42 NW12 Abstracts

IP0 IP2 Martin D. Kruskal Lecture - Pattern Quarks and Kramers’ Law - Validity and Generalizations Leptons Consider the overdamped motion of a Brownian particle in Martin Kruskal was one of a kind, a pure scientist who a multiwell potential. Kramers’ Law describes the small- followed roads less travelled by and whose new discoveries noise limit of the particle’s mean transition time between made all the difference in so many ways. Once a question local minima. We will outline several approaches to the intrigued and gripped him, he never let go. The discovery problem of describing such noise-induced transitions, dis- of solitons and the uncovering of whole new classes of inte- cuss recent generalizations of Kramers’ Law to potentials grable systems have led to major advances in the worlds of with nonquadratic saddles and stochastic partial differen- Hamiltonian partial differential equations, in exactly inte- tial equations as well as the limitations of the law such as grable models in statistical physics, and to the fascinating the cycling effect. Generalizations and refined results on and more recently discovered properties of random matri- cycling are joint work with Nils Berglund (Orleans). ces. And there may be yet many more windfalls to come. In this Kruskal lecture, I will give a brief overview of some Barbara Gentz of the early days, of how Martin took on the Fermi, Pasta, University of Bielefeld Ulam conundrum which had defied explanation by some of Bielefeld, Germany the best minds of the time and, with the help of colleagues [email protected] Zabusky, Miura, Gardner and Greene, turned it into one of the most important discoveries of the last half of the 20th century. As for my own contributions to today’s lecture, IP3 I have no illusions that the story I shall tell will have any Fluid Ratchets and Biolocomotion of the same impact but hope that it will reflect some of the same pioneering spirit that Martin displayed in pursu- In this talk, I will discuss a few laboratory experiments ing the unconventional. What I plan to do is sketch some where moving structures freely interact with the surround- ideas I have been mulling in connection with the Standard ing fluid. These moving structures, or boundaries, behave Model and show that quark and lepton like objects with in asymmetric ways - either because of their anisotropic different masses and all the invariants associated with spin construction or by the spontaneous breaking of symmetry (+/-1/2) and charge (+/-1/3,+/-2/3,+/-1) can occur nat- in their response to the fluid. When subjected to reciprocal urally as the result of phase transitions in far from equi- forcing, their motion might be described as a ratcheting be- librium, pattern forming systems. In contrast to the Stan- havior. In one case, a fluid is forced through a corrugated dard Model, they arise without any a priori introduction channel that is under shaking. In another, a solid body of the SU(2) and SU(3) symmetries into the governing free hovers stably in an oscillatory airflow, mimicking a hover- energy. The invariants can each be connected to the con- ing insect. Lastly, a symmetric wing leaps into a forward densation of mean and Gaussian curvature of the constant flight when flapped up and down, following a symmetry phase surfaces along certain defect cores. breaking bifurcation. These phenomena can be viewed as the starting points for understanding the effect of increas- Alan Newell ing biological realism. University of Arizona Department of Mathematics Jun Zhang [email protected] Courant Institute [email protected]

IP1 IP4 Stability and Synchrony in the Kuramoto Model Semiclassical Computation of High Frequency The phenomenon of the synchronization of weakly coupled Waves in Heterogeneous Media oscillators is a old one, first been described by Huygens in his “Horoloquium Oscilatorium.’ Some other exam- We introduce semiclassical Eulerian methods that are effi- ples from science and engineering include the cardiac pace- cient in computing high frequency waves through hetero- maker, the instability in the Millenium Bridge, and the geneous media. The method is based on the classical Liou- synchronous flashing of fireflies. A canonical model is the ville equation in phase space, with discontinous Hamiltoni- Kuramoto model ans due to the barriers or material interfaces. We provide physically relevant interface conditions consistent with the correct transmissions and reflections, and then build the in- dθ  terface conditions into the numerical fluxes. This method i ω γ θ − θ dt = i + sin( j i) allows the resolution of high frequency waves without nu- j merically resolving the small wave lengths, and capture the correct transmissions and reflections at the interface. This method can also be extended to deal with diffraction and We describe the fully synchronized states of this model quantum barriers. We will also discuss Eulerian Gaussian together with the dimensions of their unstable manifolds. beam formulation which can compute caustics more accu- Along the way we will encounter a high dimensional poly- rately. tope, a Coxeter group, and a curious combinatorial iden- tity. This leads to a proof of the existence of a phase tran- Shi Jin sition in the case where the frequencies are chosen from an Shanghai Jiao Tong University, China and the iid Gaussian distribution. University of Wisconsin-Madison [email protected] Jared Bronski University of Illinois Urbana-Champain Department of Mathematics [email protected] NW12 Abstracts 43

CP1 with equal nonzero amplitudes in all channels. Disordered Beam Propagation in a Focusing Non- linear Medium Avner Peleg State University of New York at Buffalo We calculate both analytically and numerically the proba- apeleg@buffalo.edu bility distribution for the intensity of a disordered optical beam propagating in a focusing media within the frame- Quan Nguyen work of the 1+1 Nonlinear Schrodinger equation. The far Vietnam National University at Ho Chi Minh City field intensity pattern is formed by a random number of [email protected] interfering optical solitons with random parameters. We obtain the statistics of these parameters (and hence the Yeojin Chung field intensity distribution) via the disordered Zakharov- Southern Methodist University Shabat eigenvalue problem. [email protected] Stanislav A. Derevyanko Aston University, Birmingham, United Kingdom CP1 [email protected] Nonlinear Refraction Versus Solitonic Fission in Optical Lattices

CP1 Within the nonlinear Shr¨odinger equation we consider the Nonlinear Dynamics and Normal Forms in the soliton beam passing through the periodic or random opti- Time Dependent Nonlinear Schrodinger/Gross cal lattice. We address how the lattice shape and geometry Pitaevskii Equations affect the refraction of the soliton. It is shown that the soliton-wave interference can play an important role in de- We consider the nonlinear dynamics of solutions to the termining the refraction type. Then we demonstrate that NLS/GP equations with potential supporting three or four the interplay between the beam refraction and reflection linear bound states. We compute and prove the existence of can bring about the splitting of a single coherent state into previously-unreported periodic orbits and routes to chaos two ones. in an ODE model problem. Normal forms are computed using Lie transforms, and the finite-dimensional solutions Yaroslav Prylepskiy are shown to be shadowed by solutions to NLS/GP. Com- Nonlinearity and Complexity Research Group parisons are made to NLS trimer and tetramer problems. Aston University [email protected] Roy Goodman New Jersey Institute of Technology Stanislav A. Derevyanko [email protected] Aston University, Birmingham, United Kingdom [email protected]

CP1 Sergei Gredeskul Adaptive Construction of Surrogates for Parameter Department of Physics Inference in Nonlinear Wave Equations Ben Gurion University of Negev, Beer Sheva, Israel [email protected] Surrogate models are often used to accelerate Bayesian inference in computationally intensive systems. Yet the construction of globally accurate surrogates can be pro- CP1 hibitive and in a sense unnecessary, as the posterior dis- Chaos Control in a Transmission Line Model tribution typically concentrates on a small fraction of the prior support. We present a new adaptive approach that We present a model of an electromagnetic system con- uses stochastic optimization (the cross-entropy method) to sisting of a transmission line oscillator coupled to termi- construct surrogates that are accurate over the support of nating nonlinear electrical boundary components. Using the posterior distribution. The method is then applied to the method of characteristics spatio-temporal chaos is ob- parameter inference in nonlinear wave equations. served. A chaos control method is presented, that uses small parameter perturbations to adjust the resistance at Jinglai Li one of the boundaries. The control is based on a Poincare Massachusetts Institute of Technology section technique that does not require a fully developed [email protected] simulation in time. Ioana A. Triandaf CP1 Naval Research Laboratory Crosstalk Dynamics of Optical Solitons in Broad- Plasma Physics Division band Optical Waveguide Systems [email protected] We investigate the dynamics of soliton parameters in broadband optical waveguide systems, induced by energy CP1 and momentum exchange (crosstalk) in pulse collisions. Josephson Oscillations in Dissipative Optical Sys- Using single-collision analysis along with collision rate cal- tems culations, we show that dynamics of soliton amplitudes in N-channel waveguide systems can be described by N- We have studied Josephson oscillations in an active fully dimensional Lotka-Volterra models. By finding the equi- optical system with defocusing Kerr nonlinearity. The con- librium states of these models and analyzing their stability, sidered system consists of a lasing cavity emitting pho- we are able to obtain ways for achieving stable transmission tons into a waveguiding structure having strong but narrow 44 NW12 Abstracts

variation of the refractive index. It has been shown that CP2 Josephson-like oscillations appear in the system when the Perturbed Magnetic Droplet Solitons flux though the barrier exceeds the threshold value. Dif- ferent regimes of the oscillations have been studied thor- The Landau-Lifshitz equation with uniaxial anisotropy in oughly. one and two spatial dimensions admits a two-parameter family of solitary wave solutions called magnetic droplets. Alexey Yulin Physically relevant perturbations due to weak damping, a Centro de F slowly varying external magnetic field, and spin torque are ’isica Te considered in the context of soliton perturbation theory. A ’orica e Computacional, dynamical systems analysis of the modulation system and Universidade de Lisboa direct numerical simulations of the governing PDE demon- [email protected] strate that physical droplets can be nucleated, accelerated, and controlled. Vladimir V. Konotop Universidade de Lisboa, Mark A. Hoefer Portugal North Carolina State University [email protected] [email protected]

Matteo Sommacal CP2 Northumbria University, Newcastle, UK Envelope Quasi-Solitons in Dissipative Systems [email protected] with Cross-Diffusion We consider two-component nonlinear dissipative spatially CP2 extended systems of reaction-cross-diffusion type. Previ- Amplification of Synaptic Inputs by Dendritic ously, such systems were shown to support “quasi-soliton’ Spines pulses, which have fixed stable structure but can reflect from boundaries and penetrate each other. Presently we Dendritic spines are the site of contact for the majority of demonstrate a different type of quasi-solitons, with a phe- excitatory synapses in the mammalian brain, and as a re- nomenology resembling that of the envelope solitons in sult are the first step in the signaling between dendritic Nonlinear Schroedinger equation: spatiotemporal oscilla- inputs and neuronal actional potential output. In this tions with a smooth envelope, with the velocity of the os- combined theoretical and experimental study, it is demon- cillations different from the velocity of the envelope. strated that spines provide a uniformly high impedance compartment across the dendritic arbor that amplifies lo- Vadim N. Biktashev cal depolarization. This spine amplification increases non- Dept of Mathematical Sciences linear voltage-dependent conductance activation and pro- University of Liverpool motes electrical interaction among coactive inputs, enhanc- [email protected] ing neuronal response.

