and Some Recent Results in Black Hole Simula- tial salinity. Internal waves of various types are commonly tions generated in the worlds oceans, and large amplitude, long wavelength nonlinear waves can be produced in the inter- In this talk I will give a review of the field of numerical face and propagate over large distances. In some physically relativity. I will discuss the 3+1 split of spacetime and realistic instances, the visible signature of internal waves on the basic equations that need to be solved. I will briefly the surface of the ocean is a band of roughness which prop- discuss the problem of finding initial data and of choosing agates at the same velocity as the internal wave. Several appropriate coordinate (gauge) conditions, concentrating of the earliest observations are the most striking, consist- particularly in the case of black hole spacetimes. I will ing of long brightly shining strips of many kilometers in also briefly discuss the recent spectacular breakthroughs extent, visible through the effect of the reflection at an in the simulation of binary black hole systems, and some oblique angle of the setting sun, and photographed from of the more interesting and exciting physical results that the Space Shuttle. I will discuss the asymptotic analysis of have been obtained from those simulations. the coupling between the interface and the free surface of a Miguel Alcubierre NW10 Abstracts 43

Nuclear Sciences Institute and new error bounds are derived. Furthermore, the CFM Universidad Nacional Autonoma de Mexico is applied to Bose-Einstein condensates(BEC), and numer- [email protected] ical results are coincident with the theoretical analysis.

Zi Cai Li IP7 National Sun Yet-sen University Lax Equations and Kinetic Theory for Shock Clus- [email protected] tering and Burgers Turbulence Cheng Sheng Chien Much of our current understanding of statistical theories Ching Yun University of turbulence relies on vastly simplified caricatures. One [email protected] such caricature is Burgers turbulence. This is the study of the statistics of shocks in Burgers equation with random initial data or forcing. This model also arises in statis- CP1 tics, combinatorics, and models of coagulation and surface A Surface Tracking Algorithm for Fast Computa- growth. It is of wide interest as a benchmark, even if it de- tion of Lagrangian Coherent Structures in 3D scribes phenomena that are not entirely turbulent. I will describe a kinetic theory for shock clustering that applies Lagrangian Coherent Structures (LCS) have recently to all scalar conservation laws with convex flux and a basic emerged as a powerful tool for analysing transport and mix- class of random initial data. A remarkable feature of the ing in fluid flows with arbitrary time dependence. However, kinetic theory is that it is presented as a Lax pair, admits the large computational time required to compute LCS re- remarkable exact solutions, and seems to have deep connec- mains a major hurdle. In this talk, we outline a surface tions with completely integrable systems. The talk relies tracking algorithm for LCS extraction. By performing cal- heavily on joint work with Bob Pego and Ravi Srinivasan. culations only near the LCS surfaces (instead of over the entire flow domain) large speed ups are achieved.

Govind Menon Douglas M. Lipinski, Kamran Mohseni Brown University University of Colorado at Boulder [email protected] [email protected], [email protected]

CP1 CP1 Three-Dimensional Vortex Soliton Interaction in Bifurcation Analysis of a Coupled Landau-Lifshitz- Waveguide Arrays Slonczewski Equations Three-dimensional vortex propagation in two-dimensional We consider a system of two Landau-Lifshitz equations arrays of optical waveguides presenting dispersion, nonlin- with spin-transfer (Slonczewski) terms that model dynam- earity and discrete diffraction, is investigated by means of ics of two macrospins in a spin-torque device. Uniaxial both variational analysis and numerics. Several structures symmetry conditions are assumed. We perform stability with vorticity S=1 or 2 are studied, among which some analysis based on reduction of this system of four ODEs vortices with S=1 are stable. Interactions are numerically to just one complex equation that can be solved explicitly. investigated, four typical outcomes are identified, depend- This allows to avoid a formidable task of finding eigenvalues ing on the initial relative velocity: rebound of slow solitons, of four-dimensional matrix with six parameters. Analytic fusion, splitting, and quasielastic interactions of fast soli- conditions of stability are obtained and the bifurcation di- tons. agram is constructed. To find periodic solutions, another reduction of the original system is derived. Fixed points Herve Leblond of this reduced system determine circular periodic orbits LPhiA, EA 4464 with phase locking. University of Angers [email protected] Lydia S. Novozhilova Western Connecticut State University Boris Malomed [email protected] Department of Physical Electronics, Faculty of Engr. Tel Aviv University, 69978 ISRAEL Sergei Urazhdin malomed@eng.tau.ac.il West Virginia University [email protected] Dumitru Mihalache Horia Hulubei National Institute for Physics and Nuclear Engineering, Bucharest CP1 [email protected] A Class of Vortex Filament Solitons in Fluids, Plas- mas and Superconductors

CP1 We consider the Uby-Isichenko-Yankov equation (UIY) for Collocation Fourier Methods for Bose-Einsterin vortex filament dynamics in plasmas and superconductors. Condensates This is a perturbation of the Localized Induction Equa- tion (LIE), which is itself an integrable model of filament There seems to exist few reports for non-periodical solu- motion in an ideal fluid.We present a novel class of solutions by spectral Fourier methods under the Dirichlet con- tions for a time-modulated modification of UIY; namely, ditions. In this paper, we will explore the spectral Fourier vortex configurations which are evolving spherical curves method(SFM) and the collocation Fourier methods(CFM) of varying radius. These solutions can be considered as for linear and nonlinear elliptic and eigenvalue problems, generalizations of a well-known class of soliton solutions 44 NW10 Abstracts

for LIE. [email protected]

Ronald Perline Drexel University CP2 [email protected] Exact Hamiltonian Equations for Fluid Films

Two dimensional models for hydrodynamic systems, such CP1 as soap films, have been studied for hundreds of years. Yet A Reaction-Diffusion Model of Persistent Bacterial there has not existed a fully nonlinear system of dynamic Infection equations analogous to the classical Euler equations. We propose an exact nonlinear system for the dynamics of a Persisters are phenotypic bacterial variants which exhibit fluid film. The system is derived in the classical Hamilto- ”antibiotic tolerance”: they survive when exposed to bacte- nian framework and neither the velocities nor the deviation ricidal antibiotics, but the bacterial colonies regrown from from the equilibrium are assumed small. We discuss the persister cells remain susceptible to further antibiotic treat- properties of the proposed equations and results of numer- ment. Exactly how bacteria switch into a persister state is ical simulations. unknown, but persister formation is linked to quorum sensing (the ability of bacteria to detect the local population Pavel Grinfeld density). Persisters are also implicated in the formation Drexel University and maintenance of biofilms, and they are believed to play [email protected] an important role in chronic bacterial infections that resist antibiotic treatment. Extending a prior model, we present a reaction-diffusion PDE model of bacterial growth which CP2 incorporates nutrient- and population-density dependent Extracting Solitons from Perturbed and Noisy Sig- mechanisms of persister formation. The functional form nals of this dependence is chosen to match experimentally observed relationships between nutrient availability, popula- When studying optical solitons in the presence of perturbation growth rates, and incidence of persister phenotypes. tions, it is often required to extract the underlying soliton These relationships involve a separation of temporal and from a perturbed signal. We propose an iterative method spatial scales in the formation of persister vs. regular bac- for doing this, and show it converges for sufficiently small teria. We investigate the pattern formation properties of perturbations. In addition, we show the specific pertur- the model as a number of experimentally accesible param- bations that most easily cause divergence of the iteration. eters are varied. We also present examples showing improved agreement (in comparison with more commonly used methods) between William E. Sherwood results from soliton perturbation theory and numerical sim- Center for Biodynamics ulations. Boston University [email protected] Jinglai Li Department of Engineering Sciences and Applied John Burke Mathematics Boston University Northwestern University [email protected] [email protected] William Kath CP2 Northwestern University Existence and Stability Analysis of 0 − π − 0 Finite Engineering Sciences and Applied Mathematics Josephson Junctions [email protected]

We investigate analytically and numerically a Josephson junction on finite domain with two π- discontinuity points CP2 characterized by a jump of π in the phase difference of the Bifurcation Analysis of a Model in Population Dy- junction, i.e. a 0 − π − 0 Josephson junction. The sys- namics tem is described by a modified sine-Gordon equation. We show that there is an instability region in which semiflux- Bifurcation analysis of a model in population dynamics ons will be spontaneously generated. Using a Hamiltonian with explicit construction of the Poincare map. We per- energy characterization, it is shown how the existence of form bifurcation analysis of a logistic model with periodic static semifluxons depends on the length of the junction, harvesting using transformation of the model to Mathieu the facet length, and the applied bias current. The critical equation. This transformation allows to explicitly find eigenvalue of the semifluxons is discussed as well. Numeri- the Poincare map, which is a very rare case for nontriv- cal simulations are presented, accompanying our analytical ial models. We construct a bifurcation diagram in the results. two-dimensional parameter space showing regions with two and zero periodic solutions separated by transition curves Saeed Ahmad where only one periodic solution exists. Stability of the pe- The University of Nottingham, Park Campus, NG7 2RD riodic solutions is also determined. Using the same ideas, Nottingham, UK more general and realistic model with periodic coefficients [email protected] can be analyzed. For this model, the same transformations lead to Hill’s equation that can be solved approximately Hadi Susanto, Jonathan A.D Wattis using Hill’s determinant. The University of Nottingham United Kingdom Ryan Doychak, Lydia S. Novozhilova [email protected], Western Connecticut State University NW10 Abstracts 45

[email protected], [email protected] ports a family of solitary waves.

Philippe Caillol CP2 University of Concepcion, C-160, Chile Dynamical System Analysis of a Transmission Line [email protected] Connected to Nonlinear Circuits R. H. Grimshaw In a transmission line oscillator a linear wave travels along University of Loughborough, UK a piece of cable, the transmission line, and interacts with [email protected] terminating electrical components. Diodes are integrated into almost all electronic devices as a means of protect- ing the logic circuits from destructive outside signals and CP3 high-voltage discharges. In the simple network model that Gasdynamic Aspects of the Analytical Passage we will present, we show nonlinear and chaotic effects as- from Isentropic to Anisentropic: Contrasts, Con- sociated with diodes that if transmitted into the primary sonances, and Classifying Remarks circuitry will disrupt or possibly damage the device. Two genuinely nonlinear approaches [isentropic (of a Bur- Ioana A. Triandaf nat type), respectively anisentropic (of a minimal Martin Naval Research Laboratory type) are associated each with a pair of regular classes Plasma Physics Division of solutions [waves, wave-wave interactions]. We use [email protected] two concurrent parallels [isentropic-anisentropic, respectively anisentropic-anisentropic (for two distinct anisentropic constructions)] to describe, by identifying some sig- CP3 nificant highly nontrivial contrasts and consonances, the Generalization of the Double Reduction Theory fragility of the analytical passage, on the mentioned pairs of classes, from an isentropic context to an anisentropic In a recent work [1, 2] Sjoberg remarked that generaliza- context. tion of the double reduction theory to partial differential equations of higher dimensions is still an open problem. In LiviuFlorinDinu this note we have attempted to provide this generalization Institute of Mathematics of the Romanian Academy to find invariant solution for a non linear system of qth or- [email protected], [email protected] der partial differential equations with n independent and m dependent variables provided that the non linear system of Marina Ileana Dinu partial differential equations admits a nontrivial conserved Polytechnical University of Bucharest, Dept. of Math. III form which has at least one associated symmetry in every ROMANIA reduction. In order to give an application of the proce- [email protected] dure we apply it to the nonlinear (2 + 1) wave equation for arbitrary function f (u) and g(u). CP3 Ahmad Y. Al-Dweik King Fahd University of Petroleum and Minerals Instability of the Split-Step Fourier Method on the [email protected] Background of a Solitary Wave We analyze a numerical instability that occurs in the split- Ashfaque H. Bokhari step method on the background of a soliton. This instabil- King Fahd University of Petroleum and Minerals ity is very sensitive to small changes of parameters of both Dhahran 31261, Saudi Arabia the numerical grid and the soliton, unlike the instability [email protected] of most finite-difference schemes. Moreover, the principle of ”frozen coefficients”, in which variable coefficients are Fiazud Din Zaman treated as ”locally constant”, is strongly violated for this Mathematical Sciences Department instability. (That is, the instability on the background of King Fahd Univeristy of Petroleum & Minerals a plane wave having the same amplitude as the soliton, [email protected] would be drastically different.) Our analysis is enabled by the fact that the period of oscillations of unstable Fourier Abdul Hamid Kara, Fazal Mahomed modes is much smaller than the width of the soliton. University of the Witwatersrand. [email protected], [email protected] Taras Lakoba University of Vermont Department of Mathematics and Statistics CP3 [email protected] Internal Solitary Waves with a Weakly Stratified Critical Layer CP3 This study considers the evolution of long internal waves. High-Order Discontinuous Galerkin Schemes for The weak amplitude assumption breaks down due to the 2+2 Vlasov Models on Unstructured Grids presence of a critical layer at a certain height which strongly modifies the flow. Nonlinear effects are invoked The purpose of this work is to explore mesh-based alterna- to resolve this singularity. This theory is relevant for the tives to the Particle-In-Cell (PIC) methods that are cur- phenomenon of internal-wave breaking and eventual sat- rently favored in many plasma physics applications. In uration. Spatially localized solutions are described by an particular, we present here a technique for solving the integrable modified Korteweg-de-Vries equation which sup- Vlasov-Poisson system that is based on operator splitting in time and high-order discontinuous Galerkin discretiza- 46 NW10 Abstracts

tions in space. The efficiency of the method is increased by CP4 considering semi-Lagrangian approaches for the advection Internal Waves and Wave Turbulence in phase space. The flexibility of the approach is enhanced by allowing the mesh in physical space to be unstructured. In this talk I will explain how one can use wave turbulence to attempt to explain the spectral energy density of David Seal, James A. Rossmanith internal waves in the ocean. I will explain basics of wave University of Wisconsin turbulence and explain how to use it oceanographic inter- Department of Mathematics nal waves. I will demonstrate that using traditional for- [email protected], [email protected] mulation of wave turbulence one runs to internal logical contradictions. Namely the results of the theory (strong nonlinearity) contradict the underlying assumptions (weak CP4 nonlinearity) used to build the theory. I will demonstrate Nonlinear Waves Interaction and Defects Dynam- possible directions out of the puzzle and will elaborate on ics in Complex (dusty) Plasmas open questions and challenges. Complex plasmas are suspensions of micro-particles in Yuri V. Lvov plasmas. They can be solid, liquid, or gaseous and ex- Rensselaer Polytechnic Institute hibit dynamic effects. The micro-particles can be traced [email protected] individually yielding full kinetic information about the system. Thus complex plasmas can be used as experimental model systems of solids and liquids to understand funda- CP4 mental properties of matter. Here we report an experimen- Kolmogorov-Like Cascades of Displayed and Hid- tal and numerical study of soliton interaction and steepen- den Perturbations of the Couette Flow ing, shock wave propagation, and defect dynamics in mono- layer lattices. The Kolmogorov-like cascades of velocity and shear produced by displayed and hidden perturbations of the Cou- Celine Durniak ette flow are treated by the multiscale n-section method Department of Electrical Engineering and Electronics, for resolving fluid-dynamic clusters of zeros and extrema. The University of Livepool The cascades are computed by a new approach of numerical [email protected] computing of high-order derivatives of generating functions with an arbitrary precision, which is adapted for trigono- Dmitry Samsonov, Paul Harvey metric and hyperbolic invariant differential structures. Dept. of Electrical Engineering and Electronics, The University of Liverpool, Liverpool, L69 3GJ, UK. Stanislav V. Miroshnikov [email protected], [email protected] Syncsort Inc. USA Sergey Zhdanov, Gregor Morfill [email protected] Max-Planck-Institut für Extraterrestrische Physik, D-85740 CP5 Germany [email protected], [email protected] Soliton Reflection in the Defocusing Nonlinear Schroedinger Equation with Non-Zero Boundary Conditions CP4 We discuss boundary value problems (BVPs) for the de- Investigation of Rarefaction Waves in Discrete Ma- focusing nonlinear Schroedinger equation with non-zero terials with a Strain-Softening Behavior boundary conditions (BCs) at infinity and linearizable BCs We investigate the stationary propagation of rarefaction at the origin. We show that, unlike the initial-value prob- waves in a discrete system with an abnormal (softening) lem, in these BVPs discrete eigenvalues appear in sym- interaction force that supports rarefaction waves as sta- metric pairs, and we derive the corresponding symmetries tionary disturbances in nondissipative chains. However, for the norming constants. The apparent reflection expe- stationary compression shock or solitary waves are not sup- rienced the solitons is due to the presence of a “mirror’ ported. Experimental observations of rarefaction waves in soliton located beyond the edge of the spatial domain. one-dimensional systems composed of steel cylinders and Anh M. Bui Boi layers of strongly nonlinear materials with abnormal be- State University of New York at Buffalo havior are compared to theoretical and numerical analysis ambuiboi@buffalo.edu of stationary solutions for rarefaction shock and solitary waves. Gino Biondini Eric B. Herbold State University of New York at Buffalo Georgia Inst. of Tech., Atlanta, Georgia Department of Mathematics High Explosives Research and Development Branch, Eglin biondini@buffalo.edu AFB [email protected] CP5 Vitali Nesterenko Steady Flow of a Falling Jet Hitting a Horizontal University of California at San Diego Wall [email protected] We consider a steady two-dimensional jet falling from a vertical pipe. Depending on the elevation of the pipe relative to an infinite horizontal wall and the Froude number, NW10 Abstracts 47

we study ﬂows with or without stagnation points. The [email protected] problem is reformulated using conformal mappings and is solved by a collocation Galerkin method. A particular form Jieqiang Tan, Andy Chan is assumed for the solution satisfying Bernoullis equation at The University of Nottingham Malaysia Campus certain discrete points. The resulting equations are solved [email protected], [email protected] by Newtons method.

Paul Christodoulides CP6 Cyprus University of Technology The Propagation of Finite Amplitude Pressure Faculty of Engineering and Technology Waves Through a Gas with a Step Temperature [email protected] Proﬁle

Frederic Dias A numerical study of wave propagation through gases with University College Dublin a step temperature distribution will be presented. This School of Mathematical Sciences is done to identify the physical mechanisms underlying [email protected] the reflection and transmission that occur when a high- intensity pressure wave interacts with a temperature gradient. Particular emphasis is paid to the effects of non- CP5 linearity through comparison with the acoustic case. An Random Attractors for Stochastic Sine-Gordon understanding of these mechanisms will aid in the evalua- Lattice Systems tion of thermal barrier effectiveness in attenuating nonlinear waves. The Sine-Gordon equation is a nonlinear hyperbolic differential equation which has a wide range of applications Nicholas Dizinno in physics. We study the asymptotic behavior of solutions Polytechnic Institute of NYU to the stochastic sine-Gordon lattice equations with mul- [email protected] tiplicative white noise. We firrst prove the existence and uniqueness of solutions, and then establish the existence of tempered random bounded absorbing sets and global CP6 random attractors. Radiating Instabilities of Zonal Jets in a Rotating Ocean Xiaoying Han Auburn University We consider interaction of internal gravity waves and a [email protected] zonal surface current (such as the Gulf Stream) within the framework of the reduced gravity shallow water model for a rotating ocean. Two aspects of the problem are studied: CP5 the scattering of each individual wave by the jet and the Black Holes As Inducers of Light collective effect that weak wave turbulence impose on the mean flow. The wave equation contains so-called critical Soon gravitational waves will be another tool for observing layer where momentum exchange between the mean flow our cosmos. Among interesting systems, those that can and waves occurs resulting in over-reflection or absorption radiate in both gravitational and electromagnetic waves of the latter. Inertially unstable flows generate propagating stand out as ideal candidates to probe deeply into our un- wave instabilities where gravity waves are spontaneously derstanding of physics at extreme regimes. In this talk i’ll emitted. discuss a few systems, efforts to study them and significant results obtained through different efforts. Vladimir N. Lapin University of Limerick Luis Lehner [email protected] Perimeter Institute for Theoretical Physics and University of Guelph [email protected] CP6 Transversal Waves of Dual Perturbations of the Poiseuille Flow CP5 Surface Wave Packet Propagation over Flat and Spatiotemporal cascades of dual perturbations of the tran- Slowly Varying Bottoms sitional Poiseuille flow are considered. Three alternative approaches to the cascade description through a spatial The initial value problem of the one-dimensional nonlin- cascade, a temporal cascade, and a spatiotemporal cascade ear Schrödinger (NLS) equation with constant coefficients are compared. Duality of displayed and hidden perturba- can be solved by using compact finite difference scheme. tions is resolved through invariant differential structures of In this talk, the similar scheme is implemented in the sig- Jacobian elliptic functions and justified by an asymptotic naling problem of the one-dimensional NLS equation with Hamiltonian approach. constant coefficient where it describes the propagation of surface gravity wave packet over flat bottom. Various ex- Victor A. Miroshnikov amples are illustrated. The similar compact finite differ- College of Mount Saint Vincent ence scheme is applied further in the signaling problem [email protected] of the one-dimensional NLS equation with variable coefficients which models the surface wave packet propagation over slowly varying bottom. CP6 Singularities in the Complex Physical Plane for Natanael Karjanto University of Twente 48 NW10 Abstracts

Deep Water Waves Department of Mathematics [email protected] Results from boundary integral techniques for deep water waves indicate the presence of singularities in the complex arclength plane. These singularities move about the com- MS2 plex plane and can approach closely to the real axis and are Computing Asymptotic Behavior for Singularly associated with the high curvature at the tip of a plunging Perturbed Semilinear Partial Differential Equa- breaker. Moreover, direct confirmation has been estab- tions lished between our numerical result and Tanveer’s theoretical prediction of square-root singularities. In addition, We describe the Chebyshev spectral integration method square-root singularities are also present in the wave ele- for computing high resolution numerical solutions to par- vation as a function of the complex horizontal coordinate. tial differential equations with non-periodic boundary con- These singularities reach the real axis in finite time when ditions. We then examine numerical solutions to singu- the wave breaks. larly perturbed equations. The first set of equations are damped nonlinear equations arising from viscoleastic mod- Chao Xie, Gregory Baker els for solids, these models include fourth order derivatives The Ohio State University which are multiplied by a small coefficient. We show that [email protected], [email protected] it is possible to compute the dynamics of these models and to compute the theoretically predicted scaling with the coefficient of the highest order term for minimizers of MS1 the energies of these models. The second set of equations Modulation Induced Photonic Transition: Optical are integrable wave equations, and include the KdV and Isolation nonlinear Schrödinger equations. Numerical solutions are demonstrated in one and two dimensions in the two di- Dynamic refractive index modulation can be used to induce mensional case, a Fourier-Chebyshev discretization is used. transitions between photonic states. We show that such Portions of this are joint work with Christian Klein and modulation can be used to achieve complete, linear, optical Kristelle Roidot. isolation, as well as achieving single-pole optical resonances with completely tunable linewidth and resonant frequency. Benson K. Muite Department of Mathematics S. Fan University of Michigan Stanford University [email protected] [email protected]

MS2 MS1 Interaction of Highly Oscillatory Waves in Weakly Nonlinear Digital Holography at the Micro- and Nonlinear Schroedinger Equations Nano-scale We consider weakly nonlinear Schroedinger equations in a Digital holography is a hybrid technique that combines semi-classical scaling and study the interaction of highly physical measurement of a wave ﬁeld with numerical prop- oscillatory waves within this context. A geometrical de- agation. Here, we extend the propagation kernel to include scription of possible resonances for cubic case is given. A the eﬀects of spatial nonlinearity. Nonlinearity mixes high- rigorous semi-classical approximation result is stated, using frequency spatial modes with low-frequency ones, so that the analytical framework of Wiener algebras. Finally, some all linear limits on imaging are superseded. In this talk, we applications and extensions (for example Hartree type non- describe the experimental and theoretical issues involved linearities) are discussed. with nonlinear digital microscopy in the non-paraxial and sub-wavelength regimes. Christof Sparber University of Cambridge, Dept. Applied Mathematics Jason Fleischer, Christopher Barsi United Kingdom Princeton University [email protected] [email protected], [email protected]

MS2 MS1 Initial Value Problem of the Whitham Equations Novel Lasing Structures and Phenomena from for the Camassa-Holm Equation Steady-State Ab Initio Theory Like the KdV equation, the Camassa-Holm equation with Abstract not available at time of publication. small dispersion can be viewed as a dispersive approxima- Douglas Stone tion to the inviscid Burgers equation. Although the zero Yale University dispersion limit for the Camassa-Holm equation has not [email protected] been established, the modulation equations have been derived. These modulation equations are ﬁrst order quasilinear hyperbolic partial diﬀerential equations. Unlike the MS2 KdV case, the modulation equations for Camassa-Holm are Critical Asymptotics for the KdV Equation in the non-strictly hyperbolic. In this talk, we will discuss the ini- Small Dispersion Limit tial value problem related to these modulation equations. Abstract not available at time of publication. Fei-Ran Tian Tom Claeys Ohio State University Department of Mathematics NW10 Abstracts 49