Mikhail Tsyganov William Kath Institute of Theoretical and Experimental Biophysics Northwestern University Pushchino Research Centre, Russian Academy of Science Engineering Sciences and Applied Mathematics [email protected] [email protected]

CP2 CP2 Evolution of Spiral and Scroll Waves of Excitation Impact of Randomness on Solitary Wave Propaga- in a Mathematical Model of Ischaemic Border Zone tion in Granular Systems We use asymptotics based on response functions to pre- We study the effect of randomness on the propagation of dict dynamics of re-entrant excitation waves in a moving solitary waves in granular media composed of spherical par- boundary of a recovering ischaemic cardiac tissue, due to ticles. Randomness in grain properties and size is con- gradients of cell excitability and cell-to-cell coupling, and sidered. The study shows the presence of two regimes of heterogeneity of individual cells. In three spatial dimen- decay in peak force and kinetic energy, and a gradual trans- sions, theory predicts conditions for scrolls to escape into fer from potential to kinetic energy of the system. In 2D the recovered tissue, where they are either collapse or de- square packed systems, we also investigate the anisotropy velop fibrillation-like state, depending on filament tension. of the randomness induced decay. We confirm these predictions by direct simulations. Mohith Manjunath,AmnayaP.Awasthi,PHILIPPEH. Irina Biktasheva Geubelle University of Liverpool University of Illinois at Urbana-Champaign [email protected] [email protected], [email protected], [email protected] Vadim Biktashev University of Liverpool CP2 Department of Mathematical Sciences [email protected] Target Detection in Non-Uniform Slowness Fields Using Time-Reversed Nonlinear Solitary Waves

Narine Sarvazyan Radar problems include propagation of short waves with George Washington University large scale realistic effects, such as refraction by complex [email protected] NW12 Abstracts 45

slowness fields. Our approach is based on dissipative non- We have now derived an exact solution for the propagation linear solitary waves centered on wave centroid surfaces, velocity of these waves and show that these waves fall into which propagate on characteristics. This approach allows a class of traveling wave phenomena encountered in other efficient propagation both forward in time, and backward systems. The results are confirmed numerically. in time (for target and source detection). Unlike ray trac- ing in complex domains, there do not appear to be any Mark Deinert multiparticle chaotic effects. The University of Texas at Austin [email protected] Subhashini Chitta UTSI subha@flowanalysis.com CP3 Rogue Waves in Plasmas John Steinhoff University of Tennessee Space Inst. Rogue waves (freak waves) are space- and time-localized Goethert Pkwy extreme events, recently arising as a paradigm in vari- [email protected] ous fields. The occurrence of such excitations in charged matter (plasmas) is investigated, from first principles. A multiscale technique for the derivation of a nonlinear CP3 Schr¨odinger (NLS) model for plasma waves is adopted, and Solitons in Dipolar Bose-Einstein Condensates the result is analyzed, in terms on intrinsic plasma parame- with Trap and Barrier Potential ters. A set of exact solutions of the NLS equation has been argued to provide a plausible model for rogue waves in the The propagation of solitons in dipolar BEC in trap poten- ocean. These include rational solutions, the Ma breather, tial with a barrier potential is investigated. The regimes the Akhmediev breather and the Peregrine soliton. We re- of soliton transmission, reflection and splitting in depen- view the properties of these solutions, and investigate their dence on the ratio between the local and dipolar nonlocal occurrence and relevance in plasmas. Our research covers interactions are analyzed analytically and numerically. The both electrostatic and electromagnetic modes, and provides conditions for a fusion of splitted solitons are found. In ad- a self-contained framework for future investigations of the dition the delocalization transition governed by strength of dependence of such modes on intrinsic plasma properties. the nonlocal dipolar interaction is presented. Ioannis Kourakis Fatkhulla Abdullaev Queen’s University Belfast Uzbek Academy of Sciences Centre for Plasma Physics (CPP) Uzbekistan [email protected] [email protected] Vikrant Saxena, Sharmin Sultana Valeriy Brazhnyi Centre for Plasma Physics (CPP) Faculdade de Ciencias, Universidade de Porto, Portugal Queen’s University Belfast, UK [email protected] [email protected], [email protected]

Jafar Borhanian CP3 Department of Physics, Faculty of Science Steady Structures and Rheology of Active Complex University of Mohaghegh Ardabili, Iran Fluids [email protected] Bacteria suspensions, active gels and assemblies of motors and filaments are active complex fluids which differ from CP3 their passive counterparts in that particles absorb energy A Characterization of Dispersive Shock Waves in and generate motion. They are interesting from a more the Magma Equation fundamental perspective as their dynamic phenomenons are both physically fascinating and potentially of great bio- In 1-D and under reasonable simplifications and assump- logical significance. In this talk, I will present a continuum tions, the full governing equations of subterranean verti- model for such systems. Hydrodynamics, stability and rhe- cal magma transport reduce to a degenerate nonlinear dis- ology will be discussed. persive partial differential equation for the porosity φ of n n −m the form: φt +(φ )z − (φ (φ φt)z)z =0,wheren, m Zhenlu Cui are physically derived parameters. For an arbitrary ini- Department of Mathematics and Computer Science tial step and physical parameter values, the properties of Fayetteville State University magma dispersive shock waves (speeds, leading amplitude) [email protected] are constructed asymptotically. Results are corroborated by comparisons with an accurate pseudospectral numerical code for long-time integration. CP3 Propagation of a Constant Velocity Fission Wave Nicholas Lowman Department of Mathematics The mechanism for the formation of a fission wave is sim- North Carolina State University ple: a large cylinder of fertile material is subjected to a [email protected] neutron source at one end. The neutrons transmute mate- rial downstream and produce plutonium, which itself un- Mark A. Hoefer dergoes fission and produces neutrons. If the conditions are North Carolina State University right, a self-sustaining reaction wave would form. Numeri- [email protected] cal studies have shown that waves of this type are possible. 46 NW12 Abstracts

CP3 the ecient nonlinear local absorbing boundary conditions On Seismic Wave in Magnetoelastic Low Velocity for the nonlinear wave equation, and give the stability anal- Crustal Layer ysis of the re- sulting boundary conditions. Finally, several numerical examples are given to demonstrate the eective- We aim to study the propagation pattern of shear type ness of our method. (This talk is based on the joint work waves or G-type waves in magnetoelastic crustal layer with Prof. Xiaonan Wu and Dr. Jiwei Zhang.) which is highly anisotropic in nature. The layer has been taken self-reinforced which is lying over a heterogeneous Hongwei Li semi-infinite medium. B. Gutenberg investigated this spe- Hong Kong Baptist University cial class of waves in low velocity zone between the Earth [email protected] crust and upper mantle. Along with the period equation Condition for a large amount of energy to be confined near the surface has been obtained. We observed that a large CP4 portion of energy can flow along the interface for horizon- A Discontinuous-Galerkin Spectral Wave Model tally polarized shear waves with group velocity lower than the shear wave velocity in the upper mantle. Effects of Spectral wind-wave models have successfully been used in reinforcement, magnetoelastic coupling parameter and in- the ocean for many different applications. We implement homogeneity on the phase velocity of shear wave have been a new numerical modeling approach using a discontinuous- obtained separately and depicted by means of graphs. The Galerkin method in both geographic and spectral space. surface plot of group velocity is drawn for wave number This allows us to have an unstructured-triangulation in and depth parameter. geographic space and a structured spectral space with the ability to use higher-order approximations when needed. Sanjeev A. Sahu We verify and validate the model on test bed cases and Indian School of Mines, Dhanbad, India achieve higher accuracy when using higher-orders in spec- [email protected] tral space. Jessica D. Meixner Amares Chattopadhyay Institute for Computational Engineering and Sciences Indian School of Mines, Dhanbad-826004, India University of Texas at Austin [email protected] [email protected]

CP3 Casey Dietrich Radiation by Superfluid Vortices in the Lighthill Insitute for Computational Engineering and Sciences Regime University of Texas at Austin [email protected] On hydrodynamic length scales, quantized superfluid vor- tices are commonly modeled as a system of coreless point Clint Dawson vortices in 2D, or vortex filaments in 3D, whose motion is Institute for Computational Engineering and Sciences determined classically through the Biot-Savart law. Yet it University of Texas at Austin is well known that a microscopic model that can describe [email protected] the motion of quantized vortices exists in the form of Gross- Pitaevskii (GP) equation which captures the structure of the vortices within the healing layers. This model has been CP4 successfully tested in predicting vortex dynamics in Bose- Numerical Modeling of Internal Waves Generated Einstein condensates. In this work, we consider the asymp- by Turbulent Wakes Behind Towed Bodies in Strat- totic regimes under which point vortex dynamics is recov- ified Media ered from the GP equation. We show that the regime un- der which this occurs corresponds to the so-called Lighthill Numerical models of an internal waves generated by wakes regime for evaluating the acoustic radiation in a classical in stratified fluids have been presented. The calculation fluid. We, therefore, formulate a corresponding theory of results show that the wake’s excess momentum of order radiation that specifies the sound emitted by the motion of from the total momentum in drag wake has a weak in- of quantized vortices in the limit of low Mach number or fluence on the internal waves and on the turbulent energy equivalently large inter-vortex separation. We demonstrate decay. As in the case of a homogeneous fluid, a more signif- our theoretical predictions with numerical simulations of icant influence of total excess momentum on the axis value the GP equation. of the defect of the longitudinal velocity component was observed. Hayder Salman University of East Anglia Nikolay Moshkin [email protected] Suranaree University of Technology [email protected], [email protected]

CP4 Gennadii Chernykh Local Absorbing Boundary Conditions for Nonlin- Institute of Computational Technologies, Russia ear Wave Equation on Unbounded Domain [email protected] The numerical solution of nonlinear wave equation on un- bounded spatial domain is considered. The articial bound- CP4 ary method is introduced to reduce the nonlinear problem Compatible Numerical Scheme for the Linear Iner- on unbounded spa- tial domain to an initial boundary value tial Waves problem on a bounded domain. Using unied approach, which is based on operator splitting method, we construct A compatible numerical model based on a discontinuous NW12 Abstracts 47

Galerkin FEM discretisation that preserves the Hamilto- and with Hurst parameter less than half. nian structure of the linear rotating Euler equations is pre- sented. Thus, it conserves the important properties of the Mahmoud M. El-Borai PDEs. Dirac theory is applied to derive the incompressible Prof.of Math. Dep. of Mathematics Faculty of Science limit of the compressible Euler or acoustic equations which Alexandria University Alexandria Egypt are used as a starting point in the numerical discretisation. m m [email protected] Several test cases will be presented to assess the quality of the numerical model. Khairia E. El-Nadi Prof of Mathematical Statistics Dep. of Math. Faculty of Shavarsh Nurijanyan Science Alexandria University Alexandria Egypt University of Twente Khairia el [email protected] [email protected]

CP5 CP4 Semicommutative Operators Associated with Dirac A Coupled Boundary-Integral/Level-Set Method Operator and Darboux Transformation for Eigenvalue Optimization Problems We clarify a class of differential operators which are semi- Eigenvalues are ubiquitous in describing wave phenom- commutative with the 1-dimensional Dirac operator P ena and thus constrained eigenvalue optimization problems defined by P = iσ2(d/dx)+σ1v(x), where σ1, σ2 are naturally arise when one seeks to control such phenomena. Pauli matrices. Such results are already known for the We study a new, coupled boundary-integral/level-set com- 1-dimensional Schr¨odinger operator H = −(d/dx)2 + u(x). putational method which demonstrates greater accuracy We extend these results to the Dirac operator P .Asare- over previously used methods while maintaining the capa- sult, we obtain the interesting formula related to the Dar- bility of handling topological changes. Results for several boux ransformation. Moreover, a new algebraic scheme for shape and structural eigenvalue optimization problems are solving the eigenvalue problem associated with the Dirac discussed, including Krein’s problem on finding the den- operator P is obtained. sity distribution of a membrane which extremizes the k-th Dirichlet-Laplacian eigenvalue. Mayumi Ohmiya, Masatomo Matsushima Doshisha University Braxton Osting [email protected], Columbia University [email protected] [email protected]

Chiu-Yen Kao CP5 The Ohio State University Interaction Properties Of Complex Solitons For [email protected]; [email protected] The Hirota MKdV Equation The Hirota-mKdV equation has general complex 1-soliton CP4 solutions that describe solitary waves with time-varying Numerical Solutions of Two-Way Propagation of phase. We use Hirota’s bilinear formalism to find the gen- Nonlinear Dispersive Waves Using Radial Basis eral complex 2-soliton solution. This solution describes a Functions collision in which a fast soliton over takes a slow soliton. By an asymptotic analysis we show that collisions produce In this talk we obtain the solutions of a Boussinesq system a shift in both the position and the phase of the fast and for two-way propagation of nonlinear dispersive waves by slow solitary waves. We observe several types of inter- using the meshless method based on collocation with radial actions during the collision depending on the speeds and basis functions. The system of nonlinear partial differential phase angles of the fast and slow solitons. equation is discretized in space by using these functions. The discretization leads to a system of coupled non-linear Abdus Sattar Mia ordinary differential equations. These equations are then Brock University, Canada solve by using an Adam-Bashforth scheme. The method sattar [email protected] is tested for exact solitary solutions to these equations. In addition four conserved quantities are calculated numeri- Stephen Anco cally. The numerical results show excellent agreement with Department of Mathematics the analytical solution and calculated conserved quantities. Brock University [email protected]

Pablo U. Suarez Mark Willoughby Delaware State University University of British Columbia, Canada [email protected] [email protected]

CP5 CP5 Fractional Partial Differential Equations Driven by Stability of Solitary Waves for the Vector Nonlinear Fractional Gaussian Noise Schroedinger Equation in Higher-Order Sobolev Some fraction parabolic partial differential equations Spaces driven by fraction Gaussian noise are considered. Initial- In this talk, a sharp form of orbital stability for the value problems for these equations are studied. Some prop- erties of the solutions are given under suitable conditions 48 NW12 Abstracts

Schr¨odinger system, also known as the vector NLS, dispersal kernels which can be different for the two popu- m lations. ∂ ∂2  i u u |u |2u , ∂t j + ∂xx j +2 i j =0 Judith R. Miller i=1 Georgetown University u x, t ∈ R2 Department of Mathematics where j are complex-valued functions of ( ) , [email protected] j =1, 2,...,m,ispresentedinL2-based Sobolev classes of arbitrarily high order. The result means practically that not only does the bulk of what emanates from the per- Huihui Zeng turbed solitary wave stay close in shape and propagation Tsinghua University and phase speeds to the original solitary wave, but emerg- [email protected] ing residual oscillations must also be very small and not only in the energy norm. CP5 Nghiem Nguyen Stability of Solitons in PT-Symmetric Nonlinear Utah State University Potentials [email protected] We consider stationary localized modes q(ξ,η)=eibξw(η) of the nonlinear Schrodinger equation iqξ = −qηη − [1 + 2 CP5 iW (η)]|q| q,whereW (η) is an odd function. We report continuous families of the nonlinear modes, which can be Vanishing Viscosity Limit for An Optimal Control b Problem of Scalar Conservation Laws in the Pres- parametrized by the propagation constant .Weinves- ence of Shocks tigate linear stability of the modes computing the Evans function of the corresponding linear operator. Our analy- To reduce the computational cost for optimal control prob- sis shows that for σ  1 the localized modes for W (η)= lems of the inviscid Burgers equation in the presence of σ sin η and W (η)=σ tanh η become unstable when b is shocks, Zuazua et al. have developed an alternating de- below a certain threshold value. However, this threshold scent method, and revisited it by the method of vanishing disappears for W (η)=ση. viscosity. In the present work, we study theoretically the vanishing viscosity limit of such a problem for 1-D scalar Dmitry Zezyulin conservation laws with a single shock or finite many non- Centro de Fsica Te´orica e Computacional interacting shocks. We employ the method of matched Universidade de Lisboa asymptotic expansions to construct approximate solutions [email protected] to the smoothed nonlinear, the linearized and the dual problems, respectively. It is then proved that the approxi- Fatkhulla Abdullaev mate solutions satisfy the corresponding equations asymp- Centro de Fisica Teorica e Computacional, totically, and converge to the solutions of the corresponding Faculdade de Ciencias, Universidade de Lisboa inviscid problems with certain rates. The discontinuities of [email protected] coefficients in those equations, especially the inverse dual equation, lead to difficulties when taking the limits. The Yaroslav Kartashov equations for the shock and the variation of its position are Institut de Ci`encies Fotniques also derived. Barcelona, Spain [email protected] Yaobin Ou University of Electronic Science and Technology of China Vladimir V. Konotop [email protected] Universidade de Lisboa, Portugal Peicheng Zhu [email protected] University of Basque Country, Spain [email protected] CP6 Inertial Waves in a Rapidly Rotating Cylinder CP5 Stability of Traveling Waves for Systems of Nonlin- Rapidly rotating flows can support waves with peculiar ear Integral Recursions in Spatial Population Biol- properties. In the inviscid limit, the equations for infinites- ogy imal disturbances about solid-body rotation reduce to a hyperbolic problem for disturbance frequencies less than We use spectral methods to prove a general stability theo- twice the background rotation rate, the characteristics of rem for traveling wave solutions to the systems of integrod- which represent discontinuities in the velocity or its gradi- ifference equations arising in spatial population biology. ent. In real life, these are regularized by viscosity, resulting We show that non-minimum-speed waves are exponentially in the observed inertial waves. We explore numerically the asymptotically stable to small perturbations in appropri- ∞ consequences of finite viscosity and nonlinearity on such ately weighted L spaces, under assumptions which ap- flows. ply to examples including a Laplace or Gaussian disper- sal kernel a monotone (or non-monotone) growth function Juan M. Lopez behaving qualitatively like the Beverton-Holt function (or Arizona State University Ricker function with overcompensation), and a constant School of Mathematical and Statistical Sciences probability p ∈ [0, 1) (or p = 0) of remaining sedentary [email protected] for a single population; as well as to a system of two pop- ulations exhibiting non-cooperation (in particular, Hassell and Comins’ model) with p = 0 and Laplace or Gaussian NW12 Abstracts 49

CP6 Modeling Have You Seen Our Water Waves? A new method, based on a relaxed variational principle, When water flows past an obstruction such as a ship or is presented for deriving approximate equations for water a step in a channel, waves are often produced behind or waves. It is particularly suitable for the construction of ap- ahead of the disturbance. Recently, techniques in exponen- proximations. The advantages will be illustrated on numer- tial asymptotics have allowed us to predict the theoretical ous examples in shallow and deep water. Using carefully existence of new classes of gravity-capillary waves. These chosen constraints in various combinations, several model waves have never been seen before – in nature or in the equations are derived, some being well-known, others being digital world. Do they truly exist? Come and decide for new. These models are studied analytically, exact travel- yourselves! ling wave solutions are constructed, and the Hamiltonian structure unveiled. Philippe H. Trinh Program in Applied and Computational Mathematics Denys Dutykh Princeton University Universit´edeSavoie-CNRS [email protected] [email protected]

Jonathan Chapman MS1 Mathematical Institute University of Oxford A New Model of Roll Waves: Comparison with [email protected] Brock’s Experiments We derive a mathematical model of shear flows for shallow MS1 water flowing down an inclined plane. Periodic solutions to this model describing roll waves were obtained. The solu- On the Stability of Some Periodic Waves Arising tions are in good agreement with the experimental profiles in the Kawahara Equation of roll waves measured in Brock’s experiments. In particu- The Kawahara equation is a weakly nonlinear model for lar, the height of the vertical front of the waves, the shock capillarity-gravity water waves which admits solitary-wave thickness and the wave amplitude are well captured by the type solutions. For each solitary wave, there exists a family model. of periodic waves which is asymptotic to the solitary wave Sergey Gavrilyuk when the period tend to infinity. In this talk we study the University Aix-Marseille, UMR CNRS 6595 IUSTI stability of these periodic waves. [email protected] Frederic Chardard ENS Lyon Ga¨el Richard France Aix-Marseille University [email protected] Marseille , France [email protected]

MS1 Viscosity-induced Instability for Euler and Aver- MS1 aged Euler Equations in a Circular Domain Geometric Integration for Damped Hamiltonian PDEs We consider the infinite time behaviour of a family of sta- tionary solutions of Euler’s equation, which can be de- A general framework for constructing numerical meth- scribed as constrained minima of energy on level sets of en- ods that exactly preserve dissipative properties of damped strophy. For free boundary conditions, this family shadows Hamiltonian PDEs is presented in detail. These methods solutions of 2D Navier-Stokes equations. However, under are compared analytically and numerically to standard con- the no-slip and under the Navier-slip boundary conditions servative methods, which generally destroy the actual dis- and in a circular domain, the infinite time Navier-Stokes sipation rates but do retain other advantages in the dissi- evolution orbit of a starting point on the family of con- pative context. Semi-linear wave equations and nonlinear strained mimina has order 1 distance to the family, however Schrodinger equations, both with added dissipation, are small the viscosity is. The viscosity in the Navier-Stokes used as examples to demonstrate the long-time behavior of equations is a singular perturbation for Euler’s equation the numerical solutions. and one might suspect that the viscosity-induced insta- bility is related to this singularity. This is not the case: Brian E. Moore we show that the same phenomenon can be observed for University of Central Florida the averaged Euler equations and second grade fluids with USA Navier-slip boundary conditions in a circular domain. [email protected] Gianne Derks University of Surrey MS2 Department of Mathematics Asymptotic Methods for Multi-scale Solitary [email protected] Waves in Periodic Media Gap solitary waves in a one-dimensional periodic lattice MS1 are studied for the case that the solitary wave spans a Relaxed Variational Principle for Water Wave large number of lattice periods. In this limit, an analyt- ical theory utilizing exponential asymptotics is presented, that reveals the bifurcation of a countable set of solitary- wave families. The stability properties of certain of these 50 NW12 Abstracts

solution families are also discussed, and the analytical pre- the lattices also show significant differences. Comparative dictions are compared against numerical results. analysis of the properties of the nonlinear modes will be reported. Triantaphyllos R. Akylas Mechanical Engineering Vladimir V. Konotop M.I.T. Universidade de Lisboa, [email protected] Portugal [email protected]