[email protected] MS3 Spontaneous Symmetry Breaking of Bose-Fermi Mixtures in Double-Well Potentials MS3 Counterflow Induced Modulational Instability, We study the spontaneous symmetry breaking (SSB) of a Vector Solitons, and Vector Dispersive Shock superfluid Bose-Fermi (BF) mixture in a double-well po- Waves tential (DWP). The mixture is described by the Gross- Pitaevskii equation (GPE) for the bosons, coupled to an Modulational instability (MI) due to relative motion of two equation for the order parameter of the Fermi superfluid. fluid components is predicted and experimentally observed Straightforward SSB in the degenerate Fermi gas loaded for a miscible, binary Bose-Einstein condensate (BEC). into a DWP is impossible, as it requires an attractive self- Stability analysis reveals the critical relative speed for the interaction, while the intrinsic nonlinearity in the Fermi gas onset of counterflow induced MI in the two component, is repulsive. We demonstrate that the symmetry breaking repulsive (defocusing) vector Nonlinear Schrödinger equa- is possible in the mixture with attraction between fermions tion. These results are corroborated by experiment and and bosons. Numerical results are represented by depen- three-dimensional numerical simulations where a prolifer- dencies of asymmetry parameters for both components on ation of dark-dark vector solitons, possibly experiencing a numbers of fermions and bosons in the mixture, and by transverse (snake) instability, occurs. In contrast, for rela- phase diagrams which displays regions of symmetric and tive speeds below critical, there is a smooth counterflow ac- asymmetric ground states. The dynamical picture of the companied by nonlinear self-steepening which leads to the SSB, induced by a gradual transformation of the single- development of vector dispersive shock waves composed of well potential into the DWP, is reported too. An analyti- trains of dark-bright solitons. cal approximation is proposed for the case when GPE for the boson wave function may be treated by means of the Mark A. Hoefer Thomas-Fermi (TF) approximation, allowing one to reduce North Carolina State University the model to a single equation for the fermionic function, [email protected] which includes competing repulsive and attractive nonlinear terms. Chris Hamner, JiaJia Chang, Peter Engels Washington State University Boris Malomed [email protected], [email protected], Department of Physical Electronics, Faculty of Engr. [email protected] Tel Aviv University, 69978 ISRAEL [email protected]

MS3 Sadhan Adhikari Spin Dynamics in Spinor Dipolar BECs Instituto de Fisica Teorica, UNESP Sao Paulo, Brazil We study the spin dynamics in a spin-1 ferromagnetic Bose- [email protected] Einstein condensate (BEC) with magnetic dipole-dipole interaction within the mean-field theory. We find that various magnetic structures such as checkerboards and stripes Luca Salasnich, Flavio Toigo emerge in the course of the dynamics due to the rather com- Dipartimento di Fisica , Universita’ di Padova plicated combined effects of spin-exchange and magnetic Padua, Italy dipole-dipole interactions, quadratic Zeeman, and finite [email protected], fl[email protected] size effects, and non-stationary initial conditions. However, the short-range magnetic pattern observed by the Berke- MS3 ley group [Phys. Rev. Lett. 100, 170403 (2008)] is not reproduced in our calculations. In this talk, we discuss Higher-order Correction to Mean-field Evolution the agreements and discrepancies between our numerical for Interacting Bosons calculation and the experiment. I will describe recent progress, and open challenges, in the Yuki Kawaguchi study of PDEs that aim to transcend the Gross-Pitaevskii Department of Physics equation. The derivation relies on the use of Boson oper- University of Tokyo ators in a Fock space, following arguments of Lee, Huang, [email protected] Yang, and Wu, who invoked the excitation of particles in pairs from the condensate to other states. Some implications will be discussed. Hiroki Saito Department of Engineering Science Dionisios Margetis University of Electro-Communications University of Maryland, College Park [email protected] [email protected]

Kazue Kudo Ochadai Academic Production MS4 Ochanomizu University Coarse-grained Molecular Dynamics Models [email protected] We will present a coarse-grained model based on molecu- Masahito Ueda lar dynamics. The goal is to reduce the dimension of the Department of Physics problem by chooses a small set of coarse-grained variables. University of Tokyo We will discuss how to derive the eﬀective equations for the [email protected] selected variables, and how to sample random noise based 50 NW10 Abstracts

on ﬂuctuation-dissipation theorem. [email protected], [email protected]

Xiantao Li Yuri V. Lvov Department of Mathematics Rensselaer Polytechnic Institute Pennsylvania State University [email protected] [email protected]

MS5 MS4 Renormalized Resonances in Wave Turbulence Coagulation and Annihilation Reactions in Subdif- fusive Media We study the non-perturbative nature of wave turbulence in strongly nonlinear regimes. We demonstrate that non- We consider reactions between particles whose mean square linear wave interactions renormalize the dynamics, leading displacement on a lattice is subdiffusive. One subdiffusion to (i) a drastic deformation of the resonant manifold even scenario is a lattice of traps with a particular distribution at weak nonlinearities, and (ii) creation of nonlinear reso- of depths. Another is a translationally invariant lattice nance quartets in wave systems for which there would be with power law escape time distributions from each trap. no resonances as predicted by the linear dispersion relation. While both lead to similar mean square displacements, the Finally, we present an extension of the weak turbulence ki- distributions of particles over sites vs time and the effects netic theory to systems with strong nonlinearities. of reactions on these distributions may be very different in the two cases. David Cai Shanghai Jiao Tong University, China Katja Lindenberg Courant institute, New York University, USA Department of Chemistry and Biochemistry [email protected] University of California San Diego, USA [email protected] MS5 Santos B. Yuste, Juan Ruiz-Lorenzo Wave Propagation in Deformed Waveguides: A Universidad de Extremadura Unifying Approach [email protected], [email protected] Using a conformal change of coordinate, we map a PDE Igor Sokolov on a 2D complicated strip to an inhomogeneous PDE in a Humboldt University rectangle. This can be reduced to a 1D effective equation [email protected] by averaging. Using the SchwarzChristoffel transformation for polygons and its numerical implemention by Driscoll we studied this reduction for the Boussinesq system in a MS4 river with variable width and the sine-Gordon equation in Non-equilibrium Statistical Mechanics, Billiards, a deformed strip. and Large Deviations Jean Guy Caputo We discuss simple but realistic models of non-equilibrium Laboratoire de mathematiques statistical mechanics where systems of particles are submit- INSA de Rouen ted to external forces and coupled to suitable thermostats: [email protected] (a) billiard with Gaussian thermostats and (b) particle systems coupled at the boundary to heat reservoirs. We study rigorously non-equilibrium steady states for such systems MS5 focussing on the entropy production, its fluctuations, and Influence of the Condensate on the Wave Turbu- the Gallavotti-Cohen fluctuation theorem. lence Spectra

Luc Rey-Bellet During direct numerical simulation of the isotropic turbu- Department of Mathematics & Statistics lence of surface gravity waves in the framework of Hamil- University of Massachusetts tonian equations formation of the long wave background or [email protected] condensate was observed. Exponents of the direct cascade spectra at the different levels of an artificial condensate suppression show a tendency to become closer to the pre- MS4 diction of the wave turbulence theory at lower levels of A Stochastic Boundary Forcing Model for Simulat- condensate. A simple qualitative explanation of the mech- ing Wave Turbulence Systems anism of this phenomenon is proposed. The beta-Fermi-Pasta-Ulam (FPU) model can serve as Alexander O. Korotkevich an example of a discretized wave turbulence system with Dept. of Mathematics & Statistics, University of New four-wave nonlinear interactions. We present a stochas- Mexico tic boundary forcing technique for the numerical simula- L.D. Landau Institute for Theoretical Physics RAS tion of a subdomain of a periodic beta-FPU chain, with a [email protected] view toward modeling the correct time evolution of the energy spectrum more accurately than conventional periodic boundary conditions would. MS5 Strong Collapse Turbulence in Nonlinear Warren Towne, Peter R. Kramer Schrödinger Equation Rensselaer Polytechnic Institute Department of Mathematical Sciences We consider a nonlinear Schrödinger equation (NLS) with dissipation and forcing in critical dimension. Without both NW10 Abstracts 51

linear and nonlinear dissipation NLS results in a finite-time University of California, Merced singularity (collapse) for any initial conditions. Dissipation [email protected] ensures collapse regularization. If dissipation is small then multiple near-singular collapses are randomly distributed in space and time forming collapse turbulence. Collapses MS6 are responsible for non-Gaussian tails in the probabil- A System of ODEs For a Perturbation of a Minimal ity distribution function of amplitude fluctuations which Mass Soliton makes turbulence strong. Power law of non-Gaussian tails is obtained for strong NLS turbulence. We study soliton solutions to a nonlinear Schrödinger equation with a saturated nonlinearity. Such nonlinearities are Pavel M. Lushnikov known to possess minimal mass soliton solutions. We con- Department of Mathematics and Statistics sider a small perturbation of a minimal mass soliton, and University of New Mexico identify a system of ODEs similar to those from the work [email protected] of Comech-Pelinovsky, which model the behavior of the perturbation for short times. We then provide numerical Natalia Vladimirova evidence that under this system of ODEs there are two pos- Department of Mathematics and Statistics, sible dynamical outcomes, which is in accord with earlier University of New Mexico conclusions of Pelinovsky. For initial data which supports [email protected] a soliton structure, a generic initial perturbation oscillates around the stable family of solitons. For initial data which Yeo-Jin Chung is expected to disperse, the finite dimensional dynamics Department of Mathematics follow the unstable portion of the soliton curve. Southern Methodist University Sarah Raynor [email protected] Wake Forest University [email protected] MS6 Time-Periodic Solutions of Nonlinear Dispersive Jeremy Marzuola Equations Department of Applied Mathematics Columbia University Together with Jon Wilkening, the speaker has developed [email protected] a numerical method for the computation of time-periodic solutions of nonlinear systems of partial differential equa- Gideon Simpson tions. The method so far has been applied to the Benjamin- University of Toronto Ono equation and to the vortex sheet with surface ten- [email protected] sion. For Benjamin-Ono, we find bifurcation from traveling waves to nontrivial time-periodic solutions. For the vortex sheet with surface tension, we find bifurcation from the flat MS6 equilibrium. Coherent Structures in the Nonlinear Maxwell Equations David Ambrose Drexel University The primitive equations governing wave propagation in [email protected] spatially varying optical fibers are the nonlinear Maxwell equations, though this process is often modeled using the nonlinear coupled mode equations (NLCME). NLCME de- MS6 scribe the evolution of the slowly varying envelope of an Solitons Bifurcating from Band-edges in Periodic appropriate carrier wave. They are known to possess soli- Media tons, which may be of use in optical transmission. In this talk, we numerically study the evolution the NLCME soli- Nonlinear Schrodinger (NLS) / Gross-Pitaveskii (GP) ton in the primitive equations, and find them to be robust. equations with periodic potentials are useful for modeling This is highly non-trivial, as the nonlinear Maxwell equa- the nonlinear dynamics arising in various physical systems, tions are a non-convex hyperbolic system, requiring careful such as propagation laser beams, Bose-Einstein conden- treatment of the Riemann problem. Furthermore, we con- sates, and deep water waves. These equations can admit sider extensions of NLCME to a system of infinitely many positive bound state solutions (aka solitons) that bifurcate nonlinear coupled mode equations and present some results from zero as the frequency bifurcates from the edge of the suggesting this new system also possesses localized states. first spectral band into the semi-infinite gap. We prove that This work is in collaboration with D. Pelinovksy and M.I. to leading order near the band edge, the bound state is a Weinstein. product of a rapidly oscillation Bloch wave and a slowly- varying ground state solution of a homogenized equation, Gideon Simpson whose coefficients depend on the effective-mass tensor and University of Toronto coupling constants. Explicit expressions are derived for [email protected] these coefficients. In the L2-critical case, these results imply that the L2 norm (power) of the bound state is below Pelinovsky Dmitry the threshold for singularity, yet that the slope condition McMaster University for stability is violated. The ensuing dynamics of perturbed [email protected] bound states is elucidated by direct computations of NLS /GPequations. Michael Weinstein Boaz Ilan Columbia University School of Natural Sciences [email protected] 52 NW10 Abstracts

MS7 [email protected] Recent Results in Evans Function Computation Recent Results in Evans Function Computation In this talk MS7 we discuss some of our recent work in multi-D Evans func- Stability in Multi-dimensional Problems via Evans tion computation. Although our techniques can be applied Functions and Lyapunov-Schmidt broadly, we are motivated by the stability analysis of viscous shock layers in compressible fluid flow. A number of In this talk, I will give an overview of different methods for unexpected analytical and numerical wrinkles have shown assessing the stability of waves on multi-dimensional cylin- up in our multi-D studies, and we demonstrate how we drical domains and illustrate these by examples. Specif- have been able to overcome them. ically, I will discuss a stability parity index for spatially periodic waves on cylindrical domains, the stability of pla- Jeff Humpherys nar patterns via a reduction of an infinite-dimensional Department of Mathematics Evans function, and the stability of fast pulses of the dis- Brigham Young University crete FitzHugh-Nagumo equation near its singular limit jeff[email protected] using Lyapunov-Schmidt reduction based on exponential dichotomies. Blake Barker Indiana University Bjorn Sandstede BYU Brown University [email protected] bjorn [email protected]

Gregory Lyng MS8 Department of Mathematics Homoclinic Snaking with Phase-winding States University of Wyoming [email protected] Many pattern forming systems contain a multiplicity of coexisting stationary spatially localized states. The spa- Kevin Zumbrun tial dynamics of such systems is characterized by homo- Indiana University clinic orbits which are organized around heteroclinic cycles USA between a fixed point and a periodic orbit. These states [email protected] occupy an extended region in parameter space called the snaking region. A similar behavior is present in systems which contain heteroclinic orbits between two fixed point, MS7 though in this case the solutions collapse to a single point Computing Stability of Multi-dimensional Waves in parameter space. In this talk I present a novel hybrid example involving phase winding states in the forced com- We present a numerical method for computing the pure- plex Ginzburg-Landau equation, which combines aspects point spectrum associated with the linear stability of mul- from both types of behavior: homoclinic snaking associ- tidimensional travelling fronts to parabolic nonlinear sys- ated with a heteroclinic connection between a fixed point tems. Our method is based on the Evans function shoot- and a periodic orbit which collapses to a point in parameter ing approach. Transverse to the direction of propagation space. we project the spectral equations onto a finite Fourier basis. This generates a large, linear, one-dimensional system John Burke of equations for the longitudinal Fourier coefficients. We Boston University construct the stable and unstable solution subspaces asso- [email protected] ciated with the longitudinal far-field zero boundary conditions, retaining only the information required for matching, by integrating the Riccati equations associated with MS8 the underlying Grassmannian manifolds. The Evans func- Localized Patterns in Large But Finite Domains tion is then the matching condition measuring the linear dependence of the stable and unstable subspaces and thus Common sense dictates that, in large domains, boundaries determines eigenvalues. As a model application, we study have a negligible influence on pattern formation and peri- the stability of two-dimensional wrinkled front solutions to odic boundary conditions are therefore often adopted for a cubic autocatalysis model system. simplicity. For localized patterns however, this can be a dangerous modeling attitude. The effect of distant bound- Simon Malham aries balances with exponentially small pinning forces act- Heriot-Watt University ing on the edges of the patterns. Using asymptotics, we [email protected] find that they can only occupy discrete locations. This limits the information capacity of such nonlinear systems. Veerle Ledoux Ghent University Gregory Kozyreff [email protected] Université Libre de Bruxelles, Brussels, Belgium [email protected] Jitse Niesen University of Leeds Pauline Assemat [email protected] Institut de Mécanique des Fluides UMR/CNRS 5502 Toulouse [email protected] Vera Thuemmler Fakultat fur Mathematik, Universitat Bielefeld, 33501 Bielefeld, Germany Jon Chapman NW10 Abstracts 53

OCIAM, Mathematical Institute, Yurii Vlasov Oxford IBM Research [email protected] [email protected]

MS8 MS9 Defect-mediated Snaking: A New Growth Mecha- Designable Nonlinearities in Optomechanically- nism for Localized Structures active Photonic-crystal Waveguides The forced complex Ginzburg-Landau equation with 1:1 We design large, tunable optical nonlinearities in waveg- and 1:2 resonance exhibits a new type of homoclinic uides that are repositioned via optical forces. Light in snaking involving a single branch of stationary spatially the waveguides exerts a mechanical force between either localized states. The states on this branch grow from a parallel waveguides or waveguide and substrate. We cal- central defect that inserts new rolls on either side, while culate intensity-dependent changes in phase birefringence pushing existing rolls outwards. This growth mechanism up to 0.026 for pump powers of 20mW. This effect enables differs fundamentally from standard homoclinic snaking in switchable linear-to-circular polarization conversion within which new rolls are added at the fronts that connect the compact devices. We further calculate mechanical Kerr co- structure to the background homogeneous state. efficients resulting from optical forces in silicon strip waveguides and 1D photonic- crystal waveguides. We show that Yiping Ma waveguide displacements responsible for the Kerr effect are Department of Physics enhanced close to the light line and near the photonic band U. C. Berkeley gap. [email protected] Michelle Povinelli USC MS8 [email protected] Localized Structures in the Multi-dimensional Swift-Hohenberg Equation Jing Ma University of Southern California The bifurcation structure and the cessation of snaking of lo- [email protected] calized radial patterns of the Swift–Hohenberg equation are explored through numerical computations as the dimension is varied. Our findings elucidate the connection between MS9 one-dimensional pulses and 2-pulses to planar spots and Nonlinear Optics (At a Single Photon Level) in rings when the dimension is changed from one to two and Photonic Crystal Nanocavities then further to three. We also discovered a new class of planar localized spot solutions and discuss an analytical A strong localization of light inside optical nanocavities approach to establishing their existence. can be employed to demonstrate nonlinear optical effects at ultra-low thresholds, with applications ranging from op- Scott McCalla tical switches and modulators controlled with sub-fJ ener- Division of Applied Mathematics gies, to inexpensive light sources in the visible. We have Brown University employed photonic crystal nanocavities in III-V semicon- scott [email protected] ductors to demonstrate frequency conversion of weak optical beams between infrared and visible wavelengths. In Bjorn Sandstede addition, a single semiconductor quantum dot strongly cou- Brown University pled to an optical nanocavity has been employed to demon- bjorn [email protected] strate controlled amplitude and phase shifts of an optical beam reflected from such a cavity, with control beams at the single photon level. MS9 Engineering Silicon Nanophotonic Waveguides for Jelena Vuckovic, Kelley Rivoire Mid-Infrared Nonlinear Optics Stanford University [email protected], [email protected] Silicon nanophotonic waveguides can be engineered to strengthen nonlinear optical interactions up to 100000 Arka Majumdar times those found in silica optical fibers. Therefore, the Stanford silicon platform can be utilized to realize a variety of ultra- [email protected] compact low-power nonlinear optical devices, having applications including all-optical signal processing, parametric light source generation, and sensing. This talk will explore Sonia Buckley nonlinear optics in silicon waveguides, highlighting the re- Stanford University cent demonstration of a mid-infrared optical parametric [email protected] amplifier exhibiting large on- and off-chip gain.

William Green MS10 IBM Research Dissipative Droplet Solitons [email protected] Large amplitude, vectorial, localized, ”droplet solitons” in anisotropic magnetic media are studied in thin ferromag- Xiaoping Liu, Richard Osgood netic ﬁlms driven by a spin polarized current. This 2D Columbia University dissipative droplet soliton is an approximate solution of a [email protected], [email protected] modiﬁed Landau-Lifshitz equation and results from a bal- 54 NW10 Abstracts

ance between a spatially localized forcing and a uniform broken. Unstable line solitons decay spontaneously into damping in addition to nonlinearity and dispersion. Using lumps,whicharedescribedbybothnumericalandvaria- perturbative techniques and numerical calculations, prop- tional approaches. erties of this strongly nonlinear, coherently precessing state will be discussed including existence conditions, stability, Herve Leblond and hysteresis. Furthermore, motivated by a numerically LPhiA, EA 4464 observed drift instability, moving droplet solitons will also University of Angers be discussed. [email protected]

Mark A. Hoefer Miguel Manna North Carolina State University Universite Montpellier [email protected] 34095 Montpellier, cedex 5, France [email protected] Thomas Silva Magnetics Technology Group, NIST [email protected] MS10 Devil’s Staircase in Nano-magnetism: Fractional Synchronization of a Spin-torque Nano-oscillator MS10 Influence of Free and SAF Layer Oscillations When an oscillator with auto-oscillation frequency ω0 is Modes on the RF Emission Line Width in Mag- driven by a strong external periodic signal of frequency netic Tunnel Junctions ωe, it can become synchronized at any rational value of r = ωe/ω0. Here, we show that by driving a spin-torque We present a numerical study of oscillation modes nano-oscillator (STNO) with a microwave magnetic field in magnetic tunnel junctions and investigate the fre- at oblique direction with respect to the STNO symmetry quency/FWHM as a function of applied current. We show axis, it is possible to observe integer (r =1, 2, 3, 4) and that the linewidth reaches a min. of 14.5MHz at the thresh- fractional (r =3/2, 7/3, 5/2, 7/2) synchronization regimes, old current, and increases sharply to 308MHz as the cur- forming a so-called ”Devil’s staircase”. rent is increased. The line-narrowing is due to an increased coherence of the uniform precession mode, while the line- Andrei Slavin, Vasil Tiberkevich broadening above threshold arises from the intrinsic oscil- Oakland University lator nonlinearity combined with overlapping contributions [email protected], from edge modes. Sergei Urazhdin, Phil Tabor Gino Hrkac West Virginia University Department of Engineering Materials , University of Sheffield, UK g.hrkac@sheffield.ac.uk MS11 Thomas Schrefl On the Small Dispersion Limit for Camassa-Holm University of Sheffield Equation

Thibaut Devolder I shall present recent results obtained in collaboration with Institut d’Electronique Fondamentale Christian Klein (Univ. Bourgogne, France) and Tamara Grava (Sissa, Italy) on the so-called small dispersion limit of solutions of the Cauchy problem of the Camassa Holm Joo-Von Kim, Claude Chappert (CH) equation for smooth initial data such that wave- Institut d’Electronique Fondamentale breaking does not occur. For initial data in this class, the CNRS numerical solution of (CH) develops a zone of fast oscil- , lations, as for the small dispersion limit of the Korteweg- de Vries equation and asymptotic expansions shall be pro- Sven Cornelissen, Liesbet Lagae posed. IMEC, Belgium , Simonetta Abenda Department of Mathematics and CIRAM, University of Dieter Suess, Dirk Praetorius Bologna Vienna University of Technology Bologna, Italy , [email protected]

MS10 MS11 Two-dimensional Electromagnetic Solitons in the Universality Behviour of Solutions of Hamiltonian Short-wave Approximation PDEs in Critical Regimes Using a short wave approximation, the Maxwell-Landau We study solutions to a family of Hamiltonian PDEs near equations describing electromagnetic wave propagation in critical points. We show that the local behaviour of the so- a thick saturated ferromagnetic ﬁlm are reduced to two- lutions near the critical points is described by a family of dimensional asymptotic models which generalize the sine- ODE of Painleve’ type. We show that up to shifts, rescal- Gordon (sG) equation, and admit a Lagrangian in contrast ings and Galileian transformations, this local description to the so-called 2 dimensional sG equation. Depending on the orientation of the external ﬁeld, symmetry can be NW10 Abstracts 55

does not depend on the initial data nor on the equation. MS12 Dynamics of Quantized Vortices in Bose-Einstein Tamara Grava Condensates SISSA, via Beirut, 2-4, 34014, Trieste, Italy [email protected] The dynamics and interaction of quantized vortices in Bose-Einstein condensates (BECs) are presented in this talk. If all vortices have the same winding number, they MS11 would rotate around the trap center but never collide. In Numerical Treatment of the Small Dispersion contrast, if the winding numbers are different, their interac- Limit in Nonlinear PDE tion highly depends on the initial distance between vortex centers. The analytical results are presented to describe Purely dispersive equations as the Korteweg-de Vries and the dynamics of the vortex centers in non-interacting BEC. the nonlinear Schrödinger equation and two dimensional While in interacting BEC, there is no analytical result but generalizations in the limit of small dispersion have solu- some conclusive numerical findings are provided for the fur- tions to Cauchy problems with smooth initial data which ther understanding of vortex interaction in BECs. Finally, develop a zone of rapid modulated oscillations in the re- the dynamic laws describing the relation of vortex interac- gion where the corresponding dispersionless equations have tion in nonrotating and rotating BECs are presented. shocks or blow-up. Fourth order time-stepping in combination with spectral methods is beneficial to numerically re- Yanzhi Zhang solve the steep gradients in the oscillatory region. We com- Florida State University pare the performance of several fourth order methods for [email protected] exact solutions and the small dispersion limit: exponential time-differencing fourth-order Runge-Kutta methods (Cox and Matthews, Krogstad, Hochbruck and Ostermann) in MS13 the implementation by Kassam and Trefethen, integrating Existence and Stability of Fully Localised Three- factors, time-splitting, Driscoll’s composite Runge-Kutta dimensional Gravity-capillary Solitary Water scheme, and an ODE solver in Matlab. Waves

Christian Klein A solitary wave of the type advertised in the title is a crit- Institut de Mathématiques de Bourgogne ical point of the Hamiltonian, which is given in dimension- 9 avenue Alain Savary, BP 47870 21078 DIJON Cedex less coordinates by [email protected] 1 1 2 2 2 H(η,ξ)= ξG(η)ξ + η + β 1+ηx + ηz − β , 2 2 2 MS12 R Quantum Many-body Systems, the Nonlinear subject to the constraint that the impulse Schrodinger Equation, and Beyond I η,ξ η ξ Only recently has the rigorous connection been made be- ( )= x 2 tween the many-body physics of Bose-Einstein condensa- R tion and the mathematics of the macroscopic model, the is fixed. Here η(x, z) is the free-surface elevation, ξ is the cubic nonlinear Schrodinger equation. I’ll discuss work trace of the velocity potential on the free surface, G(η)is on two-dimensional cases for Bose-Einstein condensation– a Dirichlet-Neumann operator and β>1/3 is the Bond especially the periodic case, because of techniques from number. In this talk I show that there exists a minimiser analytic number theory and applications to quantum com- of H subject to the constraint I =2μ,where0<μ puting. I’ll also mention work on a high-probability phase 1. The existence of a solitary wave is thus assured, and transition for invariant measures of the NLS. since H and I are both conserved quantities its stability follows by a standard argument. ‘Stability’ must however Kay Kirkpatrick be understood in a qualified sense due to the lack of a global Courant Institute well-posedness theory for three-dimensional water waves. New York University [email protected] Mark D. Groves Universität des Saarlandes [email protected] MS12 Dynamics of Solitary Waves in Double Well Poten- tials Motivated by BEC MS13 Instability of Periodic Water Waves We present some models motivated by studies of Bose- Einstein condensates trapped in double-well potentials. We Periodic traveling waves exist in 2D water waves and lots extend the analysis of the prototypical 1D model with lo- of dispersive models, such as Stokes waves of deep water cal nonlinearity to spatially varying collisional interactions, (Stokes, 1847) and Cnoidal waves of KDV equation. I will long-range interactions, and also the 2D environment in discuss an unified approach to study the stability and in- the setting of a four-well potential. By a Galerkin-type stability of periodic waves of water waves and a large class few-mode approach, we study the existence, stability and of dispersive models, under perturbations of the same pe- dynamics of the nonlinear localized steady states and pre- riod. The results include a sharp instability criterion for dict the bifurcation diagrams. KDV and BBM type models, and a proof of the existence of unstable Stokes waves under some natural assumptions. Chenyu Wang Department of Mathematics and Statistics Zhiwu Lin University of Massachusetts Georgia Institute of Technology [email protected] School of Mathematics 56 NW10 Abstracts

[email protected] trap. The results to be reported are obtained in collaboration with Yu. V. Bludov, G. L. Alﬁmov, and D. Zezyulin.

MS13 Existence and Stability of Solitary Waves on Water Vladimir V. Konotop Universidade de Lisboa, Lisbon The talk will discuss most recent development on the exis- [email protected] tence and stability of two- and three-dimensional waves on the surface of water of finite depth with or without suface tension using various model equations or exact Euler equa- MS14 tions. It will be shown that in some special cases, these Architectural and Functional Connectivity in equations have solitary-wave solutions. Moreover, various Scale-free Integrate-and-fire Networks stability results for these waves will be addressed, such as transverse stability, spectral stability or conditional stabil- We present a study of scale-free networks of identical, ity. conductance-based, stochastically driven, integrate-and- fire excitatory neurons. Using the mean-field approach, we Shu-Ming Sun show that the firing rates of neurons in scale-free networks Virginia Tech themselves follow scale-free distribution. At the same time Department of Mathematics we show that the firing rate in such networks is strongly [email protected] dependent on the degree correlation function. The analytical results are compared to the direct numerical simulations of the coupled integrate-and-fire neurons. Ultimately, MS14 our analysis provides a link between the functional and Localized and Periodic Patterns in nonlinear anatomical connectivity in scale-free networks. Schrödinger Equation with Complex Potentials Maxim S. Shkarayev Exact solutions for the generalized nonlinear Schrödinger Mathematical Science Department, Rensselaer Polytech (NLS) equation with inhomogeneous complex linear and Inst. nonlinear potentials are found. We have found localized [email protected] and periodic solutions for a wide class of localized and periodic modulations in space of complex potential and nonlinearity coefficient. Applications for the case of PT MS14 symmetry are discussed. Bubble Break-up as a Two-dimensional Free Hy- drodynamics Problem Fatkhulla Abdullaev Uzbek Academy of Sciences Break-up of gas bubbles immersed in liquids is a natural Uzbekistan event occurring in many multiphase systems. Associated to [email protected] this event is a process of gas neck reconnection. Recent experiments performed in University of Chicago have shown Vladimir V. Konotop that neck reconnection is a non-universal process with the Universidade de Lisboa, Lisbon outcome strongly dependent on the initial conditions. In [email protected] this talk I will show how the neck reconnection dynamics can be described with a two-dimensional free surface hy- Mario Salerno drodynamic model. This model can be e?ectively solved Dipartimento di Fisica E. R. Caianiello, and numerically with conformal mapping techniques that were Consorzio Nazionale Interuniversitario per le Scienze introduced in the works of S. Tanveer and V. Zakharov Fisich et.al. I will present the results of numerical simulations [email protected] which show that the smooth contact of bubble surface is a generic outcome of the dynamics, but the ?nal shape of the neck is strongly dependent on the initial conditions. I will Alexey Yulin also discuss possible extensions of this model that include Centro de F the e?ect of large scale velocity and non-trivial topologies ’isica Te of the interface. ’orica e Computacional, Universidade de Lisboa Konstantin Turitsyn [email protected] CNLS, Los Alamos National Laboratory, MS B258, Los Alamos, New Mexico, 87545, USA [email protected] MS14 Matter Waves in Complex Potentials L. Lai Inelastic interactions of condensed atoms with an external Department of Physics, trap in the mean-field approximation lead to the Gross- University of Chicago ˜ Pitaevskii equation with a complex potential. If the atomic http://home.uchicago.edu/lplai/ losses are compensated by a linear gain, stable patterns of the matter emerge. Such nonlinear dissipative structures Wendy W. Zhang appear as attractors and are characterized by the balance Physics Department & James Franck Institute of the dispersion and the nonlinearity due to two-body in- University of Chicago teractions, one the one hand, and on the other hand due to [email protected] equilibrium between the losses and gain. In the talk there will be reported the results on dynamics of matter waves in dissipative optical lattices and in a dissipative parabolic NW10 Abstracts 57

MS15 MS15 Quantum Decay Rates for Manifolds with Hyper- Collapsing Vortex Soliton Blowups for L2-critical bolic Ends NLS Mathematically, quantum decay rates appear as imaginary The focusing cubic nonlinear Schrodinger equation in two parts of poles of the meromorphic continuation of Greens dimensions admits vortex solitons with spin m.Theseare functions. As energy grows, decay rates are related to prop- generalizations of the standard soliton (spin m =0)and erties of geodesic flow and to the structure at infinity. For have spatial structure, Qm(r, θ)=eimθRm(r). In the a cusp, infinity is small, which typically slows decay. How- case of no spin, it is well known [Merle & Raphael] that ever, I will present a class of examples for which decay rates there exists an open class of solutions in H1 that blowup go to infinity with energy even in the presence of a cusp. at the log-log rate by concentrating precisely the L2 norm This is part of a more general investigation of resonances of the soliton into a point. We prove that, in the case on manifolds with hyperbolic ends. of spin m = 1, there is a relatively open class of data that blowup at the log-log rate by concentrating the L2 Kiril Datchev norm of the Q1 vortex soliton into a point. These are University of California, Berkeley the first examples of log-log blowups that focus strictly [email protected] more than the L2 norm of the soliton. Our proof is in two parts. We show that, when restricted to functions of the same spin, the degeneracies and algebraic structure of the MS15 Hamiltonian near the vortex soliton are the same as in the H 1 Improved Regularity for log-log Blow-up Solu- spinless case. In particular, a similar modulation argument L2 tions to the Critical NLS is possible, assuming a crucial spectral property holds. In the case m = 1, we use numerical techniques to show that We prove that if u(t) is a log-log blow-up solution, of the 2 the spectral property is indeed true. type studied by Merle-Raphael (2001–2005) to the 1-d L i∂ u ∂2u |u|4u critical focusing NLS equation t + x + =0with Ian Zwiers, Gideon Simpson u ∈ H 1 u t H 1 initial data 0 then ( ) remains bounded in away University of Toronto from the blow-up point. This is obtained without assuming 1 [email protected], [email protected] that the initial data u0 has any regularity above H .Asan application, we construct an open subset of initial data in 1 3 1 3 Hrad(R )(radialH (R ) functions) with corresponding MS16 solutions that blow-up on a sphere at positive radius for 1 Eigenvalues of Embedded Solitons the 3d quintic (H˙ -critical) focusing NLS equation i∂tu + Δu + |u|4u =0. Abstract not available at time of publication. Justin Holmer Roy Choudhury Department of Mathematics, Brown University, University of Central Florida Box 1917, 151 Thayer Street, Providence, RI 02912, USA [email protected] [email protected]

Svetlana Roudenko MS16 Arizona State University A Connection Between Parametric Resonance and [email protected] Bike Tracks This talk describes a surprisingly close connection between MS15 objects mentioned in the title. Estimates on the Strichartz Norm for Small Solu- L2 Mark Levi tions of the -critical NLS Equation Department of Mathematics We consider the mass-critical Schroedinger equation in Pennsylvania State University 1 and 2 space dimensions (both focusing and defocusing [email protected] cases). By Strichartz estimates for the linear problem, solutions with small L2 norm are globally deﬁned and belong Sergei Tabachnikov to an Lp (Strichartz) space (here, p = 6 in 1d and p =4 Penn State University in 2d). We show that for small L2 norm the maximum [email protected] of the Lp (Strichartz) norm is attained, and give a precise estimate of this maximum as the mass tends to 0. In Robert Foote particular, in the focusing case, it is greater than the cor- Wabash College responding maximum for the linear equation, which was [email protected] computed by Foschi and Hundertmark-Zharnitsky, and it is smaller in the defocusing case. This is a joint work with T. Duyckaerts and F. Merle. MS16 Stability and Instability of Kink-wave Solutions to Svetlana Roudenko the Sine-Gordon Equation Arizona State Universtity Department of Mathematics and Statistics We give a proof for the conditions of stability and of insta- [email protected] bility of certain traveling wave solutions to the sine-Gordon equation. For a traveling kink wave solution of speed c,the 2 Thomas Duyckaerts, Frank Merle kink wave is stable if and only if c − 1 < 0. The proof Universit´e de Cergy-Pontoise uses the Maslov index as a means for determining both [email protected], [email protected] the unstable and stable case. Ricatti equations and fur- 58 NW10 Abstracts

ther geometric considerations are also used in establishing superposition of two pulses, so long as they move along stability. trajectories which are not parallel. In particular we prove that if the initial data for the equation is close to the sum of Robert Marangell two separated pulses, then the solution converges exponen- Warwick Mathematics Institute tially fast to such a superposition so long as the distance [email protected] between the two pulses remains suﬃciently large.

J. Douglas Wright MS16 Drexel University On a DNA Model based on Coupled Oscillators Department of Mathematics with Interactions up to Three Neighbors Away [email protected] We will introduce a DNA model based on coupled oscillators interacting with neighbors up to three bases away. MS17 First, we will show results regarding the hyberbolicity of Fronts in Heterogeneous Media the fixed point at the origin. We will proceed with the stability analysis of static solutions and will show new types of In this presentation, we analyze the influence of hetero- solutions, in comparison with the nearest-neighbor model. geneities on localized solutions to a three-component sys- Finally, we will present results on the modulational insta- tem of FitzHugh-Nagumo type. It turns out that the het- bility of plane waves and the stability of breathers. erogeneity pins the solutions. More specifically, traveling front solutions can be pinned at the heterogeneity and from Zoi Rapti a 1D-familiy of stationary pulse solutions (in the homoge- University of Illinois at Urbana-Champaign neous case), the heterogeneity case only picks a few. [email protected] Peter van Heijster Division of Applied Mathematics MS17 Brown University, Providence RI Renormalization Group Apprach to Semi-strong [email protected] Interactions

This talk is based on joint work with P. v.Heijster, A. Doel- MS18 man, and K. Promislow. We develop a renormalization Noise Induced Pulse Interaction: The Casimir Ef- group method to rigorously derive equations of motion for fect in Mode Locked Lasers the positions of N fronts interacting semi-strongly in the three-component extended FitzHugh-Nagumo system. Ultrashort pulses in a multi-pulse passively mode locked lasers exhibit rich dynamics, behaving as a system of inter- Tasso J. Kaper acting particle-like entities. An important mechanism for Boston University the inter-pulse interactions is continuum light radiated and Department of Mathematics absorbed by the pulses. Statistical light-mode dynamics [email protected] (SLD) is the established method for analyzing the fluctu- ating dynamics of the pulse-continuum optical field of the mode locked laser; using SLD we identify a noise-induced MS17 long-range attractive interaction between pulses that is an Wave Stability in Systems with a Non-strictly Hy- entropic effect, quite reminiscent of the Casimir attraction perbolic Part in quantum electrodynamics. In combination with a gain- induced repulsive attraction we present the first complete In this talk we present resolvent estimates for hyperbolic- theory of the formation of steady-state pulse ’molecules’ parabolic systems, where the hyperbolic part need not be with stable inter-pulse spacings in multi-pulse mode locked strictly hyperbolic (e.g. Hodgkin-Huxley equation). It is lasers. shown that these imply the stability of traveling waves, fronts and pulses, as well as the convergence of the freez- Omri Gat ing method [Beyn, Thmmler 2004]. Our results generalize Hebrew University of Jerusalem, Department of Physics those of [Kreiss, Kreiss, Peterson 1994] and also include Jeruslam, ISRAEL systems that are not covered by the invariant manifold ap- [email protected] proach of [Bates, Jones 1989]. Numerical experiments will be presented for illustration. Rafi Weill Jens Rottmann-Matthes Dept. Electrical Engineering Department of Mathematics Technion - IIT, Haifa, Israel 3200 Bielefeld University, Germany rafi[email protected] [email protected] Baruch Fischer Department of Electrical Engineering MS17 Technion Interaction Manifolds in Reaction Diffusion Sys- fi[email protected] tems We consider a general planar reaction diffusion equation MS18 which we hypothesize has a localized traveling wave solu- General Features of Dissipative Soliton Reso- tion. Under assumptions which are no stronger than those nances: A Roadmap for High-energy Optical needed to prove the stability of a single pulse, we prove that the PDE has solutions which are roughly the linear NW10 Abstracts 59

Pulses? MS19 Resonant Nonlinear Damping of Quantized Spin A Dissipative Soliton Resonance (DSR) consists in the in- Waves in Ferromagnetic Nanowires crease to infinity of the energy of dissipative solitons when the propagation equation parameters converge to a specific We use ferromagnetic resonance excited by spin transfer set of values. Using the complex cubic-quintic Ginzburg- torque from alternating spin-polarized current to measure Landau equation, we confirm the existence of DSRs for the spectral properties of dipole-exchange spin waves in wide ranges of parameters in both chromatic dispersion permalloy nanowires. Our measurements reveal that geo- regimes, and establish general features of the ultra-high metric confinement has a profound effect on the nonlinear energy pulses found close to a DSR. Relationship to high- damping of spin waves in the nanowire geometry. The energy mode-locked fiber oscillators is proposed. damping parameter of the lowest-energy quantized spin wave mode depends on applied magnetic field in a resonant Philippe Grelu way and exhibits a maximum at a field that increases with ICB - UMR5209 CNRS - Universite de Bourgogne, Dijon, decreasing nanowire width. This enhancement of damping France originates from a nonlinear resonant three-magnon conflu- [email protected] ence process allowed at a particular bias field value determined by quantization of the spin wave spectrum in the Wonkeun Chang nanowire geometry. Max Planck Institute for the Science of Light Erlangen, Germany Ilya Krivorotov,CarlBoone [email protected] University of California, Irvine [email protected], [email protected] Nail Akhmediev Optical Sciences Center, RSPhysSE Jordan Katine, Jeffrey Childress Australian National University Hitachi Global Storage Technologies, San Jose [email protected] [email protected], jeff[email protected] Jose Soto-Crespo Instituto de Optica, Madrid, Spain Vasil Tiberkevich, Andrei Slavin [email protected] Oakland University [email protected], [email protected] Adrian Ankiewicz Optical Sciences Center, RSPhysSE Jian Zhu, Xiao Cheng Australian National University University of California, Irvine [email protected] [email protected], [email protected]

MS18 MS19 Amorphous Photonic Lattices: Fundamentals and Ginzburg-Landau Model of Bose-Einstein Conden- Applications sation of Magnons

We present, theoretically and experimentally, a new con- We introduce a system of phenomenological equations cept in photonics: amorphous photonic lattices. These for Bose-Einstein condensates of magnons in the one- structures exhibit a band gap and negative effective mass, dimensional setting. The nonlinearly coupled equations, without any Bragg diffraction. We show that in these sys- written for amplitudes of the right- and left-traveling tems the bands are comprised of localized Anderson states, waves, combine basic features of the Gross-Pitaevskii and but defect states residing in the gap are much stronger complex Ginzburg-Landau models. They include localized localized. The underlying fundamental physics and po- source terms to represent the microwave magnon-pumping tential applications are discussed, including the intriguing field. With the source represented by delta-functions, we possibility of fabricating a hollow-core disordered photonic find analytical solutions for symmetric localized states of band-gap fiber. the magnon condensates. We also predict the existence of asymmetric states with unequal amplitudes of the two Alexander Szameit components. Numerical simulations demonstrate that all Technion - Israel Institute of Technology analytically found solutions are stable. With the delta- [email protected] function replaced by broader sources, the simulations reveal a transition from the single-peak stationary symmetric Mikael C. Rechtsman states to multipeak ones, generated by the modulational in- Courant Institute stability of extended nonlinear-wave patterns. In the sim- New York University ulations, symmetric initial conditions always converge to [email protected] symmetric stationary patterns. On the other hand, asymmetric inputs may generate nonstationary asymmetric lo- Felix Dreisow, Matthias Heinrich, Robert Keil, Stefan calized solutions, in the form of traveling or standing waves. Nolte Comparison with experimental results demonstrates that Friedrich-Schiller-Universität the phenomenological equations provide for a reasonably [email protected], [email protected], good model for the description of the spatiotemporal dy- [email protected], [email protected] namics of magnon condensates. Boris Malomed Moti Segev Department of Physical Electronics, Faculty of Engr. Technion - Israel Institute of Technology Tel Aviv University, 69978 ISRAEL [email protected] 60 NW10 Abstracts

[email protected] [email protected]

O. Dzyapko, V. E. Demidov, S. O. Demokritov Robert Buckingham University of Muenster University of Cincinnati ,, [email protected]

MS19 MS20 Chaotic Solitons in Magnetic Thin Films On Meromorphic Solutions to the KdV Equation

This presentation reports the observation of chaotic soli- We show that under the KdV flow an initial profile sup- tons in a magnetic film strip-based active feedback ring. ported on (-8,0) from a very broad function class (without At some ring gain level, one observes the self-generation of any decay assumption or certain type of behavior at mi- a spin-wave pulse whose amplitude changes with time in a nus infinity) instantaneously evolves into a meromorphic chaotic manner. The pulse has a hyperbolic secant shape function with no poles on the real line. Our treatment is and a flat phase. The time-domain signal exhibits a finite based on a suitable modification of the inverse scattering dimension and a positive Lyapunov exponent. The results transform and a detailed investigation of the Titchmarsh- were confirmed by simulations based on the gain-loss non- Weyl m-function. As by-product, we improve some related linear Schrodinger equation. results of others. Mingzhong Wu, Zihui Wang, Aaron Hagerstrom, Justin Alexei Rybkin Q. Anderson Department of Mathematics and Statistics Colorado State University University of Alaska, Fairbanks [email protected], , , [email protected]

Lincoln D. Carr Colorado School of Mines MS20 The Zero-dispersion Limit of the Benjamin-Ono Wei Tong, Richard Eykholt Equation Colorado State University We investigate the zero dispersion limit of the Cauchy , problem of the Benjamin-Ono equation and provide the first rigorous results about it in this paper. We show this Boris Kalinikos limit exists in the weak L2(R) sense and can be explicitly St. Petersburg Electrochemical University written in terms of the multivalued solution of the inviscid Burgers equation with the same initail condition. MS20 Zhengjie Xu On the Weakly Dispersive Shocks of the KP Equa- Department of Mathematics tion University of Michigan [email protected] Recently, a large number of soliton solutions of the KP equation has been found and classified. In this talk, I will Peter D. Miller consider the weak dispersion limit of the KP equation, and University of Michigan, Ann Arbor discuss the weakly dispersive shock waves based on the [email protected] classification of the soliton solutions.

Yuji Kodama MS21 Ohio State University Department of Mathematics Pulse Switching in Nonlinear Optical Networks [email protected] An area of intense research is that of photonics, where light propagation features are controlled by clever engineering MS20 of periodic optical structures. In this presentation after we concentrate on one model: a novel photonic crystal cou- Fluxon Condensates and the Semiclassical Sine- pler. The coupler is a novel system aim at producing con- Gordon Equation trolled optical pulse delays. Here we demonstrate that the The semiclassically scaled sine-Gordon equation is consid- interplay of asymmetry in the design of each port with non- ered with (relatively) slowly varying initial data. Consid- linear eﬀects produces optical pulses where we can control ered as functions of x, the initial conditions parametrize a the group velocity. curve in the phase portrait of the simple pendulum. We Alejandro Aceves study the solution of the initial-value problem and show x Southern Methodist University that away from points where the initial data crosses [email protected] the pendulum separatrix the solution is asymptotically described by modulated elliptic functions characterizing either breather trains or kink trains with superluminal ve- MS21 x locities. Near points where the initial data crosses the Bisolitons in Optical Fibers with Dispersion Man- pendulum separatrix the behavior is more subtle. This is agement joint work with Robert Buckingham. Optical ﬁber links with dispersion management are known Peter D. Miller to be supporting propagation of stable coupled solitary University of Michigan, Ann Arbor NW10 Abstracts 61

wave solutions bisolitons. We theoretically demonstrated of the frequency profile of the net gain/loss coefficients. existence of an additional family of such solutions for the same set of system parameters. Energy of these solutions is Avner Peleg shown to be larger then energy of conventional bisolitons. State University of New York at Buffalo Direct numerical simulations demonstrated their stability. [email protected]ffalo.edu

Yeojin Chung Ildar R. Gabitov Southern Methodist University Department of Mathematics, University of Arizona [email protected] [email protected] Quan M. Nguyen Pavel M. Lushnikov State University of New York at Buffalo Department of Mathematics and Statistics qnguyen2@buffalo.edu University of New Mexico [email protected] MS22 Maxim S. Shkarayev Stability of Traveling Water Waves: Resonant Per- Mathematical Science Department, Rensselaer Polytech turbations Inst. [email protected] A boundary perturbation algorithm will be presented for computing the spectral data of periodic traveling wave so- Mikhail G. Stepanov lutions to the potential flow equations. This algorithm ex- Department of Mathematics, University of Arizona tends the transformed field expansion technique of Nicholls [email protected] (2002) to compute stability with respect to perturbations which resonate with the linear spectrum. The transformed field expansion approach will be motivated. The algorithm MS21 will be evaluated both as a modern tool, and with historical Methods for Large Deviations and Rare Events in perspective. Optical Fibers Benjamin Akers In optical fiber systems, amplified spontaneous emission University of Illinois - Chicago noise causes signal fluctuations that lead to errors in those [email protected] rare cases when the noise-induced changes are large. We discuss methods capable of determining large deviations in- MS22 duced by noise in such systems. First, the large deviations are found using a constrained optimization problem that The Modulational Instability in the Presence of exploits the mathematical structure of the governing equa- Dissipation and Forcing tions and a numerical implementation of the singular value Within the framework of the fully nonlinear water waves decomposition. The results of the optimization problem equations, we consider a Stokes wavetrain modulated by then guide importance-sampled Monte-Carlo simulations the Benjamin-Feir instability in the presence of both vis- to determine the events’ probabilities. We show that the cous dissipation and forcing due to wind. The wind model method works for a general class of intensity-based optical corresponds to the Miles’ theory. By introducing wind ef- detectors and for arbitrarily shaped pulses. fect on the waves, the present paper extends the previous William Kath works of [Segur et al, 2005, “Stabilizing the Benjamin-Feir Northwestern University instability”, J. Fluid Mech.,539] and [Wu et al, 2006, “A Engineering Sciences and Applied Mathematics note on stabilizing the Benjamin-Feir instability”, J. Fluid [email protected] Mech., 556] who neglected wind input. The marginal stability curve derived from the fully nonlinear numerical simulations coincides with the curve obtained by [Kharif et Jinglai Li al, 2010, “The modulational instability in deep water un- Department of Engineering Sciences and Applied der the action of wind and dissipation” (submitted)] from Mathematics a linear stability analysis. Furthermore, it is found that Northwestern University wind input goes in the subharmonic mode of the modula- [email protected] tion whereas dissipation damps the fundamental mode of the initial Stokes wavetrain.