MS2 Dmitry Zezyulin Light Localization in Specially Designed Photonic Centro de Fsica Te´orica e Computacional Lattices Universidade de Lisboa [email protected] Optically-induced photonic lattices have served as an ideal platform for exploring discretizing light behaviors. In this talk, we provide a brief overview of our recent work on MS2 spatial control of light propagation in specially-designed Stability Analysis of Solitons in PT Symmetric Lat- photonic lattices established by the optical induction tech- tices nique. In particular, we present our experimental re- sults on controlled Bragg reflection and anomalous diffrac- Parity-time (PT)-symmetric potentials are complex poten- tion/refraction in ionic-type lattices, surface localization tials in the Schr¨odinger equation whose real and imaginary and edge states in photonic superlattices and honey-comb parts are symmetric and anti-symmetric respectively. PT lattices, image transmission through 3D photonic lattices potentials have the surprising property that even though with engineered coupling, along with recent results on dis- they contain gain and loss in the potential, the linear spec- ordered lattices. trum of the Schr¨odinger operator can still possess all-real eigenvalues, thus allowing solitons to exist over a contin- Zhigang Chen uous range of propagation constants in the presence of San Francisco State University nonlinear effects. In this talk, we investigate the stabil- [email protected] ity of solitons in PT-symmetric periodic potentials (op- tical lattices) in both one and two dimensional systems. MS2 First we show analytically that when the strength of the gain-loss component rises above a certain threshold (phase- Multivortex Discrete Solitons in Coupled Nonlin- transition point), an infinite number of Bloch bands turn ear Waveguides and Photonic Lattices complex simultaneously. Second, we show numerically that We introduce discrete solitons with globally linked mul- stable families of solitons exist in PT lattices. In one di- tiple vortices in a ring of nonlinear oscillators coupled to mension the fundamental solitons in the semi-infinite gap a central site. The system is described by the nonlinear remain stable up until the phase transition point, however, Schroedinger equation and supports a variety of multivor- in higher bandgaps and higher dimensions we find that in- tex solitons with complex phase structures. We study these creasing the gain-loss component has a destabilizing effect multivortex solitons and determine their stability analyti- on soliton propagation. The parameter range of stable soli- cally and numerically. We show that these solitons may be tons shrinks as new regions of instability appear. In fact, in perturbed to produce stable “breather’ states with novel two dimensions stable solitons only exist for a bounded set vortex dynamics, such as coordinated charge flipping and of propagation constants in the semi-infinite gap, even for vortex spiraling. very small imaginary components. Thirdly, we investigate the evolution of unstable solitons under perturbation. Daniel Leykam The Australian National University Sean Nixon [email protected] Department of Mathematics and Statistics University of Vermont [email protected] Anton S. Desyatnikov Australian National University [email protected] Lijuan Ge University of Vermont [email protected] MS2 Nonlinear Modes in Finite-dimensional Jianke Yang PT-symmetric Systems Department of Mathematics and Statistics University of Vermont Linear parity-time symmetric lattices below the point of [email protected] the symmetry breaking posses pure real spectra. Such models can be reduced by a unitary transformation ei- ther to Hamiltonian systems or to dissipative systems not MS3 obeying the symmetry but still having pure real spectrum Space-time Statistics of Capillary Wave Turbulence (termed pseudo-Hermitian). This property, valid for linear systems does not hold when the nonlinearity is taken into We report on the experimental observation of space-time account. While Hamiltomian and PT symmetric lattices resolved pure capillary wave turbulence. The wave sys- allow for existence of the continuous branches of the so- tem is studied experimentally with two immiscible ?uids lutions, the pseudo-Hermitian models allow for existence of almost equal densities where capillary surface waves are of only a discrete set of the localized modes. The stabil- excited by a low-frequency random forcing. The probabil- ity properties of localized modes in theses three types of ity density function of the wave amplitude shows a quasi- Gaussian behavior and the power spectral density is shows NW12 Abstracts 51

a power-law behavior in frequency with a slope of α =- of ”inverse cascade” may be observed. 3.0 and in wave number with a slope β =-4.0. These results agree with theoretical predictions on Kolmogorov- Sergio Rica Zakharov spectra and wave amplitude statistics for capil- Universidad Adolfo Ibanez lary waves. Facultad de Ingenieria y Ciencias [email protected] Claudio Falcon Departamento de Fsica Universidad de Chile, Santiago, Chile MS3 [email protected] Wave Turbulence in Nonlinear Fiber Optics: Ex- periments and Theory Andr´es Franco Departamento de Fsica, Universidad de Chile We present a theoretical and experimental study of the [email protected] nonlinear propagation of partially incoherent optical waves in single mode optical fibers. We revisit the traditional treatment of the wave turbulence theory to provide a statis- MS3 tical kinetic description of the integrable scalar NLS equa- Non Linear Dynamics of Flexural Wave Turbulence tion. We theoretically and numerically show the existence of an irreversible evolution toward a statistically station- We report a direct measurement of the nonlinear timescale ary state. We report the experimental observation of this TNL related to energy transfer in wave turbulence of flexion peculiar relaxation process of the one dimension NLS equa- waves on a thin elastic plate. This time scale is extracted tion. from the space-time measurement of the deformation of the plate by studying the temporal dynamics of wavelet Pierre Suret coefficients of the turbulent field. We discuss the separation Universite de Lille 1 between the relevant time scales which is at the core of the [email protected] weak turbulence theory. A. Picozzi Benjamin Miquel Universit´e de Bourgogne Laboratoire de Physique Statistique [email protected] Ecole Normale Sup´erieure/CNRS/Univ. P. & M. Curie [email protected] Stephane Randoux Universit´e de Lille 1 Nicolas Mordant [email protected] Laboratoire des Ecoulements G´eophysiques et Industriels Univ. J. Fourier / CNRS / Grenoble INP/IUF [email protected] MS4 A Fast and Accurate Absorbing Boundary Condi- tion for the Helmholtz Equation MS3 Wave Turbulence: A Story Far From Over Constructing accurate absorbing or radiating boundary conditions for the Helmholtz equation in heterogeneous When is the wave turbulence closure valid and when is it media is difficult and costly. We propose here a general not ? When are the Kolmogorov-Zakharov solutions valid framework for rapidly constructing and evaluating ABC’s and when are they not ? This talk will discuss these chal- by compressing the Dirichlet to Neumann map, using ma- lenges. trix probing. This fits the DtN map to a few well-chosen matrices and thus has greater potential for accuracy and Alan Newell flexibility in variable media. This can be used as precom- University of Arizona putation for solving the wave equation multiple times. Department of Mathematics [email protected] Rosalie Belanger-Rioux,LaurentDemanet Massachusetts Institute of Technology Department of Mathematics MS3 [email protected], [email protected] Can One Hear a Kolmogorov Spectra? Wave Tur- bulence on a Thin Elastic Plate MS4 We study the long-time evolution of waves of a thin elastic Dispersionless Time-domain PDE Solvers for High- plate in the limit of small deformation so that modes of os- frequency Problems and General Spatial Geome- cillations interact weakly. According to the theory of weak tries turbulence (successfully applied in the past to plasma, op- tics, and hydrodynamic waves), this nonlinear wave system We present fast time-domain high-order PDE solvers that evolves at long times with a slow transfer of energy from address some of the main difficulties associated with simu- one mode to another. A kinetic equation for the spectral lation of realistic scientific and engineering systems under transfer in terms of the second order moment is derived. high-frequency and nonlinearity. Based on a novel Fourier- We have shown that such a theory describes the approach Continuation (FC) method for the resolution of the Gibbs to an equilibrium wave spectrum and represents also an en- phenomenon, these solvers can deliver unconditional sta- ergy cascade, often called the Kolmogorov-Zakharov spec- bility, essentially dispersionless numerics, high order accu- trum. We have realized numerical simulations that con- racy and perfect parallel scaling for challenging linear and firmed this scenario moreover under some conditions a kind nonlinear problems involving high frequencies and general 52 NW12 Abstracts

spatial domains. MS4 Global Geometrical Optics Method Oscar P. Bruno California Institute of Technology We develop a novel approach, named global geometrical op- [email protected] tics method, for the numerical solution to wave equations in the high-frequency regime. The initial Cauchy data is assumed in the WKB form. The basic idea of this approach MS4 is to reformulating the governing equation in a moving The Gaussian Beam Method for the Dirac Equa- frame, and deriving a globally valid WKB-type function tion in the Semi-classical Regime with the aid of partition of unity. This WKB-type func- tion is merely defined on the Lagrangian manifold induced The Dirac equation is an important model in relativis- by the Hamiltonian flow, and from which, the wave solu- tic quantum mechanics. In the semi-classical regime  tion can be retrieved by a coherent state integral within 1, the best existing method time splitting spectral first order accuracy. The merit of the proposed approach method [Z.Y. Huang, S. Jin, P.A. Markowich, C. Sparber is manyfold. First, compared with the thawed Gaussian and C.X. Zheng, A time-splitting spectral scheme for the beam approaches, it presents an approximate wave solu- Maxwell-Dirac system, Journal of Computational Physics, O tion with first order asymptotic accuracy uniformly, even 208(2005), pp. 761-789.] require the mesh size to be ( ), around caustics. Second, compared with the canonical which makes the direct simulation extremely expensive. operator method, this approach does not require any a In this paper, we present the Gaussian beam method for priori knowledge about the structure of Lagrangian man- such problem. With the help of suitable space decomposi- ifold. Third, compared with the frozen Gaussian beam tion technique, the Gaussian beams can be independently approaches such as Herman-Kluk semi-classical propaga- evolved and summed to constructed the solution of the tor method, the proposed approach involves an integral on Dirac equation efficiently and accurately. Moreover, the a manifold of much lower dimension. We report numerical proposed Eulerian Gaussian beam keep the advantages of tests on both Schr¨odinger and Helmholtz equations. constructing the Hessian matrices by using level set func- tions’ derivatives. Finally we test our method by several Chunxiong Zheng numerical examples. Tsinghua University [email protected] Zhongyi Huang Tsinghua University, China [email protected] MS5 Coherent Structures, Transport and Invariant Hao Wu Manifolds in Unsteady Flows Department of Mathematical Sciences Tsinghua University Lagrangian coherent structures (LCSs) are ubiquitous in [email protected] fluid flows at all scales, and are strongly influential in de- termining scalar transport in unsteady flows. This talk Shi Jin will outline theoretical, diagnostic and numerical issues re- Shanghai Jiao Tong University, China and the lated to characterizing both LCSs and consequent trans- University of Wisconsin-Madison port, and introduce the areas to be examined in more de- [email protected] tail by the minisymposium speakers. Invariant manifolds (as defining LCSs) will also be addressed under general Dongsheng Yin time-dependence, along with explicit examples in 2D and Department of Mathematical Sciences 3D. Tsinghua University Sanjeeva Balasuriya [email protected] Connecticut College Department of Mathematics MS4 [email protected] Frozen Gaussian Approximation for High Fre- quency Wave Propagation MS5 We propose the frozen Gaussian approximation for the Data Mining Remotely Sensed Image Sequences computation of high frequency wave propagation. It pro- and Transport Analysis of Spatiotemporal Dynam- vides a highly efficient computational tool based on the ical Systems asymptotic analysis on phase plane. Compared to geo- Scientific fields, such as climatology, and oceanography metric optics, it provides a valid solution around caustics. produce large data sets from spatiotemporal video data Compared to the Gaussian beam method, it overcomes the as remotely sensed hyperspectral satellite data. Varia- drawback of beam spreading. We will present several nu- tional methods for image processing suited to complex dy- merical examples as well as preliminary applications in ge- namical systems typical of fluid dynamics, and the tools ology to verify the performance of the method. of dynamical systems such as transfer operators have not Xu Yang been brought to bear on data inferred directly from movies. Courant Institute of Mathematical Sciences We discuss modeling and transport analysis for remotely New York University sensed ecological systems such as biological products in- [email protected] cluding algae blooms. Erik Bollt Clarkson University [email protected] NW12 Abstracts 53