MS21 Christian Kharif ImpactofRamanCrosstalkonOOKandDPSK Institut de Recherche sur les Phenomenes Hors Equilibre Massive Multichannel Transmission Systems [email protected] We investigate the effects of Raman crosstalk on the dynamics of pulse amplitudes in fiber optics communications Julien Touboul systems with a large number of frequency channels. For LSEET OOK transmission we show that the nth normalized mo- [email protected] ments of the probability density function of the amplitude grow exponentially with both propagation distance and n. MS22 For DPSK transmission we find that the dynamics is described by an N-dimensional predator-prey model and show Linear Stability of Periodic Traveling Water Waves that stable transmission can be achieved by a proper choice in Two and Three Dimensions Several numerical methods have been devised to simulate 62 NW10 Abstracts

not only traveling waveforms on the surface of an ideal fluid MS23 (the water wave problem) but also their dynamic stability. Transition Metamaterials In this talk we will describe two robust and spectrally accurate Boundary Perturbation approaches to the problem of Metamaterials open unparalleled opportunities for ”engi- computation and linear stability of periodic water waves. neering” previously inaccessible values of refractive indices We will present results in both two (Stokes waves) and and new ways for light manipulation. In this talk, we dis- three (short-crested waves) dimensions. cuss a new class of graded-index metamaterials, transition metamaterials, whose effective dielectric permittivity and Dave Nicholls magnetic permeability gradually change from positive to University of Illinois, Chicago, negative values. These materials reveal several peculiar USA optical phenomena, including anomalous electromagnetic [email protected] field enhancement and resonant absorption, thus opening new opportunities for linear and especially nonlinear optical applications. MS22 Stability in the Euler Equations Natalia M. Litchinitser The State University of New York at Buffalo Euler’s equations describe the dynamics of gravity waves natashal@buffalo.edu on the surface of an ideal fluid with arbitrary depth. In this talk, I discuss the stability of one-dimensional travel- Irene Mozjerin, Tolanya Gibson ing wave solutions for the full set of Euler’s equations via a State University of New York at Buffalo generalization of a non-local formulation of the water wave irenemoz@buffalo.edu, tjgibson@buffalo.edu problem due to Ablowitz, Fokas and Musslimani [Ablowitz et al 2006, J. Fluid Mech.]. Transforming the non-local formulation into a traveling coordinate frame, we obtain a Matthew Pennybacker new scalar equation for the stationary solutions using the University of Arizona original physical variables. Using this new equation, we [email protected] develop a numerical scheme to determine traveling wave solutions by exploiting the bifurcation structure of the non- Ildar R. Gabitov trivial periodic solutions. Next, we determine numerically Department of Mathematics, University of Arizona the spectral stability for the periodic traveling wave so- [email protected] lution by extending Fourier-Floquet analysis to apply to the non-local problem. We can generate the full spectra for all traveling wave solutions. In addition to recovering MS23 well-known results such as the Benjamin-Feir instability Excited States of the Gross-Pitaevskii Equation for deep water waves, we confirm the presence of high- with a Harmonic Potential in the Thomas-Fermi frequency instabilities for shallow water waves. Finally, I Limit discuss preliminary stability results of a two-dimensional surface with respect to two-dimensional perturbations. Excited states of Bose-Einstein condensates are considered in the semi-classical (Thomas-Fermi) limit of the Gross– Katie Oliveras Pitaevskii equation with repulsive inter-atomic interactions Seattle University and a harmonic potential. The relative dynamics of dark Mathematics Department solitons (density dips on the localized condensate) with re- [email protected] spect to the harmonic potential and to each other is approximated using the averaged Lagrangian method and the Bernard Deconinck method of Lyapunov-Schmidt reductions. This permits a University of Washington complete characterization of the equilibrium positions of [email protected] the dark solitons as a function of the chemical potential parameter. It also yields an analytical handle on the oscillation frequencies of dark solitons around such equilibria. MS23 The asymptotic predictions are generalized for an arbitrary Justified Asymptotics for Gap Soliton Bifurcations number of dark solitons and are corroborated by numerical from Gap Edges in Cubic 2D Schroedinger and computations for 2- and 3-soliton configurations. This is a Maxwell Models joint work with M. Coles and P. Kevrekidis.

Gap solitons in nonlinear periodic structures near spec- Dmitry Pelinovsky tral gap edges are formally approximated by slowly vary- McMaster University ing envelope models, coupled mode equations (CMEs). We Department of Mathematics provide a general technique for their derivation in the 2D [email protected] Schrödinger and vector Maxwell models in Bloch variables. Using the Lyapunov-Schmidt reduction, we show which MS23 class of CME solutions ensures the existence of a corresponding solution of the original model and provide H s On the Blow up Behavior of Solutions to the Fo- bounds on the asymptotic approximation. cusing Schrodinger Equation Tomas Dohlan We review the global behavior of solutions to the 3d fo- IWRMM, Karlsruhe Institute of Technology, cusing (cubic) nonlinear Schrödinger equation with finite Kaiserstr. 12, 76128 Karlsruhe, Germany energy initial data and discuss blow up criteria as well as [email protected] various dynamics of blow up solutions. Svetlana Roudenko Arizona State Universtity NW10 Abstracts 63

Department of Mathematics and Statistics flow (for the non-smoothed noise.) This involves in apply- [email protected] ing Bourgain’s idea on invariant measures for deterministic PDEs to stochastinc PDEs. This is a joint work with J. Quastel and G. Richards. MS24 Near Soliton Evolution in 2-d Schrödinger Maps Hiro Oh, Jeremy Quastel University of Toronto We consider the Schrödinger Map equation in 2 + 1 di- [email protected], [email protected] mensions, with values into §2. This admits a lowest energy steady state Q, namely the stereographic projection, which extends to a two dimensional family of steady states MS25 by scaling and rotation. We prove that Q is unstable in Stability of Transition Waves in Cahn-Hilliard the energy space H˙ 1. However, to achieve this we show Equations and Systems that within the equivariant class Q is stable in a stronger topology X ⊂ H˙ 1. This is joint work with Daniel Tataru. I’ll discuss stability of transition waves for the Cahn- Hilliard equation on Rn and for Cahn-Hilliard systems on Ioan Bejenaru R. In the case of Cahn-Hilliard equations, the underlying University of Chicago Hamiltonian structure allows for a relatively straightfor- [email protected] ward spectral analysis, and we find that transition waves are spectrally stable under quite general circumstances. Daniel Tataru Pointwise semigroup methods can then be applied to es- University of California, Berkeley tablish that spectral stability implies nonlinear stability. I [email protected] will review these results and discuss recent progress in the case of systems.

MS24 Peter Howard Texas A&M Concentration Compactness for Wave Maps [email protected] We will discuss a recent adaptation of the concentration compactness method for critical nonlinear waves due to Bongsuk Kwon Bahouri Gerard and Kenig Merle to the context of critical Texas A&M University wave maps with hyperbolic target. This is joint work with [email protected] W. Schlag.

Joachim Krieger MS25 University of Pennsylvania Coherent Structures in Recurrent Precipitation [email protected] We discuss propagation into unstable states in a simple, yet Wilhelm Schlag surprisingly rich, 2-species reaction diffusion system that University of Chicago arises as a model for recurrent precipitation and for solidi- [email protected] fication in undercooled liquids. We show that the invasion of unstable states can create homogeneous bulk states or transient periodic patterns. Transitions between different MS24 invasion regimes can be interpreted as heteroclinic bifur- Long-time Existence for Quasilinear Wave Equa- cations. The results are a crucial building block for a con- tions in Exterior Domains ceptual understanding of Liesegang pattern formation. We shall explore some new long time existence results for Arnd Scheel quasilinear wave equations with small initial data in exte- University of Minnesota rior domains. In particular, we focus on results where the School of Mathematics nonlinearity is permitted to depend on the solution not just [email protected] its derivatives. Jason Metcalfe MS25 University of North Carolina, Chapel Hill Conditional Stability of Special Solutions of the [email protected] Klein-Gordon Equation We explicitly construct the center-stable manifold for the MS24 steady state solutions of the Klein-Gordon equation, in- Well-posedness and Invariant Measure for the cluding the one dimensional equation. The main difficulty Stochastic KdV-Burgers Equation in the one-dimensional case is that the required decay of the Klein-Gordon semigroup does not follow from Strichartz es- In this talk, we consider the periodic stochastic KdV- timate alone. In this talk, I will explain how to resolve this Burgers equation (SKdVB) with additive noise. The noise issue by proving an additional weighted decay estimate, is white is time. However, it is a derivative of the white which will allow us to close the argument. noise in space. First, we establish local well-posedness via the Fourier restriction method by discussing how to treat Milena Stanislavova the rough noise. By establishing an a priori bound on the University of Kansas, Lawrence L2 norm (when the noise is controlled to be in L2 in space), Department of Mathematics we extend local-in-time solution to global ones. Lastly, we [email protected] show that the (spatial) white noise is invariant under the 64 NW10 Abstracts

MS25 [email protected] Stationary Radial Spots in a Planar Three- component Reaction-diffusion System MS26 In this presentation, we analyze the existence and stabil- Stress-driven Organization of Melt and Strain in ity of stationary radial spots of a planar three-component Deforming Partially Molten Rocks: A Synthesis of system of FitzHugh-Nagumo type. In particular, we re- Observations port on results on the most unstable eigenvalues that de- cide through which type of instability the stationary struc- We present experimental observations of deforming vis- tures become unstable. Numerical simulations suggest cous granular material (rocks) with small but intercon- that the third component is necessary to stabilize trav- nected melt fractions, in which melt organizes to form net- eling spot solutions as these solutions are unstable for its works of shear zones (waves of enhanced strain and melt). two-component analogue. These networks reduce effective viscosity and energy dissipation under a variety of boundary conditions. We con- Peter van Heijster strain kinetics by tracking time-space evolution of pattern Division of Applied Mathematics formation. Two-phase flow theory requires nonlinearities Brown University, Providence RI in the constitutive relation (in stress, melt fraction and [email protected] anisotropy) to fit observed patterns.

Bjorn Sandstede Ben Holtzman Brown University Lamont Doherty Earth Observatory bjorn [email protected] Columbia University [email protected]

MS26 Dan King Compaction-dissolution Waves in Magma Dynam- University of Minnesota ics Department of Geology and Geophysics [email protected] Viscous deformation of the solid matrix coupled to reactive porous flow of the buoyant melt in the interior of the Earth David Kohlstedt gives rise to a rich set of instabilities. We present a stabil- University of Minnesota ity analysis and high resolution simulations. A regime dia- [email protected] gram identifies four dynamic regimes: high-porosity channels form; compaction-dissolution waves form; both channels and waves form; system is stable. This wave instability MS26 leads to a migrating checkerboard pattern in the porosity, Rigorous Results on Model Equations for Magma and depletion of the soluble material along nodal lines, and Migration may offer an alternative explanation for observed depletion patterns. An outstanding problem in Earth science is to understand the migration of magma in the Earths interior. One pro- Marc A. Hesse posed method is the percolation of the melt through the University of Texas porous rock. We survey models for the transport of melt Department of Geological Sciences through this viscously deformable porous medium. Fur- [email protected] thermore, we present results on existence, uniqueness, and stability for reduced versions of these systems, which are Alan Schiemenz, Yan Liang dispersive, nonlinear wave equations. This work is joint Brown University with Marc Spiegelman and Michael I. Weinstein. [email protected], Yan [email protected] Marc Spiegelman Marc Parmentier Columbia University Brown University Lamont-Doherty Earth Obs. Geological Sciences [email protected] em [email protected] Michael I. Weinstein Columbia University MS26 Dept of Applied Physics & Applied Math Melting and Two-phase Flow in the Earth and [email protected] Planets Gideon Simpson This presentation reviews the influence of melt nanostruc- University of Toronto ture on migration, storage, and evolution of magma in the [email protected] deep interior of the Earth and other planets. Structure of intergranular contact controls the average surface tension, melt mobility, and effective elastic properties. In the plane- MS27 tary scale, such changes influence rate of magma transport, Collapse and Light Bullets in the Few-cycle Regime location of magma storage, seismic signature of melt reser- voir, and tidal dissipation in eccentric satellites. Few-cycle pulse propagation in a two level medium is considered in the two limits of low and high resonance fre- Saswata Hier-Majumder quency. Two different (2+1)-dimensional models of gen- University of Maryland eralized Kadmotsev-Petviashvili and sine-Gordon type are Department of Geology derived and studied. In one case collapse is evidenced, in NW10 Abstracts 65

the other case light bullets form. University of Washington [email protected] Herve Leblond LPhiA, EA 4464 J. Nathan Kutz University of Angers University of Washington [email protected] Dept of Applied Mathematics [email protected] Dumitru Mihalache Horia Hulubei National Institute for Physics and Nuclear Engineering, Bucharest MS28 [email protected] Shape Optimization of Helical Propellers for Low Reynolds Number Propulsion David Kremer LPhiA EA 4464, University of Angers, Certain bacterial species utilize helical flagella for propul- [email protected] sion in low Reynolds number environments. Recent experiments replicating this locomotion strategy use magnetic fields to rotate colloidal helices. We investigate MS27 the dependence of the performance and efficiency of such Modeling Photon Generators and Frequency Con- micro-swimmers on their geometry, via shape optimization vertors on helical bodies. We identify geometries that maximize rotation-translation coupling and efficiency. From this, Abstract not available at time of publication. we may draw conclusions regarding the performance of bacterial flagella and provide insights into effective micro- Colin McKinstrie swimmer design. Lucent Technologies [email protected] Eric E. Keaveny, Shawn Walker Courant Institute New York University MS27 [email protected], [email protected] Pedestal-Free Pulse Compression in Nonlinear Fibers and Nonlinear Fiber Bragg Gratings Michael Shelley Courant Institute of Mathematical Sciences We studied theoretically pedestal-free optical pulse com- New York University pression using self-similar chirped optical solitons in non- [email protected] linear optical fibers and near the photonic bandgap structure of Fiber Bragg gratings with exponentially decreasing dispersion. In addition to compression of self-similar MS28 fundamental solitons, we also studied optical pulse com- A Second Order Virtual Node Algorithm for Pois- pression using higher order chirped solitons using the son Interface Problems Bäcklund transformation method. The special chirped soliton breathers can achieve both the high degree compression Abstract not available at time of publication. and high quality compression. Joseph Teran Alexander Wai UCLA Hong Kong Polytechnic University [email protected] Dept. of Electronic and Information Engineering [email protected] MS28 Qian Li Dynamics and Transport of Nano-shuttle and Hong Kong Polytechnic University Nano-motors Department of Electronic and Information Engineering [email protected] Bio-polymers such as actin filaments or microtubules are important in many cell functions. Through interactions with molecular motors, the chemical energy of ATP MS27 (adenosine triphosphate) hydrolysis is converted to me- Two-dimensional Mode-locking in Planar Waveg- chanical work and the biopolymers can move or transport uide Arrays materials in the cells. In vitro motility assays have been used to characterize the dynamics of the biopolymers and A theoretical proposal is presented for the generation the molecular motors that drive the motions. These re- of mode-locked light-bullets in planar waveguide arrays, sults have been utilized for nano-cargo transport in cellu- extending the concept of time-domain mode-locking in lar size synthetic devices. In this talk we present a contin- waveguide arrays to spatial mode-locking in slab waveg- uum slender-body model for such bio-polymers immersed uides. The model presented yields three-dimensional local- in viscous fluid. We will show that this model gives rea- ized states that act as global attractors to the waveguide sonable agreement with the experiments. Simulations also array system. Single-bullet stationary and time-periodic show novel wave dynamics of these bio-polymers in differ- solutions are observed to be stabilized in this system. The ent forcing landscape. transition from single bullet to multiple bullet solutions is also studied. Yuan-Nan Young Department of Mathematical Sciences Matthew O. Williams NJIT Applied Mathematics [email protected] 66 NW10 Abstracts

Michael Shelley parametrix for the analysis of the NLS near the point of Courant Institute of Mathematical Sciences gradient catastrophe (following up the talk by A. Tovbis New York University in the same minisimposium). In particular we are faced [email protected] with the unusual problem of constructing and alternative parametrix in a neighbourhood of the points in the parameter space where the usual solution to the relevant RHP MS28 is known to not exist. The solution of this problem relies Algorithms for Simulating Vesicle Flows upon modifying the Riemann Hilbert problem for the Lax matrix of the PainlevéIequation. Vesicles are locally-inextensible fluid membranes that can sustain bending. We consider the dynamics of flows of Marco Bertola vesicles suspended in Stokesian fluids. We use a boundary Concordia University, Canada integral formulation for the fluid that results in a set of [email protected] nonlinear integro-differential equations for the vesicle dynamics. The motion of the vesicles is determined by bal- Alexander Tovbis ancing the nonlocal hydrodynamic forces with the elastic University of Central Florida forces due to bending and tension. Numerical simulations Department of Mathematics of such vesicle motions are quite challenging. On one hand, [email protected] explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives and a milder constraint due to a transport-like MS29 stability condition. On the other hand, an implicit scheme Semiclassical Focusing NLS with Square Barrier can be expensive because it requires the solution of a set Initial Data of nonlinear equations at each time step. We present a set of numerical techniques for efficient simulation of vesicle In this paper we present results on the small dispersion flows. The distinctive features of these numerical methods limit of the focusing nonlinear Schrödinger equation for include using boundary integral method accelerated with step initial data. Using Riemann-Hilbert techniques we fast multipole method, spectral (spherical harmonic) dis- derive rigorous pointwise asymptotics for the solution glob- cretization of deforming surfaces in space, and a careful ally in space and for times less then a O(∞) maximal time. choice of semi-implicit time-stepping scheme. We demon- In particular we observe that each discontinuity is regular- strate numerical schemes that experimentally are uncondi- ized by the immediate onset of genus-one oscillations in a tionally stable, and have low cost per time step. growing region of space time emanating from each discontinuity. To leading order these oscillations are described Denis Zorin by a slowly modulated one phase wave whose evolution is Computer Science Department given by the corresponding Whitham equations. Courant Institute, New York University [email protected] Robert Jenkins Department of Mathematics Shravan Veerapaneni University of Michigan Courant Institute [email protected] New York University [email protected] MS29 Abtin Rahimian, George Biros Universality of Gradient Catastrophe Approxima- Georgia Institute of Technology tion for the Semiclassical Focusing NLS, or how to [email protected], [email protected] see the Poles of the Tritronquee Solution of P-I. We present a complete description of the nondegenerate MS29 point of gradient catastrophe for the semiclassical limit of the focusing Nonlinear Schrödinger equation (NLS), which Numerical Studies of Breaking in the Semiclassi- include both the oscillatory and the nonoscillatoty parts of cal Solution of the Focusing Nonlinear Schrödinger a vicinity of the gradient catastrophe point. In particular, Equation we prove that: i) there is a one to one correspondence be- I will discuss parametric dependence of Riemann-Hilbert tween the spikes in the oscillatory part and the poles of the problems in connection with focusing NLS. The result is special tritronquée solution of the Painleve I (P1); ii) the based on numerical observations and it is obtained in an height of each spike is (asymptotically) 3 times the abso- abstract setup of Riemann-Hilbert problems. This is joint lute value of the backgrond; iii) Each spike has the uni- work with A.Tovbis and S.Venakides. versal shape of the (scaled) rational breather solution to the NLS; iv) all error estimates expressed in terms of Sergey Belov the tritronquée solution. All results were obtained through Department of Mathematics the nonlinear steepest descent method for Riemann-Hilbert Rice University problems and discrete Schlesinger transformations. There [email protected] seem to be an interesting connection with ”rogue” waves theory.

MS29 Alexander Tovbis Parametrices of Riemann-Hilbert Problems When University of Central Florida They Do Not Exist: Painlev´e I and the Shape of Department of Mathematics the First Spike at the Gradient Catastrophe of NLS [email protected] The talk will delve in the construction of the proper Marco Bertola NW10 Abstracts 67

Concordia University, Canada MS30 [email protected] A Lie-Transform Based Idea to Treat Weakly Stochastic Hamiltonian Systems MS30 Weakly stochastic Hamiltonian systems arise naturally in A Hybrid Waveplate Model for the Description of the context of fiber optics and soliton dynamics. The noise Polarization-mode Dispersion in Installed Optical interacts with the deterministic trajectories in a highly Fiber Transmission Systems complicated way and it is, in general, difficult to develop a systematic perturbation expansion due to operator asym- I will present a new mathematical model for polarization- metry in the associated Fokker-Planck equation. The pur- mode dispersion (PMD) in installed optical fiber transmis- pose of this talk is to introduce entirely new idea based on sion systems. The model is a generalization of various vari- Lie transforms to treat such systems. ants of the so-called hinge model of PMD, and it reduces to these variants in appropriate limits. I will then describe an Tobias Schaefer adaptive variance reduction technique that combines im- Department of Mathematics portance sampling and the cross-entropy method and that The College of Staten Island, City University of New York can be used to assess PMD-induced transmission impair- [email protected] mentsinthismodel.

Gino Biondini MS31 State University of New York at Buffalo A Boundary Integral Method for the Irrotational Department of Mathematics Water Wave biondini@buffalo.edu We present a boundary integral method for the efficient computation of 2D irrotational water waves. The fluid do- MS30 main considered is of finite depth, with a variable bottom. Capacity Limits of Fiber Optics Communication The height of the free surface, as well as the height of the Systems fixed bottom, can be multi-valued. Results of the simulations will be shown. Determining the maximum quantity of information that can be transported over optical fibers is a very challenging David Ambrose problem of both fundamental and practical interests. An Drexel University important challenge in establishing a “fiber capacity’ orig- [email protected] inates from the difficulty of dealing with fiber propagation governed by the stochastic nonlinear Schrödinger equation. Jon Wilkening In this symposium, we will describe an approach we devel- UC Berkeley Mathematics oped to calculate a capacity estimate of “fiber channels’ [email protected] and present capacity estimates results. We will also discuss the physical origins of capacity limitations. MS31 René-Jean Essiambre Absorbing Boundaries for 2D Free Surface Flow Bell Labs, Alcatel-Lucent [email protected] The evolution of water waves has long been modeled using nonlinear potential flow. In [S. Wu, Well-posedness in Gerhard Kramer Sobolev spaces of the full water wave problem in 2-D, In- University of Southern California vent. Math, 130 (1997), pp. 39-72], S. Wu proved the well- Department of Electrical Engineering posedness of the two-dimensional water wave problem, and [email protected] formulated a PDE that is equivalent to the incompressible irrotational Euler system. To leading order, the equation Gerard Foschini, Peter Winzer reduces to a linear nonlocal wave-like scalar PDE involving Bell Labs, Alcatel-Lucent the Hilbert transform. We are concerned with numerical [email protected], [email protected] solutions for water wave problems in unbounded domains, using this new formulation of the water wave equation. The computational domain is truncated to a finite size, MS30 and boundary conditions are required to make the domain Resolving the Raman-induced Cross-frequency boundaries transparent. An idea that has proven successful Shift in Soliton Collisions with other wave-like equations is to modify the equation in a region near the outer boundary in order to absorb outgo- Raman-induced frequency-shift in optical pulse collisions ing waves without generating reflections. In this talk, we (Raman cross frequency-shift) is a nonlinear process that will give an overview of the new formulation, introduce a plays an important role in the dynamics of ultra-short opti- one way version of the water wave equation, and incorpo- cal pulses. We present the results of high-resolution numer- rate wave damping. We will present numerical approaches ical simulations, which enable an accurate measurement of to solving the equation in both Fourier and physical space. the Raman cross frequency-shift in soliton collisions. The Numerical examples will be shown. measurements are based on a fine frequency grid, with frequency spacing as low as 10−3. The results of the simula- Geri I. Jennings tions are compared with approximate analytic predictions. University of Michigan [email protected] Quan M. Nguyen,AvnerPeleg State University of New York at Buffalo Smadar Karni qnguyen2@buffalo.edu, [email protected]ffalo.edu University of Michigan 68 NW10 Abstracts

Department of Mathematics assume some regularity of the exact solution. Numerical [email protected] experiments to demonstrate the stability and high accuracy of the proposed methods for chemotaxis models and Jeffrey Rauch comparison with other methods will be presented. University of Michigan [email protected] Yekaterina Epshteyn Carnegie Mellon University [email protected] MS31 An Efficient Boundary Integral Method for 3D In- MS32 terfacial Flow with Surface Tension Three Wave Interaction in Negative Index Meta- An efficient, non-stiff boundary integral method for 3D materials porous media flow with surface tension is presented. Sur- face tension introduces high order (i.e., high derivative) Three wave interaction in negative index material takes terms into the evolution equations, which leads to severe place in the presence of contra-propagating waves, which stability constraints for explicit time-integration methods. results from opposite directionality of wave and Poynting Furthermore, the high order terms appear in nonlocal oper- vectors. We studied parametric amplification and second ators, making the application of implicit methods difficult. harmonic generation in the presence of material losses and Our algorithm employs a special representation of the in- detuning from resonance phase matching condition. The- terface which enables efficient application of implicit time- oretical study is verified by direct numerical simulations. integration methods via a small-scale decomposition. The algorithm is found to be effective at eliminating the severe Ildar R. Gabitov time-step constraint that plagues explicit time-integration Department of Mathematics, University of Arizona methods. [email protected] Michael S. Siegel New Jersey Institute of Technology Zhaxylyk Kudyshev Department of Mathematical Sciences Department of Physics, Al-Farabi Kazakh National [email protected] University, Almaty, Kazakhstan [email protected] David Ambrose Drexel University Andrei Maimistov [email protected] Department. of Solid State Physics, MEPI, Russia Dept. of General Physics/REC Bionanophysics, MIPT, Svetlana Tlupova Russia Department of Mathematics [email protected] University of Michigan [email protected] MS32 Effective Dynamics of Double Solitons for Per- MS31 turbed mKdV Computation of Time-periodic Water Waves We show that an interacting double soliton solution to the H I will describe a spectrally accurate method for computing perturbed mKdV equation is close in 2toadoublesoli- time-periodic water waves of finite or infinite depth. We ton following the effective dynamics obtained as Hamilton’s use adjoint methods to minimize a functional (of the initial equations for the restriction of the mKdV Hamiltonian to condition and period) that is positive unless the solution is the submanifold of solitons. The interplay between alge- periodic, in which case it is zero. We find solutions of the braic aspects of complete integrability of the unperturbed true water wave resembling multi-phase cnoidal solutions equation and the analytic ideas related to soliton stability of KdV and observe interesting disconnections in the bi- is central in the proof. furcation diagrams due to nonlinear resonances at critical Justin Holmer bifurcation parameters. Department of Mathematics, Brown University, Jon Wilkening Box 1917, 151 Thayer Street, Providence, RI 02912, USA UC Berkeley Mathematics [email protected] [email protected] Galina Perelman Jia Yu Ecole Polytechnique UC Berkeley [email protected] [email protected] Maciej Zworski University of California, Berkeley MS32 [email protected] Numerical Methods for Chemotaxis Models In our work we propose a family of stable and highly ac- MS32 curate numerical methods, based on interior penalty dis- Dissipative Effects in Positive-negative Index Cou- continuous Galerkin schemes for the Keller-Segel chemo- pler taxis model. We prove hp error estimates for the proposed schemes. Our proof is valid for pre-blow-up times since we The intriguing example of the negative-positive refraction NW10 Abstracts 69

medium is the coupler, where one of the waveguides is fab- [email protected] ricated from a material with a negative refractive index. This device acts as a (distributed) mirror. The radiation Aaron Hoffman entering one waveguide leaves the device through the other Boston University waveguide at the same end but in the opposite direction. Department of Mathematics and Statistics The opposite directionality of the phase velocity and the [email protected] energy flow in this channel results in the gap soliton formation in a uniform nonlinear coupler without periodicity or feedback mechanism. The corresponding analytical so- MS34 lution is found and it is used for numerical simulation to Supercritical Fronts for Reaction-diffusion Equa- illustrate that the results of the solitary wave collisions tions in Infinite Cylinders are sensible to the relative velocity of the colliding solitary waves. In this talk I will present some recent results concerning the analysis of fronts invading into unstable equilibrium Andrei Maimistov in infinite cylinders. It will be shown that supercritical Department. of Solid State Physics, MEPI, Russia fronts of invasion can be viewed as critical points of cer- Dept. of General Physics/REC Bionanophysics, MIPT, tain functional which allows to use machinery of calculus of Russia variations. Consequently, the study of superctitical fronts [email protected] can be performed in a rather general framework with no structural assumptions on nonlinearity apart from minimal regularity requirements. This is a joint work with C.B. Mu- MS33 ratov and M. Novaga Well-posedness for Gauged Schrodinger Equations Peter Gordon We discuss conditions on the gauge potential under which NJIT we can have a blowup or global existence and scattering. [email protected] We show applications to some physical systems.

Magdalena Czubak MS34 University of Toronto Linear Stability for Solutions to the Vortex Fila- [email protected] ment Equation

In its simplest form, the self-induced dynamics of a vortex MS33 filament in a perfect fluid is governed by the Vortex Fil- Numerical Simulation of Resonant Tunneling for ament Equation (VFE), which is related to the Nonlinear Cubic Schrodinger Equation Schrödinger (NLS) equation via the Hasimoto map. The NLS is integrable and admits a corresponding AKNS lin- In this work, we show numerically the phenomenon of res- ear system. We show how the squared eigenfunctions of the onant tunneling for fast solitons through large potential AKNS system and the Hasimoto map can be used to con- barriers for the cubic Nonlinear Schrödinger equation in struct solutions of the linearized VFE and obtain stability one dimension with external potential. Resonant tunnel- properties of vortex filaments. ing is well known in linear scattering theory and refers to a situation where the reflection coefficient vanishes at cer- Stephane Lafortune, Annalisa M. Calini tain energies of incoming waves.We consider two classes College of Charleston of potentials, namely the ‘box’ potential and a repulsive 2- Department of Mathematics delta potential under certain conditions. We show that the [email protected], [email protected] transmitted wave is close to a soliton, calculate the transmitted mass of the solution and show that it converges to Scotty Keith the total mass of the solution as the velocity is increased. Department of Mathematics Xiao Liu University of North Carolina University of Toronto [email protected] [email protected] MS34 MS33 On the Traveling Waves in the Gray-Scott Model Weak Interactions Between Fronts in LDE For a wide range of parameter values we show the existence In order to better understand interactions between fronts in of rich families of traveling wave solutions of the Gray- lattice differential equations, we construct solutions which Scott model. We find pulse solutions, periodic wave trains, are roughly the superposition of two well-separated fronts. families of fronts that connect constant states, constant In particular, we construct a two-dimensional manifold of states to a periodic wave train, two periodic wave trains. such solutions (the two parameters should be thought of In certain singular limits, by using rescaled versions of the as the location of each of the fronts) and show that this equations, we pinpoint the structure of the traveling waves. manifold is dynamically stable. In particular, we discuss The results are anchored in geometric singular perturba- how our methods, initially developed for reaction-diffusion tion theory. PDE, carry over to lattice systems. This work is joint with Vahagn Manukian Aaron Hoffman. Miami University Doug Wright [email protected] Drexel University Mathematics 70 NW10 Abstracts

MS34 [email protected] Infinite Dimensional Evans Function and the Sta- bility Index MS35 In this talk, an over view is presented for a topological Traveling Waves in Thermally Driven Optical Para- frame work defined for eigenvalue problems of elliptic op- metric Oscillators erators. This theory is an infinite dimensional generalization of that of the classical Sturm-Liouvill theory to non The mean-field behavior of detuned optical parametric self-adjoint operators. Starting from the classical one, how oscillators subject to absorption-induced heating can be the theory is generalized to non self-adjoint problems and described using a parametrically forced scalar nonlinear to infinite dimensional setting will be explained. Schrödinger equation, which is shown to support tightly bound asymmetric traveling waves in addition to weakly Shunsaku Nii bound ‘multipole-mode’ solutions. We discuss the exis- Kyushu University tence and stability of these traveling waves in the scalar [email protected] equation and in the mean-field system from which it is derived.

MS35 Richard O. Moore Bifurcations and Stability of Boundstates in NLS New Jersey Institute of Technology via Reduced Equations [email protected]

The talk will focus on the existence and orbital stability of periodic in time, localized in space solutions (boundstates) MS35 of nonlinear Schrödinger equations. I will discuss recently Transitional Dynamics in the Reduced Model of a obtained necessary and sufficient conditions for the exis- Mode-locking Waveguide Array tence of bifurcation points along branches of boundstates, then I will classify the possible bifurcations and their ef- Recent studies on mode-locking in waveguide arrays have fect on stability of the branches via reduced equations. shown that an increase in gain can transition the system The results were obtained in collaboration with D. Peli- from N-toN + 1-pulse solution through a chaotic process novski (McMaster), P. Kevrekidis (UMass) and V. Natara- initiated by a Hopf bifurcation. This transition is ana- jan (UIUC). lyzed by projecting the governing PDEs onto finite set of orthogonal modes that capture most of the energy in the Eduard Kirr system. Bifurcation diagrams of the reduced model reveal Department of Mathematics a sequence of period-doubling bifurcations, identifying the University of Illinois at Urbana-Champaign mechanism responsible for the onset of chaos. [email protected] Eli Shlizerman University of Washington, Seattle MS35 [email protected] Long Time Dynamics Near the Symmetry Break- ing Bifurcation for Nonlinear Schrödinger/Gross- Matthew O. Williams Pitaevskii Equations Applied Mathematics University of Washington We consider a class nonlinear Schrödinger / Gross- [email protected] Pitaevskii equations (NLS/GP) with a focusing (attractive) nonlinear potential and symmetric double well linear J. Nathan Kutz potential. NLS/GP plays a central role in the modeling University of Washington of nonlinear optical and mean-field quantum many-body L2 Dept of Applied Mathematics phenomena. It is known that there is a critical norm [email protected] (optical power / particle number) at which there is a symmetry breaking bifurcation of the ground state. We study the rich dynamical behavior near the symmetry breaking MS36 point. The source of this behavior in the full Hamilto- Continuation of Solutions of the Nonlinear nian PDE is related to the dynamics of a finite-dimensional Schrodinger Equation Beyond the Singularity Hamiltonian reduction. We derive this reduction, analyze a part of its phase space and prove a shadowing theorem on The continuation of NLS solutions beyond the singularity the persistence of solutions, with oscillating mass-transport hasbeenanopenproblemformanyyears.InthistalkI between wells, on very long, but finite, time scales within will present several novel approaches to this problem, and the full NLS/GP. The infinite time dynamics for NLS/GP discuss their consequences. are expected to depart, from the finite dimensional reduction, due to resonant coupling of discrete and continuum / Gadi Fibich radiation modes. Tel Aviv University School of Mathematical Sciences Jeremy Marzuola fi[email protected] Department of Applied Mathematics Columbia University Moran Klein [email protected] Tel Aviv University [email protected] Michael I. Weinstein Columbia University Dept of Applied Physics & Applied Math NW10 Abstracts 71

MS36 photorefractive crystal. Predicting the Filamentation of High-power Beams and Pulses without Numerical Integration: A Non- Can Sun, Jason Fleischer linear Geometrical Optics Method Princeton University [email protected], [email protected] We present an analytic method for predicting the initial self-focusing dynamics of high-power beams and pulses. Using this method we study the filamentation pattern of a MS37 variety of input profiles in 1D, 2D and 3D, without solving Title Not Available At Time of Publication partial differential equations. In particular, this method explains why super-Gaussian initial conditions lead to ring- Abstract not available at time of publication. type blowup of NLS solutions and predicts a new type of George Biros temporal splitting. Georgia Tech Nir Gavish Computational Science and Engineering Mathematics [email protected] Michigan State University [email protected] MS37 Simulation of Tumor Growth - Free Boundary Gadi Fibich Problems using Octrees Tel Aviv University School of Mathematical Sciences Simulation of biological processes often require the solu- fi[email protected] tion of partial differential equations (PDEs) on large data set. I will present a novel paradigm for solving PDEs on Alexander Gaeta octree data structures. Second-order accuracy is achieved School of Applied and Engineering Physics for several classes of relevant PDEs. Cornell University [email protected] Frederic G. Gibou UC Santa Barbara Luat Voung [email protected] The Institute of Photonic Sciences [email protected] MS37 Shivers, Blisters and Waves in Eukaryotic Cilia and MS36 Flagella Cascaded Four-wave Mixing and Spatial Supercon- The undulating motions of eukaryotic flagella and cilia pro- tinuua duce forces that cause locomotion. While it is known that We experimentally and numerically study nonlinear light these oscillations (bending waves) form as a result of the propagation in a fractal waveguide array. We consider a coupling between the filament backbone (geometry and me- nested set of periodic arrays and examine energy transport chanics) and active forces due to molecular motors (kinet- as a function of band structure, nonlinearity, and probe ics), the mechanisms that result in sustained oscillations beam geometry. Experimentally, we observe the behavior are unclear and are still intensely researched. I will de- directly in position and momentum spaces. The results scribe a minimal model for the onset and propagation of are fundamental to nonlinear wave dynamics in self-similar motor driven bending waves that is motivated by recent ex- structures and hold potential to improve the efficiency and perimental results. Equations that describe the evolution sensitivity of fractal photonic devices. of the force generating active motors are coupled to filament configuration and elasticity. Appropriate ensemble Shu Jia averaged force-velocity relationships for the motors com- Princeton University pletes the set of equations. These equations allow us to Department of Electrical Engineering understand and interpret three types of flagellar motions [email protected] - 1) shivers - localized shape perturbations that decay in time and space, 2) blisters - a single pronounced bump that Jason Fleischer travels along the filament and 3) waves - sustained bending Princeton University oscillations that emerge under favorable conditions. [email protected] Arvind Gopinath MIT MS36 Mechanical Engineering [email protected] Condensation of Random Classical Waves he nonlinear propagation of random fields can be described MS37 by thermodynamic arguments. For quantum fields, a major consequence is (Bose-Einstein) condensation, in which An Integrated Model of Microtubule-based Pronu- the ground state is macroscopically occupied. Here, we clear Motion in the Single-celled C. Elegans Embryo show using wave turbulence theory that classical fields can We present an integrated model of microtubule-based also condense, with thermalization manifesting itself by pronuclear motion in the single-celled C. elegans embryo. means of an irreversible evolution towards the state of max- This theoretical model determines how the male pronucleus imum disorder. The theory is confirmed by observing the is transported through the single-celled embryo in the early condensation of classical optical waves in a self-defocusing stages of development. In this model, centrosomes initiate 72 NW10 Abstracts

stochastic microtubule growth and these microtubules in- bility condition is satisfied as well as functions for which teract with motor proteins distributed in the cytoplasm. it fails. We will use steepest descent techniques for oscilla- Consequent pulling forces drag the pronucleus through the tory Riemann-Hilbert problems to obtain asymptotics for cytoplasm, here modeled as a viscous incompressible New- the solutions to the MNLS with these initial conditions. tonian fluid whose motions are constrained by contact with the cell periphery. The cell periphery also limits micro- Jeffery DiFranco tubule growth. Our computational method is based on an Seattle University immersed boundary formulation which allows for the si- [email protected] multaneous treatment of fluid flow and the dynamics of structures immersed within. We find that the constraining Peter D. Miller geometry of the shell has a large effect on the microtubule University of Michigan, Ann Arbor forces necessary to move the pronucleus on experimentally [email protected] observed time-scales, being an order of magnitude larger than those estimated in previous studies that do not take the effect of the shell into account when computing cyto- MS38 plasmic motions. Our simulations shows pronuclear migra- Chords and Solitons: The KP Solitons Revisited tion, and moreover, a pronuclear centration and rotation very similar to that observed in vivo. I will present a brief summary of the recent development of classification theorem of the KP solitons [Chakravarty and Tamar Shinar Kodama, 2008-9]. The theorem shows that each soliton Courant Institute solution can be parametrized by a unique chord diagram New York University representing a derangement of the permutation group. In [email protected] view of the theorem, I will study several real waves observed in shallow water, and show that the KP equation Fabio Piano provides an excellent model equation to describe resonant Center for Genomics and Systems Biology phenomena for those shallow water waves. I will also dis- Department of Biology, NYU cuss the differences of KP solitons from the waves in real [email protected] experiments. Yuji Kodama Michael J. Shelley Ohio State University New York University Department of Mathematics Courant Inst of Math Sciences [email protected] [email protected]

MS38 MS38 Random Normal Matrices via Riemann-Hilbert Stability of Periodic Solutions of Nonlinear Disper- Analysis sive Equations Random hermitian matrix model has been studied quite We present a theorem relating the number of potential in- successfully by applying the steepest descent analysis of the stabilities of a periodic KdV travelling wave to the Hes- corresponding Riemann-Hilbert problem. We show that sian of the classical action governing the travelling wave the same technique can be applied to random normal ma- ordinary differential equation. In the case of a polynomial trix model. This model is equivalent to certain weighted nonlinearity this can be further simplified using the fact Bergman polynomials. We perform the asymptotic analy- that the action is a hyperelliptic function and satisfies a sis on that polynomial. We consider a special case as an Picard-Fuchs relation. example. This is a joint work with Ferenc Balogh, Marco Jared Bronski Bertola and Ken McLaughlin. University of Illinois Urbana-Champain Seung-Yeop Lee Department of Mathematics California Institute of Technology [email protected] [email protected]

Mat Johnson Marco Bertola, Ferenc Balogh Indiana Indiana Concordia University [email protected] [email protected], [email protected] Todd Kapitula Calvin College Ken McLaughlin [email protected] University of Arizona [email protected] MS38 Semiclassical Analysis of the Modified Nonlinear MS39 Schrodinger Equation Diffusive Transport in Two-dimensional Nematics The Modified nonlinear Schrödinger equation (MNLS) ex- The spatially-extended Onsager-Maier-Saupe model of ne- hibits both modulationally stable and unstable behavior matic liquid crystals is a variational model in which the for different initial conditions. In this talk we will inves- free energy is a functional of a spatially-dependent prob- tigate the semi-classical asymptotics for a family of initial ability density of orientations. I will discuss the diffusive conditions that contains both functions for which the sta- transport dynamics associated with this model (Doi kinetic NW10 Abstracts 73

equation) and will derive equations governing the evolution periodic solutions and a three-parameter family of solitary of vortex-like patterns in the high concentration (long rods) wave solutions. We numerically study the linear stability limit of the two-dimensional model. of these solutions and present a parameter-dependent stability cut-off. Further, we asymptotically solve the wave Ibrahim Fatkullin shoaling problem in the slowly-sloping bathymetry limit. University of Arizona Department of Mathematics John Carter [email protected] Seattle University Mathematics Department [email protected] MS39 Dielectrophoretic Assembly of Nematic Structures Rodrigo Cienfuegos in Dispersions of Metal Nanorods Hydraulic and Environmental Engineeering Department Pontificia Universidad Catolica de Chile The greatest challenge in the development of nanoscale- [email protected] structured metamaterials is to achieve the required spatial arrangement of elements and to switch these arrangements whenever necessary. We demonstrate experimentally an MS40 approach to construct a metamaterial in which metallic An Asymptotic Expansion for Solitary Gravity- nanorods, of dimension much smaller than the wavelength capillary Waves of light, are suspended in a fluid and placed in a nonuniform electric field. The field controls the spatial distribution and In this talk, we present high-order asymptotic series for orientation of nanorods because of the dielectrophoretic ef- one dimensional gravity-capillary solitary waves; the first fect. The field-controlled placement of nanorods causes term in the asymptotic series is the well-known sech2 so- optical effects such as varying refractive index and optical lution of the KdV equation. The series is used (with nine anisotropy (birefringence). terms included) to investigate how small surface tension af- fects the height and energy of large amplitude waves, and Oleg Lavrentovich waves close to the solitary version of the Stokes’ extreme Kent State University wave. In particular, for small surface tension, the solitary [email protected] wave with the maximum energy is obtained. For large surface tension, the series is also used to study the energy of MS39 depression solitary waves. Energy considerations suggest that, for large enough surface tension, there are solitary Some Recent Results on the Smoluchowski Equa- waves that can get close to the fluid bottom. tion Arising in the Modeling of Nematic Liquid Crystals Terry Haut University of Colorado, Boulder The talk will address two recent results concerning the [email protected] Smoluchowski equation and the Onsager model for nematic liquid crystals. The first result concerns the existence of inertial manifolds for the Smloluchowski equation both on MS40 the circle and on the sphere. Inertial manifolds is a concept Rogue Waves, Dissipation and Downshifting developed in the context of parabolic dissipative PDEs, which, when they exist, allows to in a sense view the PDE Damping plays an important role in stability and down- as a finite-dimensional dynamical system. The second re- shifting of waves. The physical and statistical properties sult concerns the isotropic-nematic phase transition for the of rogue waves in deep water are investigated using the Onsager model using more complicated potentials than the focusing Nonlinear Schrödinger equation and the Dysthe Maier-Saupe potential. equation with an additional damping term. The effects of both linear and nonlinear damping on the development of Jesenko Vukadinovic rogue waves and the interaction between rogue waves and Department of Mathematics downshifting are examined using numerical investigations CUNY, Staten Island and analytical arguments based on the inverse spectral the- [email protected] ory of the underlying integrable model, perturbation analysis, and statistical methods. MS39 Constance Schober Phases and Flow Behavior of Biaxial Liquid Crys- Department of Mathematics tals University of Central Florida [email protected] Abstract not available at time of publication.

Qi Wang MS41 University of South Carolina [email protected] Solitary Waves in Non-dispersive Periodic Media Solitary waves generally arise through a balance of dis- MS40 persion and nonlinearity. A special kind of solitary waves have been discovered that arise in non-dispersive piecewise- Periodic Solutions of the Serre Equations homogeneous periodic elastic materials. These waves, The Serre equations are a model for the evolution of surface dubbed stegotons, owe their existence to an effective dis- waves on an incompressible, irrotational, inviscid, shallow persion resulting from material heterogeneity. In fact, such fluid. These equations admit a four-parameter family of waves may arise quite generally in first-order hyperbolic 74 NW10 Abstracts

systems with periodically varying (in space) coeﬃcients. [email protected] We show through computational examples the diversity of situationsthatcanleadtosuchwaves.Wealsopresentan- alytical and computational results that quantify the condi- MS42 tions leading to stegoton-like waves. Finally, we discuss an Singularities and Asymptotics for Some Dynamics interesting connection of these waves to the K(2,2) com- of Maps into the Sphere pacton equation. I will describe results on singularity (non-)formation and David Ketcheson stability, in the energy-critical 2D setting, for some non- KAUST linear Schroedinger-type systems of geometric origin – the Applied Mathematics Schroedinger map and Landau-Lifshitz equation – which [email protected] model dynamics of ferromagnets and liquid crystals. Stephen Gustafson MS41 Department of Mathematics Nonlinear, Nonlocal, Degenerate, and Dispersive - University of British Columbia What More Could You Ask For? [email protected]

The magma equations are a family nonlinear, nonlocal, degenerate, dispersive wave equations that arise from models MS42 of magma migration in the Earth’s interior. It is conjec- Internal Modes of Discrete Solitons Near the Anti- tured that for the physically meaningful range of nonlin- continuum Limit of the dNLS Equation earities, the equations do not develop ﬁnite time singularities. In this talk, we present cases for which we can prove Discrete solitons of the discrete nonlinear Schrdinger global well-posedness. We then discuss the challenges of (dNLS) equation become compactly supported in the anti- other cases, particularly for dimensions greater than one. continuum limit of the zero coupling between lattice sites. This work is joint with Marc Spiegelman and Michael I. Eigenvalues of the linearization of the dNLS equation at Weinstein. the discrete soliton determine its spectral and linearized stability. All unstable eigenvalues of the discrete solitons Gideon Simpson near the anti-continuum limit were characterized earlier for University of Toronto this model. We analyze the resolvent operator and prove [email protected] that it is uniformly bounded in the neighbourhood of the continuous spectrum if the discrete soliton is simply connected in the anti-continuum limit. This result rules out MS41 existence of internal modes (neutrally stable eigenvalues of Compactons, Solitary Waves between Solitons and the discrete spectrum) of such discrete solitons near the Localized Blow-up Solutions anti-continuum limit.