MS5 MS6 A Global Theory of Transport Barriers Aggregation and Growth of Clusters During a Thermal Quench We introduce a unified approach to locating transport bar- riers in general two-dimensional, non-autonomous dynam- The creation of clusters in the context of a quench is stud- ical systems. Using tools from continuum mechanics and ied. Two model problems are considered. First, under- differential geometry, we show that transport barriers can saturated vapor is cooled at a prescribed rate. After a be obtained as minimal geodesics under an appropriate time-lag, the nucleation rate rises exponentially. A short Riemannian metric induced by the Cauchy-Green strain burst of nucleation decreases the super-saturation and ends tensor field. As such, transport barriers can be directly the creation of new clusters. We find asymptotic descrip- computed (as opposed to just indirectly observed) as so- tions of the resulting distribution of cluster sizes and the lutions of classic Lagrangian or Hamiltonian equations of total amount of clusters generated. The time-lag is related motion. We show how these results reveal previously unde- to the quench rate by an implicit formula. Next, the flow tected transport barriers with mathematical rigour in sys- of a warm gas onto a cold wall is examined. The creation tems ranging from aperiodically forced models to satellite and growth of clusters in the boundary layer next to the observations of the ocean. cold wall are the natural extension of the first problem. We find the total amount of clusters generated, the concentra- George Haller tion of monomers in the gas and the size of the clusters as McGill University functions of the distance to the wall. We also determine Department of Mechanical Engineering the distance from the wall at which the nucleation hap- [email protected] pens, and the length of the “growth layer’, during which the growth of clusters brings the gas to equilibrium. MS5 Yossi Farjoun Lagrangian Coherent Structures and Eddy Diffu- Massachusets Institute of Technology sion [email protected]

Lagrangian coherent structures (LCS) are known as the templates for chaotic mixing in nonlinear aperiodic dynam- MS6 ical systems, such as geophysical flows. In recent years, Branching Processes and Coagulation: Asymptotic theoretical developments on the deterministic LCS have Self-similarity for a Generalized Smoluchowski allowed the objective identification of mixing barriers and Equation enhancers in geophysical flows. Stochastic transport as- sociated with LCS, on the other hand, is less studied, We investigate the asymptotic self-similarity of solutions to partly due to the inherent scale separation between coher- a new coagulation model arising in the theory of branch- ent structures and molecular diffusion. However, sub-grid ing processes. Informally, the equation models a coales- scale uncertainty of geophysical flows cannot be neglected cence process with multiple interactions, where the size when one tries to quantify diffusive transport of substances. and number of interacting clusters are sampled randomly. In this talk we will discuss some recent efforts on quantify- Under a suitable regular variation assumption on the sam- ing diffusive mixing associated with the LCS. In particular, pling measure, we characterize all nontrivial scaling limits eddy diffusivity tensors associated with advection-diffusion of solutions. Our results include, as a special case, the scal- are constructed based on Lagrangian measures from LCS. ing limits for the classical kernel K(x, y) = 2, previously Some archetypal examples will be discussed. studied by Menon and Pego, among others.

Wenbo Tang Nicholas Leger, Gautam Iyer, Robert Pego Arizona State University Carnegie Mellon University [email protected] [email protected], [email protected], [email protected] MS5 Topological Detection of Coherent Structures MS6 Coagulation Dynamics Driven by Uniform Growth In many applications, particularly in geophysics, we often have fluid trajectory data from floats, but little or no infor- We describe progress toward classifying initial data that mation about the underlying velocity field. The standard lead to self-similar growth in a coagulation process that techniques for finding transport barriers, based for example involves uniform growth of domains in one dimension. This on finite-time Lyapunov exponents, are then inapplicable. is joint work with Jack Carr (Heriot-Watt University). However, if there are invariant regions in the flow this will be reflected by a ‘bunching up’ of trajectories. We show Robert Pego that this can be detected by tools from topology. Carnegie Mellon University [email protected] Jean-Luc Thiffeault Dept. of Mathematics University of Wisconsin - Madison MS6 [email protected] Symmetry-breaking in Crystallisation

Michael Allshouse A significant stage in the formation of living systems Dept. of Mechanical Engineering was the transition from a symmetric chemistry involving Massachusetts Institute of Technology mirror-symmetric chiral species into a system involving just [email protected] one-handedness. We derive and analyse systems of cou- pled coagulation-fragmentation equations which describe crystal-grinding and exhibit symmetry-breaking, with the 54 NW12 Abstracts

aim of elucidating those mechanisms responsible for the bi- Kevin Zumbrun furcation. The talk will cover and hopefully extend work Indiana University published in Orig Life & Evol Biospheres, 41, 133–173. USA (2011). [email protected]

Jonathan Wattis School of Mathematical Sciences MS7 University of Nottingham Horseshoes and Hand Grenades: On Standing [email protected] Waves in a Gross-Pitaevskii Equation We look for standing waves in NLS-type equations with a MS6 periodic or N-well potential – a popular model for opti- Coagulation Equations with Nonhomogeneous Ker- cal propagation and Bose-Einstein condensation. We sim- nels plify a shooting strategy by detailing the construction of a horseshoe map. The geometry of the horseshoe varies Coagulation processes are to be found everywhere in na- with the nonlinearity (attracting, repulsive or competing), ture: from the coalescence of aerosols and polymers on yet in each case we instantly identify a huge assortment of the microscopic scale to the coalescence of water to form standing waves. Finally, we discuss how to quickly recover hail stones on the macroscopic scale to the formation of stability information encoded within the horseshoe. planets and stars on the cosmic scale. The earliest equa- tion to model coagulation processes was derived by Smolu- Russell Jackson chowski in 1916. The Smoluchowski coagulation equation, U. S. Naval Academy, Mathematics Department along with various extensions and generalizations, have Annapolis MD 21402-5002 been widely studied. While the general coagulation pro- [email protected] cess, namely the coming together of small particle clusters to form larger particle clusters, are common, the physics governing the coalescence of aerosols and the formation of MS7 planets is undoubtedly very different. These physical dif- Stability of Solitons Traveling on Vortex Filaments ferences are manifested in the particular coagulation kernel that is assumed in the coagulation equation. In their quest In this work, we develop a general framework for study- to understand symmetry in coagulation processes, mathe- ing the linear stability of soliton solutions of the vortex maticians and physicists have attempted to find self-similar filament equation (VFE), based on the correspondence be- solutions to Smoluchowski’s coagulation equation. When tween the VFE and the nonlinear Schrodinger (NLS) equa- looking for self-similar solutions, it is natural to consider tion provided by the Hasimoto map. In particular, we show homogeneous coagulation kernels. In this talk I hope to that the one-soliton solutions of the VFE are linearly sta- show that some interesting phenomena can be found by ble, which is in agreement with the numerical results ob- considering non-homogeneous coagulation kernels. tained by Kida in 1982. Furthermore, it is shown that a similar result hold for the planar VFE. In the planar case, Henry van Roessel the filament equation is related to the integrable modified University of Alberta Korteweg-de-Vries Equation. [email protected] Stephane Lafortune, Annalisa M. Calini College of Charleston MS7 Department of Mathematics Spectral Stability of Shock Layers in Compressible [email protected], [email protected] Fluid Flow We review the recent developments in our group’s study on MS7 the stability theory for high Mach number viscous shock Nonlinear Eigenvalues, Keldysh’s Theorem, and layers in multi-D compressible fluid flow, as well as deto- the Evans Function nation waves in reactive flow. In particular, we discuss the substantial challenges observed when computing the Evans We consider eigenvalue problems for nonlinear Fredholm function in Eulerian coordinates at high frequencies. We operator pencils, and show how to count and localize them then show how we can overcome these problems by trans- using contour integrals and an abstract Keldysh Theo- forming into canonical and somewhat general coordinates. rem. For pencils of differential operators, this allows one to The results are surprising. count the number of zeros of the Evans function by solving boundary value problems on finite intervals. Jeffrey Humpherys Brigham Young University Yuri Latushkin jeff[email protected] Department of Mathematics University of Missouri-Columbia Blake Barker [email protected] Indiana University BYU Wolf-Jergen Beyn, Jens Rottmann-Matthes [email protected] University of Bielefeld [email protected], Gregory Lyng [email protected] Department of Mathematics University of Wyoming MS7 [email protected] Fredholm Determinants and Computing the Sta- NW12 Abstracts 55

bility of Travelling Waves Schrodinger and of the Klein-Gordon type. In the former class of models, we will consider the existence, stability We are concerned with the computation of spectra of lin- and dynamics of standing waves and in the latter of time- ear operators; in particular with large scale and multi- periodic, exponentially localized discrete breather solutions dimensional non-selfadjoint spectral problems. We present for mostly local but also nonlocal models. a a general class of multi-dimensional shooting algorithms for computing on Grassmann manifolds, developed by Panayotis Kevrekidis Ledoux, Malham and Th¨ummler and Ledoux, Malham, UMass, Amherst Niesen and Th¨ummler, for this purpose. Further, Ledoux, [email protected] Malham and Marangell have revealed there is a relation be- tween singularities in the Grassmann to chart Riccati flow and isolated eigenvalues. This is because the Riccati flow MS8 generalizes the Weyl–Titchmarsh function in one spatial Dark Solitons and Soliton Complexes in Nonlocal dimension. In more than one spatial dimension, in terms Dissipative Systems of the infinite dimensional Fredholm (or Sato) Grassman- nian, the operator Riccati flow represents the Dirichlet to Abstract not available at time of publication. Neumann map. If there is time we will try to relate this work to that of Deng and Jones on the Morse/Maslov index Maxim Molchan for selfadjoint multi-dimensional elliptic operators and also University of Cape Town consider connections to Fredholm determinants by Borne- [email protected] mann and Karambal.