Compactons are solitary waves with compact support ap- Anton Sakovich pearing in evolution equations with nonlinear dispersion. A McMaster University comparison of solitons of the Korteweg-de Vries equation [email protected] and compactons of the Rosenau-Hyman K(2,2) equation is presented based on a phase plane analysis. The relation between compactons of the K(2,2) and the localized MS42 blow-up solutions of nonlinear reaction-diffusion combus- Near-linear Dynamics in KdV with Periodic tion models is presented. Shock waves, rarefaction waves Boundary Conditions and blowup solutions of the K(2,2) are also discussed. The consequence of these solutions on the numerical analysis We show that the evolution of periodic (in space) high of evolution equations with compactons is emphasized, in- frequency solutions of the Korteweg de Vries equation is cluding the apperance of numerically-induced phenomena. almost linear for large times. Francisco R. Villatoro Nikos Tzirakis, Burak Erdogan E.T.S. Ingenieros Industriales, Universidad de Málaga University of Illinois, Urbana-Champaign Dep. Lenguajes y Ciencias de la Computación [email protected], [email protected] [email protected] Vadim Zharnitsky Department of Mathematics MS41 University of Illinois Analytical Results for Equations with Nonlinear [email protected] Dispersion At this time, there is little existence theory for equations MS42 in which the mechanism which generates dispersion is it- Solitary Waves for the Hartree Equation with a self nonlinear. In this talk, I will present a new equations Slowly Varying Potential which possesses compactly supported traveling waves (a hallmark of degenerate dispersive equations) and for which We study the Hartree equation with a slowly varying the Cauchy problem has a weak solution. This work is joint smooth potential V (x√)=W (xh),andwithaninitialcon- with D. Ambrose. dition which is ≤ h away in H 1 from a soliton. We show that up to time | log h|/h and errors of size + h2 J. Douglas Wright in H 1, the solution is a soliton evolving according to the Drexel University classical dynamics of a natural effective Hamiltonian. This Department of Mathematics NW10 Abstracts 75

result is based on methods of Holmer-Zworski, who prove value to the number of eigenvalues with negative Krein a similar theorem for the Gross-Pitaevskii equation, and signature. Because the construction of the Krein matrix is on the spectral estimates for the linearized Hartree oper- functional analytic in nature, it can be used for eigenvalue ator recently obtained by Lenzmann. We also provide an problems posed in more than one space dimension. extension of the result of Holmer-Zworski to more general initial conditions. Todd Kapitula Calvin College Ivan Ventura Dept of Math & Statistics UC Berkeley [email protected] [email protected]

Kiril Datchev MS43 University of California, Berkeley An Unexpected Geometric Interpretation of Krein [email protected] Signature Krein signature is an algebraic quantity that proved to be MS43 extremely helpful in spectral stability analysis. But conve- Stability of Waves in Liquid Films on Vibrating nience of its use is often complicated by inability to deter- Substrates mine it from eigenvalue diagrams produced by numerical calculations. We demonstrate a case when the geometric We study thin liquid films on horizontally vibrating sub- interpretation of Krein signature unexpectedly appears in strates. Using an equation derived by Shklyaev, Al- a class of nonlinear eigenvalue problems with applications abuzhev, and Khenner (Phys. Rev. E 79, 051603, 2009), in hydrodynamic stability or electric power systems. Our we show that all periodic and solitary-wave solutions of technique is based on a simple geometric intuition that sug- this equation are unstable regardless of their parameters. gests to reduce an analysis of a real part of the spectrum This is a joint work with E.S. Benilov. of a nonlinear eigenvalue problem to an analysis of spectra of an one-parameter family of linear operators without Marina Chugunova extending the dimension of the underlying space. Department of Mathematics University of Toronto Richard Kollar [email protected] Comenius University Department of Applied Mathematics and Statistics [email protected] MS43 The Stability of Finite-genus Solutions of Inte- grable Equations MS44 Symmetry-breaking Bifurcation in the Gross- The last few years we have been able to establish the or- Pitaevskii Equation with a Double-well Potential bital stability of the periodic finite-genus solutions of the KdV equation, as well as the stationary periodic solutions We classify bifurcations of the asymmetric states from of the defocusing NLS equation and the MKdV equations. a family of symmetric states in the focusing (attractive) To accomplish this, we have been combining ideas from in- Gross-Pitaevskii equation with a symmetric double-well tegrability with classical methods from dynamical systems potential. Depending on the shape of the potential, both involving the Krein signature and the classical work of Gril- supercritical and subcritical pitchfork bifurcations may oc- lakis, Shatah and Strauss. In this talk I will outline how cur. We also consider the limit of large energies and one can proceed to establish the orbital stability of the pe- show that the asymmetric states always exist near a non- riodic finite-genus solutions of integrable equations which degenerate extremum of the symmetric potential. These are characterized by a Lax pair whose first component is states are stable (unstable) in the case of subcritical non- self adjoint. linearity if the extremum is a minimum (a maximum). All states are unstable for large energy in the case of super- Bernard Deconinck critical nonlinearity. This is a joint work with E. Kirr and University of Washington P. Kevrekidis. [email protected] Dmitry Pelinovsky McMaster University MS43 Department of Mathematics The Krein Signature, Krein Eigenvalues, and the [email protected] Krein Oscillation Theorem

In this paper the problem of locating eigenvalues of nega- MS44 tive Krein signature is considered for operators of the form Band Gaps, Localization and Solitons in Disor- JL,whereJ is skew-symmetric with bounded inverse and dered Materials L is self-adjoint. A ﬁnite-dimensional matrix, hereafter referred to as the Krein matrix, is constructed with the In this talk I discuss spectral gaps in Schr¨odinger operators property that if the√ Krein matrix has a nontrivial kernel with potentials that are completely disordered (no delta- for some z0,then± −z0 ∈ σ(JL). The eigenvalues of the function peaks in Fourier space). Such gaps are counterin- Krein matrix, i.e., the Krein eigenvalues, are real mero- tuitive since without periodicity, Bloch’s theorem does not morphic functions of the spectral parameter, and have the apply, and thus there is no ”band splitting” mechanism for property that their derivative at a zero is directly related to gap formation. Physically, these band gaps are exhibited the Krein signature of the eigenvalue. The Krein Oscilla- in amorphous semiconductors and insulators. Here, I will tion Theorem relates the number of zeros of a Krein eigen- present numerical and experimental results on an optical system that exhibits an amorphous gap, as well nonlinear 76 NW10 Abstracts

effects such as gap solitons in both two and three dimen- sively modelocked lasers to date can be modeled using the sions. nonlinear Schrödinger equation and its modifications; this new type of short-pulse laser must be modeled using the Mikael C. Rechtsman Maxwell-Bloch equations and their modifications. Courant Institute New York University Curtis R. Menyuk [email protected] UMBC Baltimore, MD [email protected] MS44 Solitons, Defect Modes and Effective Mass Muhammad Talukder University of Maryland Baltimore County I will discuss the bifurcation of spatially localized de- Comp.Sci.andElectricalEng.Dept. fect modes into spectral gaps for the linear and nonlin- [email protected] ear Schroedinger / Gross-Pitaevskii (NLS / GP) equations with a periodic potential. Our analysis uses a Lyapunov- Schmidt reduction based on a spectral localization ”near MS45 to” and ”far from” a spectral band edge. Effective mass Amplifier Similaritons in a Fiber Laser (associated with the homogenized periodic structure) plays an important role in the dynamic stability of soliton-like Amplifier similaritons are observed in a fiber laser. This states of NLS / GP. This talk is based on joint with B. Ilan constitutes a new pulse shaping mechanism with remark- and with M. Hoefer. able features: the parabolic pulse is a local attractor in a segment of the oscillator, the pulse evolution exhibits Michael I. Weinstein large spectral breathing, and the pulse chirp is less than Columbia University the group-velocity dispersion of the cavity. In addition, a Dept of Applied Physics & Applied Math simple source generating parabolic pulses with high ener- [email protected] gies and the shortest pulses from a normal-dispersion laser will have use in applications.

MS44 William H. Renninger, Andy Chong, Frank Wise Unified Description of Wave Envelope Dynamics in Cornell University Simple Periodic Lattices [email protected], [email protected], [email protected] Wave envelope propagation in 2-D simple periodic nonlinear media is studied. A discrete approximation, known as the tight-binding approximation, is employed to find the MS45 linear dispersion relations and the governing equations of Dynamics of Few-cycle Ti:sapphire Lasers thediscretewaveenvelopedyanmics.Aunifieddescrip- tion of the dynamics is obtained for any input envelopes We will present a one-dimensional laser model based on and for arbitrary 2-D simple periodic lattices. These gov- the Nonlinear Schroedinger Equation which accurately erning equations are discrete evolution equations which are describes the pulse dynamics of few-cycle, dispersion- highly related to the linear dispersion relations. If the en- managed Ti:sapphire lasers. This allows us to predict the velopes vary slowly in space,the continuous limits of these output spectrum and pulse shape in good agreement with discrete equations are effective nonlinear Schrödinger equa- experimental results. By extending this model, the carrier- tion. The coefficients of the dispersive terms are related envelope phase dynamics in these lasers are analyzed and to the second differentiation of the linear dispersion rela- the major contributions to the carrier-envelope phase shift tions. The case that the dispersion relations are degenerate quantitatively identified. is also studied and the corresponding governing equations are coupled effective NLS equations. Michelle Y. Sander Electrical Engineering and Computer Science Yi Zhu MIT Department of Applied Mathematics [email protected] University of Colorado at Boulder [email protected] Erich Ippen MIT MS45 [email protected] Self-induced Transparency Modelocking in Quan- tum Cascade Lasers Franz Kaertner MIT, Department of EECS Standard (interband) semiconductor lasers operate in a Boston, MA limited wavelength range, below about 4 microns. Quan- [email protected] tum cascade (intraband) semiconductor lasers (QCLs) that operate in the mid-IR and far-IR have important applications to medicine, environmental sensing, and national MS45 security. While short pulse intersubband semiconductor High-energy Fiber Lasers based on Dissipative Soli- lasers (∼ 100 fs) are available, that is not the case for tons QCLs. Standard passive modelocking is hard to do in QCLs because of their long coherence times and short gain Recent research has shown that it is possible to generate ul- recovery times. We propose a fundamentally different ap- trashort pulses in a laser with only normal-dispersion com- proach, based on the self-induced-transparency (SIT) ef- ponents. Such a pulse balances amplitude modulation as fect, that turns these weaknesses into strengths. All pas- well as the phase modulations, and so is a dissipative soli- NW10 Abstracts 77

ton. This approach allows systematic study of dissipative what has been learnt about the binary black hole prob- solitons, as well as simple and practical fiber lasers with lem over the past several years from numerical simulations unprecedented performance. Theoretical and experimen- of the Einstein field equations, focusing on the more ”ex- tal progress will be reviewed. treme” solutions obtained in the high velocity limit. This is of possible relevance to LHC and cosmic ray physics in cer- Frank Wise tain proposed large extra dimension scenarios. Some of the Cornell University interesting results include the near-Planck scale luminos- [email protected] ity in radiated gravitational waves, recoil velocities of on the order of ten thousand kilometers per second or larger, zoom-whirl orbital motion, the formation of near-extremal MS46 Kerr black holes, and that in the ultra relativistic limit the The Einstein-Maxwell System in 3+1 Form and Ini- internal nature of the colliding object, whether black holes tial Data for Multiple Charged Black Holes or not, seemingly becomes irrelevant. In this talk I discuss the Einstein-Maxwell system as an Frans Pretorius initial value problem using the 3+1 formalism. I also focus Department of Physics on the problem of finding initial data for multiple charged Princeton University black holes assuming time-symmetric initial data and us- [email protected] ing a puncture-like method to solve the Hamiltonian and Gauss constraints. I discuss the behavior of the resulting initial data families, and show that previous results in this MS46 direction can be obtained as particular cases. Mathematical and Computational Challenges in Generating Templates for Gravitational Wave De- Miguel Alcubierre tections Nuclear Sciences Institute Universidad Nacional Autonoma de Mexico The generation of gravitational wave templates is critical [email protected] for the detection of signals such as binary black hole collisions, among others. I will review the current status of numerical simulations of such scenarios, the mathematical MS46 and computational challenges, and some current work to The Initial Data Problem for Magnetized Neutron address those challenges. Stars Manuel Tiglio While magnetic fields are ubiquitous features of stellar sys- Department of Physics tems, it is only relatively recently that the evolution of University of Maryland compact objects such as neutron stars have been simu- [email protected] lated using both general relativity and ideal magnetohy- drodynamics. An important aspect of these simulations is beginning with physically reasonable initial data. To MS47 this end, we consider the problem of constructing equilib- The Semiclassical Limit of the Sine-Gordon Equa- rium configurations of axisymmetric, polytropic stars with tion strong magnetic fields in general relativity. The small-dispersion or semiclassical sine-Gordon equation Eric W. Hirschmann is an integrable model of magnetic flux propagation in Brigham Young University long Josephson junctions. While in principle the inverse- [email protected] scattering method gives the exact solution for any well- posed initial data, in practice asymptotic analysis often yields the most useful information about the behavior of MS46 the solution. We first present a recent spectral confinement Compact Systems as Inducers of Waves in Electro- result, giving conditions when the spectrum of the system magnetic and Gravitational Bands must lie in certain regions. Secondly, we discuss ongoing work on the asymptotic behavior of solutions near criti- Strongly gravitating systems can produce both gravita- cal points where the qualitative behavior of the solution tional and electromagnetic radiation. Detection and anal- changes. Parts of this work are joint with Peter Miller. ysis of them would provide a unique way to study these systems and fundamentally advance our understanding of Robert J. Buckingham highly energetic phenomena and our cosmos. This talk will Dept. of Mathematical Sciences present results related to these system and discuss their im- The University of Cincinnati plications. [email protected] Luis Lehner Perimeter Institute for Theoretical Physics and Peter D. Miller University of Guelph University of Michigan, Ann Arbor [email protected] [email protected]

MS46 MS47 High Speed Black Hole Collisions Caustics, Cluster Expansions and Counting Graphs The class of spacetimes describing the merger of two black This talk will explore the structure of the large N (genus) holes contain some of the most fascinating solutions to the expansion of the free energy for Hermetian random matri- equations of general relativity. In this talk I will review ces. In particular, we will explain that the coeﬃcients of 78 NW10 Abstracts

these expansions are generating functions for the asymp- can evolve even in the presence of simple local update totic (as the matrix size, N, grows) correlations of the Gibbs rules. Specifically, we propose a discrete-time stochastic measures underlying the random matrix ensembles. The model of the evolution of a layered neuronal network which main result is that these generating functions are in fact is capable of learning through plasticity of the connec- fairly explicit rational functions of a distinguished alge- tions between layers. The structure of the network is thus braic function that is itself a geometric generating func- modulated by the conductances of the connecting neurons, tion. Moreover, in the double scaling limit of ”2D quantum which evolve according a model mimicking long-term po- gravity” these coefficients limit, in a precise combinatorial tentiation. We demonstrate that the network is capable of sense, to the coefficients in the asymptotic expansions of rich properties (e.g. bifurcation, various forms of stability, solutions to the Paineleve I hierarchy. This result has sig- etc.) that depend on maximum possible values of conduc- nificant applications to problems of recent interest in enu- tance, the inputs to the network, and the number of levels merative combinatorics. in the integrate-and-fire output neuron model. We will also remark on the Lyapunov exponents and information- Nicholas Ercolani theoretic properties of such networks. University of Arizona [email protected] Lee DeVille University of Illinois Department of Mathematics MS47 [email protected] On Long-time Asymptotics for the KdV Equation

We will present recent results concerning the rigorous MS48 proof, via Riemann-Hilbert techniques, of the long-time Reliability of Coupled Oscillators asymptotics for the Korteweg deVries equation. We concentrate in particular on the asymptotics in the so-called This talk concerns layered networks of coupled oscillators collisionless-shock region, as well as on new techniques re- and the reliability of such networks. Reliability means that lated to Riemann-Hilbert problems which arise during our upon repeated presentations of a given stimulus, the net- derivation. work gives essentially the same response each time. I will present an analysis of how conditions within such layered Percy Deift networks aﬀect their reliability. This is joint work with Courant Institute, NYU Eric Shea-Brown and Lai-Sang Young. [email protected] Kevin K. Lin Irina Nenciu Department of Mathematics University of Illinois at Chicago University of Arizona [email protected] [email protected]

Eric Shea-Brown MS47 Department of Applied Mathematics Semiclassical Limit of the Scattering Transform for University of Washington the Focusing NLS [email protected]

The semiclassical limit of the focusing Nonlinear (cubic) Lai-Sang Young Schrödinger Equation (NLS) corresponds to the singularly Courant Institute of Mathematical Sciences perturbed Zakharov Shabat (ZS) system that defines the New York University direct and inverse scattering transforms (IST). We derive [email protected] explicit expressions for the leading order terms of these transforms, which we call semiclassical limits of the direct and inverse scattering transforms. Thus, we establish an MS48 explicit connection between the decaying initial data of the Synchrony in Stochastic Pulse-coupled Neuronal q x, A x eiS(x) form ( 0) = ( ) and the leading order term of its Network Models scattering data. This connection is expressed in terms of an integral transform that can be viewed as a complexified We investigate the interplay between fluctuations, which version of the Abel transform. Our technique is not based de-synchronize, and synaptic coupling through net- on the WKB analysys of the ZS system, but on the in- work connections, which synchronize model networks of version of the modulation equations that solve the inverse integrate-and-fire neurons. By calculating the probabil- scattering problem in the leading order. The results are ity to remain synchronized, we explore the significance of illustrated by a number of examples. the local network topology and of more physiological addi- tions to the model on its ability to maintain synchrony. We Alexander Tovbis calculate the random time between synchronous events in University of Central Florida terms of a first-passage-time problem for a single neuron. Department of Mathematics [email protected] Katherine Newhall Rensselaer Polytechnic Institute [email protected] MS48 Evolution of Structures in Neuronal Networks with Peter R. Kramer Plasticity Rensselaer Polytechnic Institute Department of Mathematical Sciences We consider neuronal network models with plasticity and [email protected] randomness; we show that complicated global structures NW10 Abstracts 79

Gregor Kovacic sheltering mechanism. The wind blowing over a strongly Rensselaer Polytechnic Inst modulated wave group due to the dispersive focusing of an Dept of Mathematical Sciences initial long wave packet increases the time duration and [email protected] maximal amplitude of the steep wave event. These results are coherent with those obtained within the framework of David Cai deep water. However, steep wave events are less unstable Shanghai Jiao Tong University, China to wind perturbation in shallow water than in deep water. Courant institute, New York University, USA Furthermore, a comparison between experimental and nu- [email protected] merical wave breaking is presented in the absence of wind. Within the framework of numerical experiments, wind is shown to speed up the wave breaking and amplify slightly MS48 the wave height. The wall pressure during the run up of the Steps Towards Coarse Graining Neuronal Network steep wave event on a vertical wall is investigated too and Dynamics within an Intermittently Desupressed a comparison between experimental and numerical results Regime is provided.

I will present some results relating to a simplified model Christian Kharif of neuronal network dynamics which is specifically de- Institut de Recherche sur les Phenomenes Hors Equilibre signed to capture some of the correlated activity observed [email protected] in real neuronal networks (e.g., in mammalian visual cortex and/or insect antennal lobe). Julien Chambarel Institut de Recherche sur les phenomenes Hors Equilibre Aaditya Rangan [email protected] CIMS New York University [email protected] MS49 Title Not Available At Time of Publication MS49 Abstract not available at time of publication. Title Not Available At Time of Publication Alfred R. Osborne Abstract not available at time of publication. University of Torino Torino, Italy Roberto Camassa [email protected] Mathematics UNC Chapel Hill [email protected] MS50 Traveling Discrete Modes in the Presence of Small Dissipation MS49 Propagation of Large Amplitude Internal Waves Nonlinear discrete systems appear in many areas of sci- over Bottom Topography ence. In the absence of losses, models like the discrete nonlinear Schroedinger equation, or oscillators under the We are interested in the evolution of large amplitude in- action of Toda or Morse potentials describe the dynamics ternal waves in the ocean which is simplified to a two-layer of traveling wave solutions (TWS) with many applications. fluid (inviscid, irrotational) with different constant densi- Typically the existence of traveling modes relates to the ef- ties. Recently, a regularized model for strongly nonlinear fect of a Peierls-Nabarro potential. Less is known of the internal waves in a two-layer system has been proposed role of dissipation. We will discuss if dissipation slows or to eliminate the shear instability at the interface of the even stops TWS. two layers. We include the effect of varying bottom topography and solve the time evolution equations numerically Alejandro Aceves using a pseudo-spectral method along with a fourth order Southern Methodist University Runge-Kutta time integration. Numerical solutions will be [email protected] presented for the interaction of large solitary waves with bottom topography. MS50 Arnaud Goullet Supersonic Davydov Soliton Propagation in a Dis- New Jersey Institute of Technology crete Chain [email protected] We investigate numerically and asymptotically the supersonic motion of a coherent Davydov type soliton. The MS49 model is formulated in terms of the cubic discrete NLS cou- Simulations of Rogue Waves in Shallow Water pled to an anharmonic lattice. The anharmonicity of the lattice allows for the propagation of supersonic compres- A series of numerical simulations considering the action of sional waves. This supersonic wave traps the Davydov like wind on steep waves and breaking waves generated through soliton producing a coherent supersonic soliton. We find the mechanism of dispersive focusing on finite depth is per- approximate solutions using the continuum approximation formed. The dynamics of the wave packet propagating and show numerically their stability. It is to be remarked without wind at the free surface is compared to the dy- that the sonic regime produces Davydov solitons with fi- namics of the packet propagating in the presence of wind. nite amplitude which are not present in the usual model. Wind is introduced in the numerical wave tank by means These results were obtained in the parameter regime apro- of a pressure term, corresponding to the modified Jeffreys’ 80 NW10 Abstracts

priate for simple models for protein chains. This shows the Tomas Dohnal posibility of supersonic propagation of coherent excitation KIT, Department of Mathematics of the peptide groups. [email protected]

Luis Cisneros ESFM, Instituto Politécnico Nacional MS51 [email protected] Fronts and Solitary Waves in Inhomogeneous Wave Equations Antonmaria Minzoni Fenomec, IIMAS-UNAM After an introduction to the topic of this mini-symposium, [email protected] this talk will focus on two models involving the inhomogeneous sine-Gordon equation. The first model describes aspects of the DNA/RNAP interaction in DNA copying. MS50 The second model involves long Josephson junctions or 0-π Refined Stability of Navier-Stokes Shocks junctions with imperfections. The existence of rich families of stationary coherent solutions will be discussed, including In the classical (inviscid) stability analysis of shock waves, stable non-monotonic fronts. the stable and unstable regions of parameter space are typically separated by an open set of parameters. Zumbrun Gianne Derks and Serre have shown that the precise location of the tran- University of Surrey sition to instability can be determined by an “effective vis- Department of Mathematics cosity’ coefficient which takes into account viscous effects [email protected] neglected in the classical theory. In this talk, we discuss the computation of the effective viscosity coefficient for the Navier-Stokes equations. MS51 Stability in Inhomogeneous Nonlinear Gregory Lyng Schroedinger Equations Department of Mathematics University of Wyoming Inhomogeneous nonlinear Schroedinger equations provide [email protected] models in a variety of settings, recently including both optics and Bose-Einstein condensation. In such models, standing waves can often be constructed via phase-plane MS50 techniques. Here, we discuss a method to quickly charac- Continuation and Bifurcations of Breathers in the terize the stability of many of these standing waves using Discrete Nls Equation geometric information available in the phase-plane. As a concrete example, we consider cubic-quintic NLS with an We present analytical and numerical results on the contin- N-well external potential. uation and bifurcations of breathers in the discrete NLS equation. The study of bifurcations concerns finite one- Russell Jackson dimensional lattice with Dirichlet boundary conditions. U. S. Naval Academy, Mathematics Department We will show evidence for subcritical fold and pitchfork Annapolis MD 21402-5002 bifurcations as the intersite coupling increases and we will [email protected] also discuss breathers that can be continued to normal modes of the weakly nonlinear system. Christopher Jones University of North Carolina at Chapel Hill Panayotis Panayotaros Department of Mathematics Depto. Matematicas y Mecanica [email protected] IIMAS-UNAM [email protected] MS51 Effects of Spatial Inhomogeneities on the Stability MS51 of Fronts in a Non-linear Wave Equation Surface Gap Solitons in the Gross Pitaevskii Equa- tion with a Nonlinear Interface The analysis of the non-linear wave equation often assumes a homogeneous potential. However in many applications The (1D) Gross Pitaevskii Equation with an inhomogene- inhomogeneities may arise. Here we consider a spatial step- ity in the form of an interface of two materials with dif- like defect and look at the stability of stationary fronts. A ferent values of the nonlinear refractive index may support condition for points (involving the length of the defect and stationary localized solutions called Surface Gap Solitons the energy of the front) where the stability of the front (SGS). We numerically compute families of SGS by using changes is presented as well as a discussion of travelling an arclength continuation method. We show that unlike fronts, including pinning of a travelling front by the intro- solitons in linearly homogeneous media, SGS are possible duction of a defect. at truly focusing/defocusing interfaces. We then investigate linear stability of SGS via the numerical Evans func- Christopher Knight tion method. Department of Mathematics University of Surrey Elisabeth Blank [email protected] KIT, University of Karlsruhe Department of Mathematics [email protected] MS52 Existence and Stability of Viscous Shock Profiles NW10 Abstracts 81

for Isentropic Mhd with Infinite Electrical Resis- tion, we derive an orientation index which yields sufficient tivity conditions for such an instability to occur. This index is geometric in nature and applies to arbitrary periodic trav- For the Navier-Stokes equations of isentropic magneto- eling waves with minor smoothness and convexity assump- hydrodynamics (MHD) with γ-law gas equation of state, tions on the nonlinearity. Using the integrable structure of γ ≥ 1, and infinite electrical resistivity, we carry out a the ordinary differential equation governing the traveling global analysis categorizing all possible viscous shock pro- wave profiles, we are then able to calculate the resulting files. For the monatomic and diatomic cases γ =5/3and orientation index for the elliptic function solutions of the γ =7/5, we carry out a systematic numerical Evans func- Korteweg-de Vries and modified Korteweg-de Vries equation analysis indicating all of the profiles are linearly and tions. nonlinearly stable. Mat Johnson Blake Barker Indiana Indiana Indiana University [email protected] BYU [email protected] Kevin Zumbrun Indiana University Olivier Lafitte USA LAGA, Institut Galilee, Universite Paris 13 [email protected] lafitte at math.univ-paris13.fr