Simon Malham MS8 Heriot-Watt University The Cauchy Problem and Traveling Waves for a [email protected] Nonlocal Gross-Pitaevskii Equation We study the Gross-Pitaevskii equation involving a nonlo- MS8 cal interaction. Our aim is to give sufficient conditions that Loops of Energy Bands for Bloch Waves in Optical cover a variety of nonlocal interactions such that the asso- Lattices ciated Cauchy problem is globally well-posed with nonzero boundary condition at infinity. We focus on even poten- We will consider Stationary Bloch waves of Bose-Einstein tials that are positive definite or positive tempered distri- condensates in an optical lattice via the Gross-Pitaevskii butions. We also provide sufficient conditions such that equation. This talk will be devoted to the bifurcation of there exists a range of speeds in which nontrivial traveling stationary states which manifests itself as loops in the en- waves do not exist. ergy bands of the Bloch waves. In particular, the bifur- cation is shown to be a supercritical pitchfork bifurcation. AndredeLaire Analytical results will be illustrated by numerical compu- Laboratoire Jacques-Louis Lions tations. Universit´e Pierre et Marie Curie [email protected] Matt Coles University of British Columbia [email protected] MS9 The Reduced Ostrovsky Equation: Integrability and Breaking MS8 Stability of Solutions to a Non-local Gross- The reduced Ostrovsky equation is a modification of the Pitaevskii Equation with Applications to Bose- Korteweg-de Vries equation, in which the usual linear dis- Einstein Condensates persive term with a third-order derivative is replaced by a linear non-local integral term, which represents the effect of The Gross-Pitaevskii equation is a common model in background rotation. This equation is integrable provided physics, but it only takes local interactions into account. a certain curvature constraint is satisfied. We demonstrate, This paper demonstrates the validity of using a nonlo- mainly through numerical simulations, that when this cur- cal formulation as a generalization of the local model. A vature constraint is not satisfied at the initial time, then large class of nonlocalities and potentials is studied. We wave breaking inevitably occurs. then establish the orbital stability of a class of parameter- dependent solutions to the nonlocal problem. Numerical Roger Grimshaw results corroborate the analytical stability results. University of Loughborough, UK [email protected] Chris Curtis Department of Applied Mathematics Karl Helfrich University of Colorado at Boulder Woods Hole Oceanographic Institution [email protected] [email protected]

MS8 MS9 Existence, Stability and Dynamics of Solitary Transverse Spectral Stability of Periodic Waves Waves in Local and Nonlocal Discrete Nonlinear Schrodinger and Klein-Gordon Lattices The Kadomtsev-Petviashvili (KP) equation possesses a four-parameter family of one-dimensional periodic travel- In this talk, we will review a number of results obtained ing waves. We study the spectral stability of the waves with over the last few years in equations of the discrete nonlinear small amplitude with respect to two-dimensional perturba- 56 NW12 Abstracts

tions which are either periodic in the direction of propaga- atomic Bose-Einstein condensate, dark-bright solitons are tion, with the same period as the one-dimensional traveling studied in the presence of impurities. We use adiabatic wave, or non-periodic (localized or bounded). We focus on perturbation theory and show that, counter intuitively, an the so-called KP-I equation (positive dispersion case), for attractive (repulsive) delta-like impurity induces an effec- which we show that these periodic waves are unstable with tive localized barrier (well) in the effective potential felt respect to both types of perturbations. Finally, we briefly by the soliton; this way, dark-bright solitons are reflected discuss the KP-II equation, for which we show that these from (transmitted through) attractive (repulsive) impuri- periodic waves are spectrally stable with respect to pertur- ties. Analytical results for the small-amplitude oscillations bations which are periodic in the direction of propagation, are found to be in good agreement with results obtained and have long wavelengths in the transverse direction. via a Bogoliubov-de Gennes analysis and direct numerical simulations. Mariana Haragus Laboratoire de Mathematiques de Besancon Vassilios Rothos Universite de Franche-Comte, France School of Mathematics, Physics and Computer Sciences [email protected] Aristotle University of Thessaloniki [email protected]

MS9 Vassos Achilleos, Dimitri Frantzeskakis Generalized Multi-symplectic Integrators for a University of Athens Class of Hamiltonian Nonlinear Wave PDEs [email protected], [email protected] Focusing on the dissipative effect of a class of PDEs with small damping in the Hamiltonian setting, a new theoreti- Panayotis Kevrekidis cal framework named generalized multi-symplectic integra- UMass, Amherst tor is proposed, extending the concept of multi-symplectic Dept of Mathematics PDEs to the non-conservative setting. To test the idea, [email protected] several numerical experiments on the compound KdV- Burgers equation are carried out. The numerical results MS10 illustrate that the generalized multi-symplectic integrator is structure-preserving and can reproduce the dissipative Stabilizing 1D Solitons Against the Critical Col- effect of the non-conservative system. lapse by Quintic Nonlinear Lattices Weipeng Hu It has been recently discovered that stabilization of two- Northwestern Polytechnical University dimensional (2D) solitons against the critical collapse in China media with the cubic nonlinearity by means of nonlinear [email protected] lattices (NLs) is a challenging problem. We address the 1D version of the problem, i.e., the nonlinear Schr¨odinger equation with the quintic or cubic-quintic (CQ) terms, the Zichen Deng coefficient in front of which is periodically modulated in Northwestern Polytechnincal University space. Stability diagrams for the solitons are produced [email protected] by means of numerical methods and analytical approxi- mations. It is found that the sinusoidal NL stabilzes soli- MS9 tons supported by the quintic-only nonlinearity in a narrow stripe in the parameter plane, on the contrary to the case The Morse and Maslov Indices for Periodic Prob- of the cubic nonlinearity in 2D, where the stabilization of lems solitons by smooth spatial modulations is not possible at For Hill’s equations with matrix valued periodic potential, all. The stability region is much broader in the 1D CQ we discuss relations between the Morse index, counting model, where higher-order solitons may be stable too. Pub- the number of unstable eigenvalues, and the Maslov in- lication: J. Zeng and B. A. Malomed, Phys. Rev. A 85, dex, counting the number of signed intersections of a path 023824 (2012). in the space of Lagrangian planes with the train of a plane. Boris Malomed Chris Jones Department of Physical Electronics, Faculty of Engr. University of North Carolina Tel Aviv University, 69978 ISRAEL [email protected] [email protected]

Yuri Latushkin Jianhua Zeng Department of Mathematics State Key Laboratory of Low Dimensional Quantum University of Missouri-Columbia Physics [email protected] Department of Physics, Tsinghua University, Beijing, China [email protected] Robby Marangell University of Sydney [email protected] MS10 Linear and Nonlinear Properties of Strained Pho- MS9 tonic Crystals Statics and Dynamics of Atomic Dark-bright Soli- Perhaps the two most important properties of photonic tons in the Presence of Localized Impurities crystals are (1) its photonic band gap, which allows for strong light confinement and very efficient lasing, and (2) Adopting a mean-field description for a 2-component its ability to strongly increase the density-of-states, en- NW12 Abstracts 57

hancing nonlinear effects. Here, we show that an inho- stability, while the stability of the two bifurcated branches mogeneous strain in a periodic two-dimensional photonic is determined by the sign of their power slopes. Fourthly, crystal structure acts to induce an effective magnetic field we show that at a transcritical bifurcation point, both near the Brillouin zone corners. This field in turn induces solitary-wave branches switch stability. Lastly, we use nu- a complete photonic band gap and gives rise to Landau merical examples to illustrate these analytical results. levels: highly degenerate energy levels with correspond- ingly high density-of-states. These Landau levels lead to Jianke Yang enhancement of nonlinear effects (such as harmonic genera- Department of Mathematics and Statistics tion). We show the presence of both defect modes and edge University of Vermont modes within the band gap (which lies in between the Lan- [email protected] dau levels). Experiments are currently on their way, using a silicon-on-insulator photonic crystal slab system. MS10 Mikael Rechtsman Nonlinear Diffraction and Interband Transitions in Department of Physics Photonic Graphene Technion University [email protected] Nonlinear diffraction and interband transitions in two- dimensional honeycomb lattices, in nonlinear optics some- Alexander Szameit times termed photonic graphene, are studied analytically Friedrich-Schiller-Universit¨at Jena and numerically. The nonlinear diffraction of the Bloch [email protected] wave envelope associated with so-called Dirac points where two linear dispersion surfaces touch each other develops a triangular pattern during propagation. This phenomenon Mordechai Segev is markedly different from conical diffractions which oc- Technion, Israel curs in the leading order ‘pure’ Dirac system in which the [email protected] diffraction pattern evolves circularly (i.e., conically). The triangular structure is related to the higher order disper- MS10 sion relation of the homeycomb lattice which has trigonal warping behavior near the Dirac points. A higher-order Multi-site Breathers in Klein–Gordon Lattices: nonlinear Dirac equation is then derived and used to de- Stability, Resonances, and Bbifurcations scribe the dynamics of the Dirac wave envelopes. Asymp- We prove the most general theorem about spectral stability totic analysis of the linear equations demonstrates the tri- of multi-site breathers in the discrete Klein–Gordon equa- angular diffraction as well as its decay rate and agrees with tion with a small coupling constant. In the anti-continuum numerical simulations. Further, nonlinear analysis is em- limit, multi-site breathers represent excited oscillations at ployed to study the energy transitions between two nearby different sites of the lattice separated by a number of bands which touch at the Dirac point. The sign of the non- ”holes” (sites at rest). The theorem describes how the linearity, focusing or defocusing, determines the direction stability or instability of a multi-site breather depends on of the triangular pattern. This is due to the preference of the phase difference and distance between the excited oscil- energy transitions from one band to another. Analytical lators. Previously, only multi-site breathers with adjacent results agree well with direct numerical simulations. excited sites were considered within the first-order pertur- Yi Zhu bation theory. We show that the stability of multi-site Zhou-Pei-Yuan Center for Applied Mathematics breathers with one-site holes change for large-amplitude Tsinghua University, China oscillations in soft nonlinear potentials. We also discover [email protected] and study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site breathers in soft quartic potentials near the points of 1:3 resonance. Mark Ablowitz Dept. of Applied Mathematics Anton Sakovich University of Colorado McMaster University [email protected] [email protected] MS11 MS10 Periodic Travelling Waves in Dimer Granular Stability Analysis of Solitary Waves Near Bifurca- Chains tion Points in Generalized Nonlinear Schrodinger Equations Bifurcations of periodic travelling waves in granular dimer chains are studied near the anti-continuum limit, when the Stability properties of solitary waves near bifurcation mass ratio between the light and heavy beads is a small points in the generalized nonlinear Schrodinger equations parameter. We show that each limiting periodic wave is with arbitrary potentials are analyzed. First, conditions uniquely continued with respect to the mass ratio parame- for three major types of solitary-wave bifurcations, namely ter and the periodic waves with the wavelength larger than saddle-node bifurcations, pitchfork bifurcations and tran- a certain critical value are spectrally stable. We also prove scritical bifurcations, are derived. Second, we show that uniqueness of the continuation of periodic travelling waves at a saddle-node bifurcation point, the stability of solitary that exist in the homogeneous granular crystals with re- waves does not switch. In particular, both branches of spect to the mass ratio parameter. Numerical computa- saddle-node bifurcations can be stable. This is in stark con- tions are developed to study connections between these two trast with saddle-node bifurcations in finite-dimensional solution families, as well as bifurcations and instability on- dynamical systems, where stability of the two branches al- sets along each family. ways switches. Thirdly, we show that at a pitchfork bifur- cation point, the continuous solitary-wave branch switches Matthew Betti 58 NW12 Abstracts

McMaster University, Canada MS12 [email protected] Large Time-step and Asymptotic- preserving Schemes for Hyperbolic Systems with Dmitry Pelinovsky Sources and their Parabolic Limits McMaster University Department of Mathematics We propose a large time step and asymptotic preserving [email protected] scheme for the gas dynamics equations with external forces and friction terms. By asymptotic preserving, we mean that the numerical scheme is able to reproduce at the dis- MS11 crete level the parabolic-type asymptotic behaviour satis Highly Nonlinear Stress Waves in 2D Granular ed by the continuous equations. By large time-step, we Crystals mean that the scheme is stable under a CFL stability con- dition driven by the (slow) material waves, and not by the Highly ordered granular systems, i.e. granular crystals, (fast) acoustic waves as it is customary in Godunov-type present a unique nonlinear dynamic behavior stemming schemes. Nonlinear stability properties are proved and nu- from the Hertzian contact potential. We investigate the merical evidences are proposed. A gain of several orders of elastic stress wave propagation through uncompressed two- magnitude in both accuracy and efficiency is showed. dimensional granular crystals with variable packing geome- tries. Experimental results are in good agreement with Christophe Chalons discrete particle model simulations. The unique dynamic Universite Paris 7 properties of these crystals can be exploited to design gran- [email protected] ular systems with predetermined stress wave paths, which could be used as impact energy mitigating devices. MS12 Andrea Leonard Multipathing Problem in Seismic Imaging and a California Institute of Technology Numerical Solution of Escape Equations [email protected] Seismic reflection imaging using the so-called Kirchhoff mi- gration approach involves the task of computing seismic MS11 traveltimes between the image point inside a model of the Analytical Study of the Interaction of Solitary Earth and points on the surface where data are collected. Waves with Defects in Granular Crystals Because of the non-linear nature of traveltime computa- tions, multiple solutions are possible and lead to the multi- It is a well known fact that uncompressed, homogeneous pathing problem. The first-arrival solution is not the most granular chains support the propagating solitary wave solu- energetic and therefore is not the most desirable for imag- tion earlier discovered by Nesterenko. In the present work ing. A possible solution to the multipathing issue is an we develop a systematic analytical approach to describe the angle-domain formulation of seismic imaging, which leads interaction of Nesterenko solitary like pulses with various to the set of escape equations. Escape equations are de- local and nonlocal structural in-homogeneities such as de- coupled first-order linear equations defined in the extended fects in nonlinear stiffness, masses of particles, inter-chain phase space. I will present a numerical algorithm for an ef- and on-site potentials. The developed analytical model is ficient solution of escape equations and its application in in a fairly good correspondence with the numerical simu- seismic imaging. lations. Sergey Fomel, Vladimir Bashkardin Yuli Starosvetsky University of Texas at Austin Technion, Israel Institute of Technology [email protected], [email protected] [email protected] MS12 MS11 Semiclassical Models for the Schrodinger Equation Equipartition of Energy in Uncompressed Granular with Periodic Potentials and Band-crossings Crystals We study the linear Schrodinger equation with a periodic Granular crystals are unique nonlinear systems that exhibit potential in the semiclassical limit. When the so called interesting properties stemming from the nonlinear contact Bloch band gap is small, the inter-band transition is sig- interactions (Hertzian) between two individual particles. nificant but can not be described by the classical transport We study numerically and experimentally the dynamic ef- model. We derive a quantum classical Liouville system to fects of spherical interstitial particles placed between two capture the inter-band transition phenomena. This system adjacent uncompressed chains of larger particles. We excite can be seen as a first order approximation to the Wigner one of the large particle chains with an impact and show equation. A classical-quantum hybrid model and the corre- energy transfer and equipartition from the excited chain to sponding domain decomposition method are presented to the adjacent chain. Experimental data is collected using solve this type of system efficiently. tri-axial accelerometers, and is compared to the numerical simulations, finding very good agreement. Shi Jin Shanghai Jiao Tong University, China and the Ivan Szelengowicz University of Wisconsin-Madison California Institute of Technology [email protected] [email protected] MS12 High Frequency Approximation in Molecular NW12 Abstracts 59

Quantum Dynamics Active Media

The time dependent Schr¨odinger equation describing nu- Resonant interaction of light with a randomly-prepared, clear quantum motion generates high frequency oscillations lambda-configuration active optical medium is described with respect to time and space. Our results are based on by exact solutions of a completely-integrable, random par- the observation that in the high frequency regime transport tial differential equation, thus combining the opposing con- equations can either be used for approximating quadratic cepts of integrability and disorder. An optical pulse pass- quantities of the wave function as well as the wave function ing through such a material will switch randomly between itself by Gaussian coherent states. The corresponding two left- and right-hand circular polarizations. Exact proba- families of particle methods are applicable for high dimen- bility distributions of the electric-field envelope variables sional systems. describing the light polarization and their switching times will be presented . Caroline Lasser Technische Universit¨at M¨unchen Gregor Kovacic [email protected] Rensselaer Polytechnic Inst Dept of Mathematical Sciences [email protected] MS12 Error Estimates for Gaussian Beam Superpositions Ethan Akins University of California, Berkeley Gaussian beams are asymptotically valid high frequency [email protected] solutions to hyperbolic partial differential equations, con- centrated on a curve through the physical domain. Su- perpositions of Gaussian beams provide a powerful tool to Peter R. Kramer generate more general high frequency solutions. In contrast Rensselaer Polytechnic Institute to the standard geometrical optics, the Gaussian beam ap- Department of Mathematical Sciences proximation does not break down at caustics. In this talk [email protected] we discuss numerical methods based on Gaussian beam su- perpositions and show error estimates in terms of the small Ildar R. Gabitov wavelength. Department of Mathematics, University of Arizona [email protected] Olof Runborg KTH, Stockholm, Sweden [email protected] MS13 Monte-Carlo Simulations of a Stochastic Maxwell- Bloch System MS13 Solutions to Maxwell-Bloch Equations with Non- The resonate interaction of light with an optical media is vanishing Boundary Conditions studied in a Maxwell-Bloch system with two levels, the lower one being degenerate.Using Monte-Carlo simulations, We formulate the inverse scattering transform with non- we are able to study the statistics of soliton polarization zero boundary conditions at infinity for (i) the focusing switching in the randomly prepared material. For simple nonlinear Schrodinger equation (ii) the scalar Maxwell- cases, analytical solutions exist for this stochastic, yet in- Bloch equations. For both models we derive soliton so- tegrable, system.Computations allow further investigation lutions and discuss their behavior. of effects, such as those stemming from non-zero boundary conditions. Gino Biondini State University of New York at Buffalo Katherine Newhall Department of Mathematics Courant Institute of Mathematical Science biondini@buffalo.edu New York University [email protected] Gregor Kovacic Rensselaer Polytechnic Inst Dept of Mathematical Sciences MS13 [email protected] Inverse-Scattering Transform for the Coupled Cou- pled Maxwell-Bloch Equations with Inhomoge- neous Broadening MS13 Light Propagation in Metamaterials with Mixed This talk will address the solution of the initial value prob- Positive and Negative Refractive Index lem for the coupled Maxwell-Bloch equations with inhomo- geneous broadening for a three level system, with generic Abstract not available at time of publication. preparation of the medium. The Inverse Scattering Trans- form for this system was first formulated by Byrne, Gabitov Alexander O. Korotkevich and Kovacic (2003). In this talk we will focus on the (non- Dept. of Mathematics & Statistics, University of New linear) evolution of the scattering data, on the problem Mexico of determining the final polarization of the medium, and L.D. Landau Institute for Theoretical Physics RAS describing soliton interactions. [email protected] Barbara Prinari University of Colorado, Colorado Springs MS13 [email protected] Integrable/Stochastic Polarization Dynamics in 60 NW12 Abstracts

Mark J. Ablowitz MS14 Department of Applied Mathematics On Global Stability for Lifschitz-Sylozov-Wagner University of Colorado at Boulder Like Systems [email protected] This talk is concerned with the stability and asymptotic Sarbarish Chakravarty stability at large time of solutions to a system of equa- University of Colorado at Colorado Springs tions, which includes the Lifschitz-Slyozov-Wagner (LSW) [email protected] system in the case when the initial data has compact sup- port. The main result is a proof of weak global asymptotic stability for LSW like systems. Comparison to a quadratic MS14 model plays an important part in the proof of the main Coagulation-Fragmentation in Alternative Telom- theorem when the initial data is critical. The quadratic ere Length Maintenance model extends the linear model of Carr and Penrose.

We present a mathematical model of alternative telomere Joseph Conlon length maintenance based on details of biophysical inter- Department of Mathematics actions. The model opens up a couple of interesting math- The University of Michigan ematical problems such as the validity of a quasi-steady [email protected] state approximation and dynamic properties of discrete coagulation-fragmentation systems with kernels out of the Barbara Niethammer typically considered classes. We also identify and estimate Mathematical Institute key factors influencing the length distribution of telom- University of Oxford eric circles using numerical simulations for different yeast [email protected] species. Katarina Bodova MS14 Comenius University Bratislava On Self-similarity in Coagulation-annihilation Sys- [email protected] tems

Richard Kollar We study the existence of self-similar behaviour in Comenius University coagulation-annihilation systems with two-species clusters, Department of Applied Mathematics and Statistics extending recent work in [Lauren¸cot and van Roessel, J. [email protected] Phys. A: Math. Theor. 43 (2010) 455210] to systems with more general rate coefficients.

Lubomir Tomaska Fernando P. da Costa Department of Genetics Universidade Aberta Lisaboa Comenius University Bratislava [email protected] [email protected] Joao Pinto Jozef Nosek Instituto Superior Tecnico, TU Lisbon, Lisbon Department of Biochemistry [email protected] Comenius University Bratislava [email protected] Henry van Roessel University of Alberta, Edmonton, Canada MS14 [email protected] Convergence to Equilibrium for the Coagulation- fragmentation Equations with Detailed Balance Rafael Sasportes and a Finite Critical Mass Universidade Aberta, Lisboa [email protected] We study the speed of convergence to equilibrium of solu- tions to the coagulation-fragmentation (CF) equations un- der the assumption of detailed balance. We show that, for MS15 a subcritical mass, the linearized equations have a spectral Modelling Retinal Waves in Starburst Amacrine gap in the natural vector space suggested by the entropy. Cells Through estimates of moments of the nonlinear equations, we are able to deduce that subcritical solutions of the CF Retinal waves are an example of spontaneous activation equations converge to equilibrium exponentially fast when- in the developing central nervous system. This activity ever uniform-in-time exponential moment bounds are avail- occurs in developing neural circuits prior to visual stimu- able (such as the Becker-Dring case). lus. The waves are the result of neighboring retinal cells spiking in a coordinated fashion which spreads across the Jose A. Ca˜nizo whole retina. We develop a continuous spatial and tempo- DPMMS, University of Cambridge ral model of this phenomenon in order to understand how [email protected] the wave properties depends on underlying parameters. We use the Fitzhugh-Nagumo model of neuron dynamics and Bertrand Lods include spatial coupling via a neurotransmitter field repre- Dipartimento di Statistica e Matematica Applicata senting a novel mechanism for generating spatiotemporal Universit`a degli Studi di Torino patterns in the developing central nervous system. [email protected] Ben Lansdell University of Washington NW12 Abstracts 61

[email protected] MS16 Emergence of Unsteady Dark Solitary Waves from Large-amplitude Periodic Patterns MS15 Spike Metric Analysis of Geometrical Effects in Ax- DSWs are most well known as exact solutions of the NLS onal Computation equation. Here it is shown that DSWs can be generated by colliding two periodic patterns. A KdV equation on a pe- Axons are much more than reliable cables in which spikes riodic background coupled to a phase shift equation is de- propagate in a stable manner. Specifically, propagation rived. The nonlinearity in the KdV equation is determined failures and reflection of pulses were observed experimen- by the curvature of the periodic states, and the dispersion tally in several neural systems. These unusual computa- is determined by the Krein signature of the periodic state. tions can be explained in classical nerve equations as a KdV planforms are also discussed. result of the interplay between the ionic currents with ge- ometrical properties of the axon. Such effects can dramat- Tom J. Bridges ically affect spike trains and consequently, neural coding. University of Surrey We compare trains before and after they are affected by [email protected] axonal inhomogeneities using a recent spike metric frame- work. MS16 Pedro Maia Weakly Nonlinear Solution of Initial Value Prob- University of Washington lem for Boussinesq-type Equations [email protected] We construct a weakly nonlinear solution of the initial- value problem for a system of coupled regularised Boussi- MS15 nesq equations on the infinite line, in terms of solutions of Disrupting Waves of Disease Spread Through a various Ostrovsky-type equations. The solution for a sin- Population: How Interventions Affect the Propa- gle Boussinesq equation is given in terms of solutions of gation of Disease Outbreak the Cauchy problems for two Korteweg-de Vries equations. We also perform relevant numerical simulations to test our Every year, the population of the world faces a number of formula. terrible diseases that spread through populations wreak- ing havoc on families and world economies. Only one dis- Kieron Moore ease has been successfully eradicated: smallpox in 1977. Loughborough University This event was a great triumph for the world and public UK health, but many more diseases still haunt humanity. The [email protected] workpresentedinthistalkwillbefocusedonhowcurrent campaign tools (routine vaccinations, behavioral changes, Karima Khusnutdinova and mass vaccination events) affect mathematical models Loughborough University, UK of disease spread through a population. The goal of this [email protected] research is to understand how to effectively and efficiently disrupt disease propagation. MS16 Joshua Proctor Whitham Modulation Equations for Korteweg-de Intellectual Ventures Vries/Kuramoto-Sivashinsky Equations