Kevin Zumbrun MS53 Indiana University New Singular Solutions of the Biharmonic NLS USA Equation [email protected] We consider blowup-type singular solutions of the fourth- order (biharmonic) nonlinear Schrödinger (BNLS) equa- MS52 tion. Our formal and informal analysis, and numerical evi- The Hamiltonian Structure of the NLS and the dence, indcate that the blowup formation resembles that of Asymptotic Stability of its Ground States the standard (harmonic) NLS. However, the lack of some key analytic tools still makes BNLS theory challenging. In Ground states satisfying conditions for orbital stability the L2-critical case, we show that the collapsing core of the by Weinstein are proved asymptotically stable for generic solution converges to a self-similar profile, and bound the equations with smooth nonlinearity, solving old problem by blowup rate. In the super-critical case we use asymptotic Soffer and Weinstein from 80’s. For Fermi Golden rule we analysis to find and characterize new peak-type and ring- implement Birkhoff normal form argument, as in solution type singular solutions of the BNLS, which also converge by Bambusi and Cuccagna on stability of trivial solution to a self-similar profile. These findings are verified numer- for NLKG. To save semilinear nature of Hamiltonian, we ically, using an adaptive mesh, at focusing factors of up to implement the Darboux Theorem. 108. Joint work with Gadi Fibich and Elad Mandelbaum. Scipio Cuccagna Guy Baruch Dipartimento di Scienze California Institute of Technology Universita di Modena e Reggio Emilia [email protected] [email protected] Gadi Fibich Tel Aviv University MS52 School of Mathematical Sciences Spectral Stability of Bose-Einstein Condensates in fi[email protected] a Periodic-Trap with Non-Local Effects

In this work, the stability of a class of solutions to a Elad Mandelbaum model of a Bose-Einstein condensate in a periodic trap with Tel Aviv University non-local effects is established numerically. Then, using [email protected] Bloch decomposition and a Lyupanov-Schmidt reduction, the spectral stability of these solutions is established. MS53 Chris Curtis Nonlinear Imaging through Random Media Department of Applied Mathematics University of Colorado at Boulder Scattering of signals through random media is a pervasive [email protected] problem in imaging. Linearly, ray diffusion leads to loss of contrast and effective noise. Nonlinearly, wave coupling between signal and noise leads to energy exchange between MS52 the components. For the right parameters, signal can grow The Transverse Instability of Periodic Traveling at the expense of noise. Here, we describe this dynami- Waves in the Generalized Kadomtsev-Petviashvili cal ”stochastic resonance” using the formalism of photonic Equation plasma. Experimentally, we demonstrate the technique by recovering diffused images using a photorefractive crystal. We consider the spectral instability of periodic traveling wave solutions of the gKdV equation to transverse perturbations in the gKP equation. By analyzing high and low Dmitry Dylov frequency limits of the appropriate periodic Evans func- Department of ELE Princeton University 82 NW10 Abstracts

[email protected] MS54 Mode-Locking Using Phase-Sensitive Amplifica- tion MS53 Dynamics and Transport of Localized Wave Pack- A method is proposed and considered theoretically ets in Random Nonlinear Systems for using phase-sensitive amplification as the intensity- discrimination (saturable absorption) element in a laser We study the long time behavior of solutions to the nonlin- cavity to generate stable and robust mode-locking. The re- ear Schrödinger equation with random initial conditions. It sulting averaged equation is a Swift-Hohenberg type model is shown that the dynamics of a random initially localized with cubic-quintic nonlinearities and a linear growth term wavepacket is either diffusive, localized or super diffusive. which couples to the nonlocal cavity energy. Ziad Musslimani J. Nathan Kutz Dept. of Mathematics University of Washington Florida State University Dept of Applied Mathematics [email protected] [email protected]

MS54 MS54 Modeling Multi-Pulsing Transition in Ring Cavity Dynamics and Timing Jitter of a Ytterbium Fiber Lasers with Proper Orthogonal Decomposition Laser

A low-dimensional model is constructed via the proper or- We study the dynamics of the high power femto-second thogonal decomposition (POD) to characterize the multi- pulses generated in lasers with normal average dispersion pulsing phenomenon in a ring cavity laser mode-locked by a that incorporate a normal Ytterbium fiber amplifier and saturable absorber. Constructing a bifurcation diagram for anomalous dispersion compensation. Stable operation is the reduced four-mode model explains the transition from permitted provided the total dispersion is normal and does a single mode-locked pulse to a double-pulse configuration. not fall below a power-dependent threshold. Whereas the It is shown that the single mode-locked pulse undergoes a chirp of a stable pulse is always positive, unstable operation Hopf bifurcation, and the created periodic solution under- occurs when the chirp undergoes large swings about zero. goes a fold bifurcation after which only the double-pulse We discuss implications for minimization of timing jitter. solution remains stable. The low-dimensional model also shows the coexistence of solutions with different number of John W. Zweck pulses. These findings are in good agreement with the full University of Maryland Baltimore County governing equation. Mathematics and Statistics [email protected] Edwin Ding Applied Mathematics Curtis R. Menyuk University of Washington UMBC [email protected] Baltimore, MD [email protected] J. Nathan Kutz University of Washington Dept of Applied Mathematics MS55 [email protected] Adaptive Space-time Discontinuous Galerkin Schemes for the Simulation of Gravitational Col- lapse MS54 The Short Pulse Master Mode-locking Equation We present a class of adaptive, high-order, discontinuous Galerkin schemes for solving the spherically symmetric We propose a new model which is valid for pulse propaga- Einstein equations. The time-stepping strategy is to allow tion in a mode-locked laser cavity in the few-femtosecond different time-steps in each element (i.e., large time-steps regime. This alternative to the master mode-locking equa- on coarse mesh elements and small time-steps on fine mesh tion is valid beyond the breakdown of the slow envelope elements). High-order in space is achieved through the dis- approximation. Perturbing the Short Pulse Equation with continuous Galerkin approach, while high-order in time is dissipative gain and loss terms allows the generation of achieved via a Lax-Wendroff-type of space-time expansion. stable ultra-short pulses from initial white-noise. This Different error indicators for the adaptive mesh refinement provides an initial theoretical framework for quantifying part of the algorithm will be considered. The proposed dynamics and stability as pulses approach the attosecond approach will be validated on a few different problems in- regime. volving the gravitational collapse of a scalar field. Edward D. Farnum James A. Rossmanith Kean University University of Wisconsin [email protected] Department of Mathematics [email protected] J. Nathan Kutz University of Washington Dept of Applied Mathematics MS55 [email protected] Numerical Simulation of a General Relativisitic Shock Wave by a Locally Inertial Godunov Method NW10 Abstracts 83

with Dynamic Time Dilation recoil of binaries from the full parameters space with sufficient accuracy for astrophysical applications. We discuss the doctoral dissertation of Zeke Vogler in which he demonstrates numerical convergence of a locally iner- Yosef Zlochower, Manuela Campanelli, Carlos Lousto tial Godunov method, which incorporates a new dynamic Rochester Institute of Technology time dilation to account for general relativistic e ffects of Center for Computational Relativity and Gravitation curvature. He gives a complete proof that pointwise a.e. [email protected], [email protected], convergence together with bounded total variation implies [email protected] convergence to a weak solution, and then demonstrates the two assumptions numerically. The convergence is for a new set of initial data that meets the constraints of the Ein- MS56 stein equations when p = 0, and solutions are interesting From the Pearcey Process to the Airy Process: A in their own right. For example, one can view the forward Fredholm Determinant Approach time evolution as resolving the secondary wave in exact models for an explosion into a static singular isothermal In this joint work with Mattia Cafasso we show how to set spherefirst constructed by Smoller and Temple. In back- up a Riemann–Hilbert formulation relevant to the Fred- ward time, the numerics demonstrate formation of a black holm determinant of the Pearcey kernels. This Riemann– hole in the evolution of a smooth solution. The components Hilbert problem is conducive to asymptotic analysis and we of the gravitational metric are only Lipshitz continuous at show how the Fredholm determinant of the Pearcey kernel shock waves, so the curvature could in principle have delta asymptotically reduces (for large values of the gaps) to the function sources at shocks, but the convergence to a weak product of two Fredholm determinants of the Airy kernel. solution rules this out. The point of departure for the work is the locally intertial Glimm scheme first introduced and analyzed by Groah and Temple. Marco Bertola Concordia University Blake Temple [email protected] Department of Mathematics University of California, Davis Mattia Cafasso [email protected] Universite de Montreal [email protected] Zeke Vogler University of California, Davis [email protected] MS56 On the Modulation Equations and Stability of Pe- riodic GKdV Waves MS55 Gravitational Waves from Extreme-mass-ratio Bi- In this talk, we will discuss recent results concerning the nary Black Hole Inspirals rigorous justification of the Whitham modulation equations for periodic traveling wave solutions of the gKdV The inspiral of stellar-mass (m ∼ 10M) compact ob- equations. We will begin by outlining how to derive the 6 jects into supermassive black holes (M ∼ 10 M) should Whitham equations in the case where the underlying peri- be a strong source of gravitational waves (GWs) for the odic wave has non-zero average. We will then discuss how planned LISA space-based gravitational-wave observatory. to prove these homogenized equations describing the mean As a step towards calculating the expected GW signal from behavior of a slow modulation (WKB) approximation of such a source, here I consider the problem of calculating the solution correctly describes the linearized dispersion the O( /M) “self-force’ on the small body due to gravita- relation near the origin. In particular, we will describe tional radiation reaction, using characteristic AMR in the two techniques for showing that the modulational stabil- Barack-Ori mode-sum regularization procedure. ity of the underlying periodic wave is equivalent with the well-posedness of the Whitham system: the first via di- Jonathan Thornburg rect Evans function calculations and the second using more Dept of Astronomy general Bloch-wave decompositions. The latter method Indiana University has the distinct advantage of applying in the more general [email protected] multi-periodic setting where no conveniently computable Evans function is yet devised. MS55 Mat Johnson Recent Advances in Modeling Gravitational Recoil Indiana Indiana from Highly-spinning Black-hole Binaries [email protected]

The recent advances in the fully nonlinear modeling Kevin Zumbrun of merging black-hole binaries led to remarkable break- Indiana University throughs in our understanding of black-hole dynamics. USA One of the most unexpected and exciting discoveries was [email protected] that when highly-spinning binaries merge the resulting black hole can recoil at up to 4000 km/s, fast enough to Jared Bronski eject the remnant from even the largest galaxies. These University of Illinois Urbana-Champain remarkable results were obtained by sampling only a small Department of Mathematics fraction of the seven dimensional parameter space of pos- [email protected] sible binaries. The challenge now is to extend these simulations to the high-spin and extreme-mass-ratio regimes in order to develop models that can accurately predict the 84 NW10 Abstracts

MS56 be discussed in a related talk in session MS47. Dispersionless Limits of Integrable Systems and Universality in Random Matrix Theory Nicholas Ercolani University of Arizona The τ-functions of the Toda and Pfaff lattice hierarchies [email protected] are given by the partition functions of the Gaussian unitary, orthogonal and symplectic ensembles of random matrices. In these cases the asymptotic expansions of the free MS57 energies given by the logarithm of the partition functions Diagonalizing Random Matrices Using Integrable lead to the dispersionless limits of the Toda and Pfaff lat- Systems tice hierarchies. There is a universality between all three ensembles of random matrices, one consequence of which The QR algorithm is known to be the time one map of an is that the leading orders of the free energy for large ma- integrable Hamiltonian system on certain classes of matri- trices agree. We show that the free energy as the solution ces. I will explain this fact and introduce the related Toda of the dispersionless Toda lattice hierarchy gives a solution algorithm. As a first step towards the probabilistic anal- of the dispersionless Pfaff lattice hierarchy, which implies ysis of these algorithms we are trying to understand the that this universality holds in general for the leading orders time it takes to identify one of the eigenvalues up to a pre- of the unitary, orthogonal and symplectic ensembles. specified tolerance given an initial matrix drawn from the 1-Hermite ensemble. Numerical simulations will be pre- Virgil Pierce sented that lead to conjectures regarding the existence of University of Texas- Pan American scaling limits for this and related random times in case of [email protected] both the QR and the Toda algorithm. Christian Pfrang Yuji Kodama Brown University Ohio State University christian [email protected] Department of Mathematics [email protected] MS57 MS57 The Spectral Edge of Beta Ensembles Log Concavity of Solutions to the Diffusive Consider n particles on the line, each subject to the same Lifschitz-Slyozov-Wagner Equation and Coarsening (convex) polynomial potential and jointly to a coulombic interaction of strength β>0. We prove that the renormal- The classical Lifschitz-Slyozov-Wagner (LSW) model is one ized law of the largest point converges to the ”Beta Tracy- of the simplest models of coarsening. Recently the speaker Widom” distribution, generalizing well known earlier re- proved point-wise in time bounds on the rate of coarsening sults from Random Matrix Theory pertaining to β =1, 2, 4 for the LSW model by using convexity methods. In this and analytic potentials, or to quadratic potential and the talk we shall discuss how these methods may be extended full range of β. This is joint work with Balint Virag and to diffusive LSW by showing that the set of log concave Manjunath Krishnapur. functions is invariant under a diffusive LSW flow with the simplest diffusion term. Brian Rider University of Colorado Joseph Conlon [email protected] Department of Mathematics The University of Michigan [email protected] MS58 Shear Stability of Stratified Flows with a Free Sur- Mohar Guha face University of Michigan [email protected] Abstract not available at time of publication. Ricardo Barros MS57 Basque Center for Applied Mathematics Scaling Limits for Deformations of Hermetian Ran- Basque Country, SPAIN dom Matrix Ensembles and their Self-Similar So- [email protected] lutions We will describe various scaling limits of the free energy MS58 for Hermetian random matrix ensembles. We derive very Internal Solitary Waves: Finite-amplitude Equilib- detailed information about the asymptotic behavior of cor- ria and Nonlinear Evolution relation functions for the underlying Gibbs measures by studying the corresponding scaling limit of the Toda lat- A new method for computing steady finite-amplitude in- tice dynamical system associated to parameter variations ternal waves in a general stably-stratified fluid is presented. of the ensemble partition functions. The continuum limit These solutions include trapped, nearly-stagnant cores of is a forced Burgers hierarchy having self-similar solutions uniform density for large amplitude. Novel nonlinear sim- that we are able to explicitly compute. Our goal will be ulations at infinite Prandtl number confirm the stability of to sketch what goes into deriving this hierarchy and con- many of these solutions. Others exhibit partial breakdown structing the self-similar solutions. Time permitting, we via Kelvin-Helmholtz instability. will also describe new implications for a related double- scaling limit that arose twenty years ago in the context of David Dritschel 2D quantum gravity. Applications of these results will also Mathematical Institute NW10 Abstracts 85

University of St Andrews MS59 [email protected] The Discrete Short Pulse Equation and its Solu- tions Stuart King, Magda Carr University of St Andrews, Scotland The short pulse (SP) equation was derived recently as a [email protected], [email protected] model equation for the propagation of ultra-short optical pulses in nonlinear media. The SP equation admits various interesting exact solutions such as loop soliton solutions MS58 and breather solutions. In this talk, we propose new in- Spectral Evolution and Instabilities of Nonlinear tegrable semi-discretization and fully discretization of the Deep-Water Ocean Waves SP equation by using the method proposed by us recently. We also discuss properties of various exact solutions of the In this work we concentrate on the development of an accu- discrete SP equation. rate and eﬃcient numerical model for a short-term prediction of evolving nonlinear ocean waves, including extreme Kenichi Maruno waves such as Rogue waves. Derivations of asymptoti- Department of Mathematics cally reduced models, based on the small wave steepness The University of Texas-Pan American assumption, as well as relatively weak transverse depen- [email protected] dence, will be presented and their corresponding numerical simulations via Fourier pseudo-spectral methods will Bao-Feng Feng be discussed. Motivated by their spectral evolution, the Department of Mathematics stability of a weakly nonlinear wave-train (Stokes wave) UTexas-Pan American on deep water subject to three-dimensional modulations is [email protected] investigated. Yasuhiro Ohta Matt Malej Department of Mathematics NJIT Kobe University [email protected] [email protected]

MS58 MS59 Title Not Available at Time of Publication On the Stability of Sonic Viscous Shock Waves Abstract not available at time of publication. In this talk I shall make simple observations leading to op- Lp Roxana Tiron timal decay rates in spaces for zero-mass perturbations Univ. of North Carolina of degenerate (or sonic) viscous shock waves. The method [email protected] uses energy estimates and weighted interpolation inequal- ities, based on the Matsumura-Nishihara weight function. The approach is elementary and applies to shocks of all MS59 order of degeneracy. Power-dependent Soliton Dynamics in Complex- Ramon G. Plaza photonic Structures Institute of Applied Mathematics (IIMAS) Abstract not available at time of publication. National University of Mexico (UNAM) [email protected] Yannis Cominis National Technical University, Greece [email protected] MS60 An Experimental Investigation of Localization in Nonlinear Granular Elastic Crystals MS59 Parity-time Symmetric Optical Lattices We experimentally investigate localization in one- dimensional nonlinear granular crystals composed of stati- Interestingly, non-Hermitian Hamiltonians that respect cally compressed spheres that interact via Hertzian con- paritytime (PT) symmetry can show entirely real spectra. tact. We show linear spectrums in monatomic and di- It has been recently realized that PT-related notions can atomic crystals. Intrinsic localization is shown in diatomic be experimentally investigated in the context of optics. We crystals resulting from a modulational instability of the report the first experimental observation of a PT optical lower optical band edge. We identify nonlinear localiza- coupled system. We also study the peculiar properties of tion arising from defects in monatomic crystals, and study PT-optical lattices such as abrupt phase transitions, power the effects of the dynamic excitation amplitude and defect oscillations, band merging, non-orthogonal Floquet-Bloch spatial proximity on the frequency of these modes. modes, nonreciprocal diffraction patterns, double refraction, phase dislocations, and solitons. Nicholas Boechler California Institute of Technology Konstantinos Makris [email protected] School of Engineering, Optics Laboratory Ecole Polytechnique Federale de Lausanne Chiara Daraio [email protected] Aeronautics and Applied Physics California Institute of Technology [email protected] 86 NW10 Abstracts

MS60 MS61 Fermi, Pasta, Ulam, and the Birth of Experimental Integration of Stationary Equations for KdV Mul- Mathematics tisolitons In May 1955, Los Alamos Report LA-1940 (”Studies of Just as the solitary-wave profiles for the KdV equation can Nonlinear Problems. I”) was published by Enrico Fermi, be obtained by integrating a second-order ordinary dif- John Pasta, and Stanislaw Ulam (FPU). Containing what ferential equation, N-soliton profiles can be obtained by is now universally known as the ”FPU problem”, the study integrating an ordinary differential equation of order 2N, initiated a revolution in modern science, ushering in the which is actually a Hamiltonian system with N degrees of age of computational science and marking the birth of non- freedeom. Gelfand, Dickey, and Dubrovin showed how to linear science as an interdisciplinary field. On one hand, explicitly integrate the system to find its periodic solutions. computational and analytical studies attempting to explain We use the same method to classify all the homoclinic or- the surprising FPU recurrences led to the discovery of soli- bits of the system, in the case N=2. Since the homoclinic tons. On the other, similar explorations of ”simple” (low- orbits are 2-solitons, this result leads to an improvement of dimensional) systems related to the FPU problem have the stability theory of Maddocks and Sachs for 2-solitons. led to the widespread modern awareness of deterministic chaos. Today, analyses of the FPU problem continue to John Albert lead to deeper insights into many areas–including the in- Department of Mathematics terplay between regular and chaotic behavior, the behavior University of Oklahoma of spatially extended but discrete nonlinear systems, heat [email protected] conduction anomalies in low-dimensional systems, and the origins of statistical mechanics. In this talk, I’ll discuss some of this history and indicate some exciting results that MS61 are helping to continue the story’s development. Hopf Instabilities of Defect Modes Mason A. Porter Localized solutions of nonlinear wave equations in the pres- University of Oxford ence of inhomogeneities–defect modes–are generally stable Oxford Centre for Industrial and Applied Mathematics at small amplitudes and unstable at larger amplitudes. A [email protected] lot of attention has been paid to symmetry-breaking bifurcations. We construct inhomogeneities that force Hamilto- nian Hopf bifurcations, and analyze the bifurcating solu- MS60 tions. Nonlinear Resonance Processes in Driven Granular Alignments Roy H. Goodman New Jersey Institute of Technology We consider axially aligned elastic spheres confined within Department of Mathematical Sciences walls. Both monosized and tapered chains are considered. [email protected] These grains repel via Hertz law. When driven at one end, the system begins to undergo continuous sequences of over- compression and dilation. The frequency of this breath- MS61 ing can be tuned by varying the geometric and mechanical Spectral Analysis for Matrix Hamiltonian Opera- properties of the elastic objects. We shall discuss how this tors system naturally allows one to detect very low frequencies and suggests a way of extracting energy from sources such In this result, we study the spectral properties of matrix as ocean waves, wind energy etc. Hamiltonians generated by linearizing about soliton solutions to nonlinear Schrödinger equations. Using a hybrid Surajit Sen analytic-numerical proof, we show that there are no em- Physics bedded eigenvalues for the three dimensional cubic nonlin- University of Buffalo earity and other nonlinearities. Though we will focus on [email protected]ffalo.edu a complete proof for the 3d cubic problem, the true nature of the work is to present a new, robust algorithm for verification of the spectral properties required for stabil- MS60 ity analysis of linearized Schrödinger operators. However, Energy Localization in Nonlinear Granular Elastic there are several cases of interest for which our current Crystals methods are inconclusive, the reasons for which we explore in as much detail as possible. We study localization of energy in one-dimensional chains of tightly packed and uniaxially compressed elastic beads. Jeremy Marzuola For monoatomic chains, the inclusion of light-mass impu- Department of Applied Mathematics rities is crucial for the appearance of energy localization. Columbia University For diatomic chains, the localization is obtained by the in- [email protected] trinsic nonlinearity, caused by the Hertzian interaction of the beads, without additional inhomogeneities. We first Gideon Simpson characterize the linear spectrum of the system. We then University of Toronto use continuation techniques to find the exact nonlinear lo- [email protected] calized solutions and study their linear stability in detail.

MS61 Georgios Theocharis Stability of Higher Order Curvature Driven Flows California Institute of Technology [email protected] We discuss the linear and nonlinear stability of a class of NW10 Abstracts 87

higher order curvature drive flows which arise in the models served. of solvated functionalized polymers. We outline the derivation of the sharp-interface reduction, and show that so long Lars Q. English as the fronts are non-self intersecting, the curvature driven Dickinson College, Pennsylvania flow gives the correct leading order normal velocity. USA [email protected] Keith Promislow Michigan State University J. Cuevas [email protected] Universidad de Sevilla [email protected] MS62 P.G. Kevrekidis Spatiotemporal Complexity of the Parametrically University of Massachusetts Driven Damped Nonlinear Schrödinger Solitons [email protected] This talk focusses on time-periodic solitons of the parametrically driven damped nonlinear Schrödinger equation. MS62 Unlike the traditional approach based on direct numerical simulations, we determine the periodic solitons as solutions A New Stabil- of the boundary-value problem on a two-dimensional do- ity Criterion for Nonlinear Schroedinger Solitons main. This allows us to obtain both stable and unstable with Spatiotemporal Driving solutions and study their bifurcations. We demonstrate the We develop a collective coordinate theory for solitons of the increasing complexity of localised attractors as the ampli- Nonlinear Schroedinger Equation (NLSE) with spatiotem- tudes of the damping and driving are raised: simple pe- poral forces of the form f(x,t) = a exp[i K(t) x]. We con- riodic solutions give way to multiple-periodic ones; free- sider constant, harmonic and biharmonic K(t). We con- standing solitons are replaced by their stable complexes. jecture a new stability criterion: if the curve P(V), where Igor Barashenkov P(t) and V(t) are the soliton momentum and velocity, has University of Cape Town a branch with negative slope, the soliton is predicted to South Africa become unstable which is confirmed by our simulations for [email protected] the perturbed NLSE. This curve also yields a good estimate for the soliton lifetime: the shorter the branch with negative slope is, the longer the soliton lives. Elena Zemlyanaya Joint Institute for Nuclear Research Franz G. Mertens Dubna, Russia University of Bayreuth [email protected] Bayreuth, Germany [email protected] MS62 Localized States in Parametrically Driven Quasi- MS63 reversible Systems A Perturbation Theory for Dispersion-managed Solitons in Femtosecond Lasers We study theoretically a family of localized states which asymptotically connect a uniform oscillatory state in the I will review a recently developed perturbation theory magnetization of an easy-plane ferromagnetic spin chain for dispersion-managed solitons. The theory, which gen- when an oscillatory magnetic field is applied and in a eralizes the familiar tools available for the nonlinear parametrically driven damped pendula chain. The conven- Schroedinger equation, does not require the underlying dy- tional approach to these systems, the parametrically driven namics to be integrable, and it can be applied to quantify damped nonlinear Schrdinger equation, does not account the effects of noise in dispersion-managed systems. for these states. Adding higher order terms to this model we were able to describe these localized structures. Gino Biondini State University of New York at Buffalo Marcel G. Clerc Department of Mathematics Department of Physics, University of Chile biondini@buffalo.edu Chile marcelclerc@dfi.uchile.cl MS63 Discussion: Future Directions and Open Problems MS62 on Mode-locked Lasers Symmetry and Mobility of Discrete Solitons in a Forced-damped Array of Coupled Pendula We provide a forum for moderated discussion among min- isymposium participants to address open problems in the We present an experimental system - a driven array of cou- modeling, analysis, and simulation of high-power short- pled pendula - in which a stability reversal between forced pulse mode-locked lasers. discrete solitons of different symmetry is observed. We compare experimental results to detailed numerical phase J. Nathan Kutz diagrams for stable on-site and inter-site centered local- University of Washington ized modes. For lattice parameters close to the transition Dept of Applied Mathematics between the two ‘phases’, the Peierls-Nabarro barrier van- [email protected] ishes and enhanced mobility of the localized mode is ob- 88 NW10 Abstracts

MS63 MS64 Modelling of Figure-eight All-fiber Laser Title Not Available at Time of Publication The figure-eight fiber laser is considered in two configura- Abstract not available at time of publication. tions, with either a unidirectional active ring cavity coupled with a nonlinear optical loop mirror (NOLM) or a unidi- Pieter Blue rectional passive cavity and a nonlinear amplifying loop University of Edinburgh, UK mirror (NALM). In each case, we derive a master equation pblue@staffmail.ed.ac.uk of cubic complex Ginzburg Landau (CGL) type, in which the coefficients explicitly depend on the characteristics of the cavity. Single-pulse and continuous wave (cw) solutions MS64 in both normal and anomalous dispersion are discussed an- Linear Stability, Price Law and Wave Decay on alytically. Black Hole Metrics Mohamed Salhi Price’s Law states that linear perturbations of a Schwarzschild black hole, depending on initial conditions, LPhiA EA 4464, University of Angers, −2l−3 −2l−2 2 Bd Lavoisier 49000 Angers, France fall off as t or t for t approaching infinity, where [email protected] l is the angular momentum. We give a rigorous proof of these decay rates in the form of weighted L1 to Linfty Foued Amrani bounds for solutions of the Regge Wheeler equation. The LPhiA EA 4464, University of Angers, proof is based on an integral representation of the solu- [email protected] tion which follows from spectral theory. We apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are Herve Leblond obtained by appropriate oscillatory integral estimate. LPhiA, EA 4464 University of Angers Avy Soffer [email protected] Rutgers University, Math Dept. soff[email protected] Fran¸cois Sanchez LPhiA EA 4464, University of Angers, 2 bd Lavoisier 49000 Angers, France MS65 [email protected] Coherent Structures and Phase Transitions in Stochastic Neuronal Network Dynamics

MS63 We consider a network of pulse-coupled oscillators con- Analysis of Dark and Bright Solitons in Mode- taining randomness both in input and in network archi- Locked Lasers tecture. We analyze the scalings which arise in certain limits, demonstrate limit theorems in the correct scaling, The formation and behavior of both dark and bright pulses and interpret various “finite-size” effects as perturbations in mode-locked lasers is studied using adiabatic perturba- of these limits. We also observe that for certain parameters, tion theory. In the anomalous regime individual pulses this network supports both synchronous and asynchronous are well approximated by soliton solutions of nonlinear modes of behavior and will switch stochastically between Schrodinger equation with key parameters evolving slowly these modes due to rare events. We also relate the analy- in time. In the normal regime both dark and bright soliton sis of this network to results in random graph theory, and pulse are found. The dark pulses are well approximated in particular, those involving the size of the “giant com- by solutions to the NLS equation with a shelf propagating ponent” in the Erd˝os-Rényi random graph and use this to away from the soliton. understand the phase transitions in the system. Sean Nixon Lee DeVille University of Colorado University of Illinois [email protected] Department of Mathematics [email protected] Mark Ablowitz Dept. of Applied Mathematics University of Colorado MS65 [email protected] Integrable Stochastic Soliton Dynamics of Light Polarization in an Active Medium

MS64 Resonant interaction of light with a randomly prepared, Title Not Available at Time of Publication active optical medium in the lambda configuration is described by exact solutions of a completely-integrable, ran- Abstract not available at time of publication. dom partial differential equation, thus combining the op- posing concepts of integrability and disorder. An optical Hans-Peter Bischof pulse passing through such a material will switch randomly Rochester University between left- and right-hand circular polarizations. Exact Computer Science probability distributions of the electric-field envelope vari- [email protected] ables describing the light polarization will be presented and their properties discussed.

Gregor Kovacic Rensselaer Polytechnic Inst NW10 Abstracts 89

Dept of Mathematical Sciences MS66 [email protected] Title Not Available at Time of Publication

Peter R. Kramer Abstract not available at time of publication. Rensselaer Polytechnic Institute Department of Mathematical Sciences Rich McLaughlin [email protected] Univ. of North Carolina [email protected] Ethan Atkins Courant Institute MS66 [email protected] Higher Order Bragg Resonance of Water Waves by Large Amplitude Bottom Corrugations Ildar R. Gabitov Department of Mathematics, University of Arizona Strong and constructive scattering occurs when water [email protected] waves propagate over bottom corrugations whose wavelength is close to an integer multiple of half a water wavelength. Exact solutions of these higher order Bragg reso- MS65 nances are developed for finite amplitude corrugations us- Parameterization of Turbulent Transport by ing a Floquet theory of linearized water waves. We ex- Mesoscale Eddies amine the ranges of water wave frequencies, within which the wave amplitude exhibits slow exponential modulation We employ homogenization theory to develop a systematic in space, and outside which slow sinusoidal modulation oc- parameterization strategy for quantifying the transport ef- curs. The effects of these higher order Bragg resonances fects of mesoscale coherent structures in the ocean which are illustrated using the normal modes of a rectangular cannot be well resolved by large-scale weather and climate tank. In collaboration with Louis N. Howard, Department simulations. We work from the ground up with simple of Mathematics, MIT. Support of J. Yu by National Sci- kinematic models and study in particular how the effective ence Foundation (Grants CBET-0756271, CBET-0845957) diffusivity depends on the governing parameters, such as is gratefully acknowledged. Strouhal number and Peclet number, in a class of random time-dependent vortex flows. Jie Yu North Carolina State University Banu Baydil jie [email protected] Dept. Mathematical Sciences Rensselaer Polytechnic Institute [email protected] MS67 Orthogonal Degree and Bifurcation in the Periodic Peter R. Kramer Nonlinear Schrdinger Lattice Rensselaer Polytechnic Institute Department of Mathematical Sciences We analyze a circular discrete NLS equation, this is, a lat- [email protected] tice of n oscillators with periodic conditions. This lattice has an explicit rotating weave, wich is a relative equilib- Shafer Smith rium, where all oscillators have the same amplitude and Courant Institute differing faces. Using the amplitude of the rotating weave Center for Atmosphere Ocean Science as parameter, we prove bifurcation of symmetric relative [email protected] equilibria from the rotating wave. In addition, we use orthogonal degree to prove bifurcation of symmetric periodic solutions. The symmetries of the periodic solutions let us MS65 conclude that these solutions are traveling waves, where Properties of Kinetic Models of Shock Clustering the faces oscillate with two different periods.

This talk is a description of some mathematical properties Carlos Garc´ıa-Azpeitia of kinetic models of shock clustering. UNAM, Mexico [email protected] Ravi Srinivasan University of Texas at Austin [email protected] MS67 Interlaced Linear-nonlinear Optical Waveguide Ar- rays MS66 A Model for Energy Dissipation due to Wave Continuous as well as periodic wave solutions in a system of Breaking coupled discrete equations describing a two component su- perlattice with interlaced linear and nonlinear constituents Abstract not available at time of publication. and the associated modulational instability are fully analytically investigated and numerically tested for focusing Wooyoung Choi and defocusing nonlinearity. This system is the basis for Dept of Mathematics investigating binary waveguide arrays. This system ex- New Jersey Institute of Tech hibits an extra band-gap controlled by a relative detuning. [email protected] A variety of stable discrete solitary modes, are found and discussed.

Kyriakos Hizanidis, Yannis Kominis 90 NW10 Abstracts

National Technical University, Greece laser induced generation of a compression solitary wave in [email protected], [email protected] two- and three-layered polymethylmethacrylate (PMMA) bars, bonded using ethyl cyanoacrylate-based (CA) adhe- Nikolaos Efremidis sive. Possible applications of the described phenomenon University of Crete, Greece include introscopy of layered structures and seismology. [email protected] Karima Khusnutdinova Loughborough University, UK MS67 [email protected] Exactly Energy-Momentum Conserving Discretiza- tions for Large, Stiff Hamiltonian Systems, Applied Galina Dreiden to Davydov-Scott Models of Energetic Pulses in α- Ioffe Physical Technical Institute, Russia Helix Protein [email protected]ffe.ru

Long-range energetic pulse propagation in α-helix protein Alexander Samsonov, Irina Semenova has been modeled by Davydov, Scott and others with Ioffe Physical Technical Institute Hamiltonian ODE systems, which in a continuum limit [email protected]ffe.ru, [email protected]ffe.ru and a singular “infinite stiffness’ limit, lead to the integrable NLS Equation. The soliton solutions of the latter suggest the possibility of coherent long-range propagation MS68 of energetic pulses in such proteins. These predictions need Pulse Propagation in Granular Chains to be tested with more accurate models, as each limit can fundamentally change the nature of solutions. As exact Granular media can support excitations ranging from the solutions are unknown, numerical solution of these large, very localized to the very dispersive. Experimental and nu- stiff Hamiltonian systems is needed. Simulation is done merical studies of pulse propagation in granular chains have by introducing a time discretization method based on dis- helped to understand energy propagation in these nonlin- cretizing the Hamiltonian form using a finite difference cal- ear discrete one-dimensional media. However, the predic- culus for gradients. This ensures exact conservation of all tive capability of most existing work has been limited by conserved quantities and allows a simple, highly stable it- the dearth of analytic results. We introduce a binary colli- erative method for solving the resulting implicit system. sion approximation that yields quantitatively accurate analytic results for chains of different granular geometries, mass, and/or size distributions. Brenton J. LeMesurier College of Charleston Katja Lindenberg Department of Mathematics Department of Chemistry and Biochemistry [email protected] University of California San Diego, USA [email protected]

MS67 Upendra Harbola Asymptotic Stability of Radiative Shocks in Multi- University of California San Diego dimension Spaces [email protected] Abstract not available at time of publication. Alexandre Rosas Toan Nguyen Universidade Federal de Paraiba Institut de Mathématiques de Jussieu arosas@fisica.ufpb.br Université Pierre et Marie Curie (Paris 6) [email protected] MS68 Periodic and Shock Waves in Strongly Nonlinear MS68 Two-Mass Chains Splitting Induced Generation of Soliton Trains in Layered Elastic Structures Periodic and shock waves were investigated in nonlinear and strongly nonlinear two-mass granular chains composed I will discuss our recent analytical and experimental stud- of steel cylinders and steel spheres. The presentation will ies of long nonlinear bulk waves in layered elastic waveg- outline the first experimental data related to the propa- uides, including wave scattering by inhomogeneities mod- gation of these waves in nonlinear and strongly nonlinear elling poor adhesion or delamination. The emphasis is on chains. The dynamic compressive forces were detected us- classical and radiating solitary waves. In particular, we ing gauges imbedded inside particles at depths equal to 4 study the dynamics of a long longitudinal bulk strain soli- cells and 8 cells from the entrance gauge detecting the in- tary wave in a split, symmetric layered bar, made of a hy- put signal. At these relatively short distances we were able perelastic (Murnaghan) material. The developed approach to detect practically perfect transparency at low frequen- is based on matching two asymptotic multiple-scale expan- cies and cut off effects at higher frequencies for nonlinear sions, integrability theory of the leading order KdV equa- and strongly nonlinear signals. We also observed transfor- tions by the Inverse Scattering Transform and some natural mation of oscillatory shocks into monotonous shocks. Sys- radiation conditions. We show that splitting of the layered tems which are able to transform nonlinear and strongly structure induces a generation of a train of secondary soli- nonlinear waves at small sizes of the system are impor- tary waves from a single incident soliton, and thus, can tant for practical applications such as attenuation of high be used to detect the defect. The theory is supported by amplitude pulses. We also perform numerical calculations experiments, performed in the Ioffe Institute in St. Pe- of signal transformation by non-dissipative granular chains tersburg (Russia), using holographic interferometry and which demonstrated transparency of the system at low fre- NW10 Abstracts 91

quencies and cut off phenomenon at high frequencies in Chemical Reactions reasonable agreement with experiments. We consider solutions of front type in a class of reaction- Vitali Nesterenko diffusion systems which represent exothermic-endothermic University of California at San Diego chemical reactions. There is a linear algebraic relation sat- [email protected] isfied along any travelling wave solution, coming from an invariant of the equations in the system. On the spectral Si Yin Wang level the front solution is known to be marginally unstable. University of California, San Diego Our recent analytic results show that on the nonlinear level [email protected] the instability of the front is of a convective nature. Anna Ghazaryan Eric B. Herbold Department of Mathematics Georgia Inst. of Tech., Atlanta, Georgia Miami University and University of Kansas High Explosives Research and Development Branch, Eglin [email protected] AFB [email protected] Yuri Latushkin University of Missouri, Columbia MS68 [email protected] Solitary Waves and Stability in Lattices and Fluids Stephen Schecter I will recount progress regarding the robustness of solitary North Carolina State University waves in nonintegrable model systems such as FPU lattices, Department of Mathematics and discuss progress toward a proof (with Shu-Ming Sun) [email protected] of spectral stability of small solitary waves for the 2D Euler equations for water of finite depth without surface tension. MS69 Stability of Planar Layers in Systems with Con- Robert L. Pego served Quantities Carnegie Mellon University Department of Mathematical Sciences We prove the stability of layers in systems exhibiting a [email protected] conservation law coupled with a reaction diffusion equation. The key element of the analysis is to find a homotopy to a lower-triangular PDE system that has stable solutions MS69 and to control the essential spectrum during the homotopy. Nonlinear Stability of Semi-discrete Shocks for The layers can destabilize during the homotopy only in the Two Sided Schemes case when a Hopf bifurcation occurs. We use a Lyapunov- Schmidt analysis in weighted spaces to show that eigen- We prove that spectrally stable Lax shocks in semidis- values can not pop out of or disappear into the essential crete conservation laws involving spatial forward-backward spectrum during the homotopy. discretization schemes are nonlinearly stable. The proof involves a construction of the resolvent kernel using ex- Alin Pogan, Arnd Scheel ponential dichotomies, recently developed in this setting, University of Minnesota and uses the associated contour integral for the Green’s School of Mathematics function to derive pointwise bounds. Previous results for [email protected], [email protected] semidiscrete shocks relied on Evans functions, which exist only for one-sided schemes. MS69 Margaret Beck Network Formation in Polymer Electrolytes Department of Mathematics and Statistics Boston University The formation of conductive networks within mixtures of [email protected] functionalized polymers and solvents is essential to many types of energy conversion devices, such as polymer elec- Hermen Jan Hupkes trolyte membrane fuel cells, bulk-heterojunction solar cells, Division of Applied Mathematics Gratzel solar cells, and lithium ion batteries, to name a Brown University few. We outline a novel energy structure which gives excel- [email protected] lent qualitative agreement with experiment, and show that the associated conservative gradient descent flows gener- Bjorn Sandstede ate pore-like networks whose meander patterns are readily Brown University predicted by the associated sharp interface reduction. bjorn [email protected] Keith Promislow Michigan State University Kevin Zumbrun [email protected] Indiana University USA [email protected] PP0 Lie point symmetries and Wave Equations on Man- ifolds MS69 Traveling Waves in Exothermic-endothermic In this presentation we discuss Lie point symmetries of a 92 NW10 Abstracts

nonlinear wave equation that arises as a consequence of a Department of Mathematical Sciences Lorentzian metric of signature -2. Apart from showing how [email protected] geometry can be responsible in giving rise to a nonlinear wave equation, we present a systematic way of using the Lie point symmetries to ﬁnd its exact solutions. Some in- PP0 teresting physical conclusions relating to conservation laws Power-Dependent Soliton Dynamics in Complex are also discussed. Photonic Structures

Ashfaque H. Bokhari In simple photonic structures with monochromatic modu- King Fahd University of Petroleum and Minerals lation of the linear refractive index, soliton dynamics do Dhahran 31261, Saudi Arabia not depend on their power and width. In this work, we [email protected] show that for more complex photonic structures where the linear and/or nonlinear refractive indices are modulated F. Zaman by more than one wavenumbers, soliton dynamics depends King Fahd University of Petroleum and Minerals on their characteristics. It is also shown that, at the in- [email protected] terface between two periodic structures, soliton reflection, transmission, and trapping depends strongly on its charac- A. H Kara teristics. Univ. Witwatersrand, Johannesburg, South Africa Yannis Kominis, Kyriakos Hizanidis [email protected] National Technical University, Greece [email protected], [email protected] MKarim St. John Fisher College, Rochester, New York [email protected] PP0 Design of the Potential in Schroedinger-Type Equations to Minimize Radiative Loss PP0 Effects of Dispersive Radiation on the Phase of We consider a quantum particle in the ground state of Stochastically Perturbed Optical Solitons in Sim- Schroedingers Equation with a localized potential. If the ulations particle is parametrically forced, the eigenfunction becomes aresonantstateanddecaystozeroaccordingtoFermis We demonstrate that soliton perturbation theory, though Golden Rule. We pose the Design Problem: what poten- widely used, does not account for dispersive radiation tial maximizes the lifetime of this resonant state? This is and therefore predicts an incorrect phase distribution formulated and studied as a constrained optimization prob- for optical solitons of the stochastically driven nonlinear lem. Optimal potentials emerge which are periodic with a Schrodinger equation (NLSE). We propose a simple varia- localized defect. We discuss extensions to the nonlinear tional model that accounts for the effect of dispersive ra- Schroedinger equation. diation on the phase evolution and correctly predicts its distribution. We then extend this model to stochastically Braxton Osting driven NLSE with dispersion management. Columbia University [email protected] Daniel Cargill,RichardO.Moore New Jersey Institute of Technology Michael I. Weinstein [email protected], [email protected] Columbia University Dept of Applied Physics & Applied Math Colin McKinstrie [email protected] Lucent Technologies [email protected] PP0 Approximation of Detailed Hodgkin-Huxley-Type PP0 Neuronal Models by Exponential Integrate-and- Vaccinating Against Hpv in Dynamical Social Net- Fire Models work Using current-voltage curves and spike metrics, develop op- We develop a dynamical network model to examine the rel- timal strategies are developed for finding the parameters ative merits of strategies for vaccinating women against the in the exponential integrate-and-fire point neuron model sexually transmitted Human Papillomavirus, which can in- with adaptation current that give the best approximation duce cervical cancer. The model community is represented to the membrane potential and firing rate dynamics of a as a sexual network of individuals with links dynamically corresponding Hodgkin-Huxley-type model. The adapta- created and destroyed through statistical rules based on tion current is modeled explicitly, but has a prescribed the node characteristics. Various strategies for distribut- jump at each spike time. The results of computations using an allotted number of doses of vaccine are tested for ing both systems give excellent agreement. Bifurcations in effectiveness in reducing the incidence of cervical cancer. both models are compared.

Pamela B. Fuller, Toni Wagner Daniel Johnson Rensselaer Polytechnic Institute Rensselaer Polytechnic Institute [email protected], [email protected] [email protected]

Peter R. Kramer Jennifer L. Moyher Rensselaer Polytechnic Institute Rensselaer Polytechnic Institute NW10 Abstracts 93

RPI when shocks develop in any of the Burgers solutions. [email protected] Robert Terrell Joshua Sauppe Mathematics Department U. Wisconsin Cornell University [email protected] [email protected]

Victor Barranca PP0 Rensselaer Polytechnic Institute Autoresonant Four-Wave Mixing in Optical Fibers [email protected] A theory of autoresonant four-wave mixing (FWM) in ta- Gregor Kovacic pered fibers is presented in application to optical paramet- Rensselaer Polytechnic Inst ric amplification (OPA). In autoresonance the interacting Dept of Mathematical Sciences waves stay phase-locked continuously despite the taper- [email protected] ing. This spatially extended phase-locking allows complete pump depletion and flat, wide frequency band am- David Cai plification. Different aspects of autoresonant OPA are de- Shanghai Jiao Tong University, China scribed including initial phase-locking, conditions for au- Courant institute, New York University, USA toresonance, stability, and spatial range of autoresonant [email protected] interaction. The theory is applicable to other autoresonant FWM processes.

PP0 Oded Yaakobi Effects of Non-Linearity and Disorder on Localiza- Racah Institute of Physics, tion of Light in Optical Fiber Arrays The Hebrew University of Jerusalem, Israel [email protected] Light propagating through optical fiber arrays tends to lo- calize in only a few center cores. Recent experiments in Lazar Friedland two-dimensional single-mode optical fiber arrays suggest Hebrew University of Jerusalem that an interplay of deterministic and random linear and Jerusalem 91904, ISRAEL nonlinear effects might be responsible for this localization. [email protected] We study this phenomenon, both analytically and numerically, in a hexagonal optical fiber array similar to that used in the experiments. Our analysis reveals that Anderson localization is evident in the linear and intermediate regimes, where a larger fraction of initial energy is returned to the center fiber due to the stochastic effects. For very high levels of input energy, the system is strongly nonlinear, with the randomness amplifying the Kerr localization. Addi- tionally, we explore the role of disorder and nonlinearity on the coherent coupling in fiber laser arrays. Gowri Srinivasan T-5, Theoretical Division, Los Alamos National Laboratory [email protected]

Alejandro Aceves Southern Methodist University [email protected]

David Moulton Los Alamos National Laboratory Applied Mathematics and Plasma Physics [email protected]

PP0 The Isentropic Euler System Admits Some Plane Wave Superpositions

A class of diﬀerentiable solutions is proved for the isentropic Euler equations in three space dimensions. The solutions are explicitly given in terms of three solutions to the inviscid Burgers equation, and three directions of propagation. The relative orientation of the directions is critical. Within the directional constraints, the Burgers solutions are arbitrary. These solutions cannot exist beyond the time