In this talk, we study the stability of periodic waves for a MS15 Korteweg de Vries/Kuramoto Sivaskinsky equation which Competing Spatio-temporal Neural Codes Evoked model a weakly unstable thin film flow down an incline. by Inhibition in the Olfactory System We will focus on the validity of Whitham’s modulation equations that describe large scale perturbations of peri- Experiments across different species have shown that per- odic waves. Of particular interest is the KdV limit where ception of odors in the olfactory system is associated with a new Whitham’s system is derived which provides new neural encoding patterns. We show that a data-driven stability criteria. computational model reduction for the antennal (olfactory) lobe (AL) of the Manduca sexta moth reveals the nature Pascal Noble, Luis Miguel Rodrigues of experimentally observed persistent spatial and tempo- University of Lyon ral neural encoding patterns and its associated decision- [email protected], [email protected] making dynamics. Utilizing the experimental data we re- duce a high-dimensional neural network model of the AL to a decision making model. Analyzing the model we conclude MS16 that the mechanism responsible for the robust and persis- Stability of Homoclinic Orbits of the Nonlinear tent appearance of neural codes is a stable fixed point. The Schrodinger Equation model is used to explain, predict and direct experiments when odors are mixed or the structure of the network is We study the linearization of the Nonlinear Schr¨odinger altered. (NLS) equation about homoclinic orbits of unstable plane wave solutions with two unstable modes. The family of Eli Shlizerman homoclinic orbits as well as a complete set of solutions of University of Washington, Seattle the associated linearized NLS equation can be constructed [email protected] using B¨acklund transformations. We show that iterating B¨acklund transformations saturates instabilities of the seed solution, making the largest dimensional homoclinic orbits 62 NW12 Abstracts

the most stable in the sense of linear stability. the temporal dynamics of self-similar scaling length L(t). We combine perturbation approach to the logarithmic cor- Constance Schober rections with the simulations using a spatially adaptive fi- Dept. of Mathematics nite difference method (a variation of adaptive mesh refine- University of Central Florida ment). The numerical method relies on self-similar prop- [email protected] erties of blowing-up solution.

Pavel M. Lushnikov MS16 Department of Mathematics and Statistics A Dimension-breaking Phenomenon for Steady University of New Mexico Water Waves with Weak Surface Tension [email protected]

Iooss and Kirchg¨assner proved that the two-dimensional Sergey Dyachenko water wave problem with weak surface tension admits two University of New Mexico families of solitary wave solutions of envelope form. The [email protected] solutions are to leading order described by the nonlinear Schr¨odinger equation. In this talk I will discuss how these waves generate families of three-dimensional periodically Natalia Vladimirova modulated solitary waves in a dimension-breaking bifurca- Department of Mathematics and Statistics, tion. The new solutions decay in the direction of propa- University of New Mexico gation and are periodic in the transverse direction. They [email protected] are related to the Davey-Stewartson equation. The proof is based on a reversible Hamiltonian spatial-dynamics formu- MS17 lation and an infinite-dimensional version of the Lyapunov centre theorem. Continuations of NLS Solutions Beyond the Singu- larity Erik Wahlen Lund University The nonlinear Schrodinger equation (NLS) is one of the Sweden canonical nonlinear equations in physics. In 1965, Kelley [email protected] showed that the NLS admits solutions that collapse (be- come singular) at a finite time (distance). Since physical quantities do not become singular, a question which has Mark D. Groves been open since 1965 is whether and how singular NLS so- Universit¨at des Saarlandes lutions can be continued beyond the singularity. A similar [email protected] situation occurs in hyperbolic conservation laws, where in the absence of viscosity, the solution can become singular Shu-Ming Sun (develop shocks). In that case, there is a huge body of Virginia Tech literature on how to continue the inviscid solution beyond Department of Mathematics the singularity. In contrast, very few studies addressed [email protected] this question in the NLS. In this talk I will present several potential continuations of the NLS beyond the singular- ity. These continuations share the universal feature that MS17 after the singularity, the solution is only determined up to Nonlinear Optics in Periodic, PT-symmetric Sys- multiplication by a constant phase term. As a result, the tems interaction between two post-collapse components (beams) is chaotic, as indeed has been observed recently in experi- In this talk I will present novel dynamics in nonlinear op- ments with high-power laser beams. tical systems with periodic and (P)arity (T)ime symmetric index of refraction. PT symmetry means a complex in- Gadi Fibich dex of refraction whose real part is an even function and Tel Aviv University whose imaginary part is odd. we will show how such sym- School of Mathematical Sciences metry adds functionality in recently proposed all optical fi[email protected] devices; for example a PT symmetric couplers produce in- put/output conditions characteristic of an all optical diode. Moran Klein Tel Aviv University Alejandro Aceves [email protected] Southern Methodist University [email protected] MS17 A Stochastic Model for Bose-Einstein Condensa- MS17 tion Logarithmic Corrections in Formation of Singular- We consider a nonlinear Schr¨odinger equation with a po- ities in 2D Radially Symmetric Reduced Keller- tential varying randomly in time, which is a model equa- Segel Model tion in Bose-Einstein condensation or in fiber optics. We The 2D Keller-Segel model (KS) can be used as a model de- study the Cauchy problem in the energy space, establish- scribing phenomena such as dynamics of bacteria in Petri ing random Strichartz estimate. We also give an initial dish and gravitational collapse of interstellar dust cloud. condition for the existence of collapsing state, and discuss We study radially symmetric solutions of 2D KS which the influence of the random potential on a stable standing blows-up in finite time. We simulate a collapsing solution wave. This is a joint work with Anne de Bouard (Ecole with high accuracy to resolve logarithmic corrections in NW12 Abstracts 63

Polytechnique, France). whoi.edu [email protected] Reika Fukuizumi Tohoku university [email protected] MS19 Hydrodynamic Solitons and Vortices in Polariton Condensates MS17 Waves in Microstructures Exciton-polaritons are composite bosons arising from the strong coupling between quantum well excitons and pho- Abstract not available at time of publication. tons confined in a semiconductor microcavity. They can be easily created by optical excitation, and they can form ex- Michael I. Weinstein tended bosonic condensates at relatively high temperatures Columbia University (10-300 K). Here we will show experiments on the transi- Dept of Applied Physics & Applied Math tion from a superfluid phase to shockwaves, vortex and [email protected] soliton formation when the condensate encounters a po- tential barrier in its flowpath at different Mach numbers, MS18 with specific features arising from the out-of-equilibrium nature of polaritons. Reduced Models of Stratified Internal Waves Alberto Amo Abstract not available at time of publication. Laboratoire de Photonique et de Nanostructures CNRS Roberto Camassa Marcoussis, France University of North Carolina [email protected] [email protected] MS19 MS18 Interactions and Asymptotics of Dispersive Shock Asymptotic Models for Large Amplitude Internal Waves Waves in Weakly Stratified Fluids The interaction of dispersive shock (DSWs) and rarefaction We derive nonlinear evolution equations for large ampli- waves (RWs) associated with the Korteweg–de Vries (KdV) tude internal waves in weakly stratified fluids. After the equation are investigated numerically and with Whitham’s linear dispersion relation of the new model is compared averaging method; the interaction of DSWs lead to multi- with that of the linearized Euler equations, the solitary phase dynamics. General non-vanishing initial conditions wave and conjugate state solutions of the model are ob- for the KdV equation are considered. KdV is solved for us- tained and compared with other theoretical solutions and ing the inverse scattering transform (IST) method. From field data. IST theory and matched asymptotics, an asymptotic solu- tion for large time is determined and it’s found that inter- Wooyoung Choi acting DSWs eventually form a single DSW. Dept of Mathematics New Jersey Institute of Tech Douglas Baldwin [email protected] Department of Applied Mathematics University of Colorado at Boulder [email protected] MS18 Internal Waves in the Ocean Seen from Spectral MS19 Perspective From Superfluid Counterflow to Novel Types of Abstract not available at time of publication. Solitons: Quantum Hydrodynamics with Dilute- gas Bose-Einstein Condensates Yuri V. Lvov Rensselaer Polytechnic Institute Dilute-gas Bose-Einstein condensates provide a versatile [email protected] tool for the investigation of superfluid hydrodynamics. We experimentally investigate the behavior of a two- component Bose-Einstein Condensate subjected to coun- MS18 terflow between the two components. The counterflow is On Stochastic Closures for Finite Amplitude Inter- found to lead to rich dynamics, including a modulational nal Waves instability, dark-bright soliton trains, and oscillating dark- dark solitons. Our recent and ongoing studies showcase Recent results concerning the analytic representation of the emergence of intriguing nonlinear behavior in a well stochastic closures for weakly interacting internal waves controlled model system using ultracold atoms. at finite amplitude will be presented and contrasted with existing analytic closures in the resonant (infinitesimal am- Peter Engels plitude) limit. The two paradigms will be studied using Washington State University ray tracing techniques that support the analytic results. [email protected] The ray tracing results further provide a concrete physi- cal interpretation for a transition from a system of coupled oscillators (the resonant limit) to a particle in a potential MS19 well (the finite amplitude limit). Dispersive Dam Breaks and Lock Exchanges in a Kurt Polzin 64 NW12 Abstracts

Two-layer Fluid University of Oxford UK Abstract not available at time of publication. [email protected] Gavin Esler UCL MS20 [email protected] On the Elliptic Sine-Gordon Equation

In this talk I will outline the problems that arise when MS19 trying to solve this equation in a convex polygon for lin- Quantum Hydrodynamics and Turbulence in Bose- earisable boundary conditions. These problem arise from Einstein Condensates the lack of compatibility of such problems at the corners of the domain, that in turn introduces a lack of integrability Quantum turbulence (QT) is comprised of quantized vor- in the spectral data. I will outline a strategy for defining an tices that are definite topological defects arising from the alternative Riemann Hilbert problem whose data are inte- order parameter appearing in Bose-Einstein condensation. grable, and discuss the use of this alternative RH problem Hence QT is expected to yield a simpler model of turbu- to solve some linearisable problems explicitly. This is joint lence than does conventional classical turbulence. A gen- work with T Fokas and J Lenells. eral introduction to this issue and a brief review of the basic concepts are followed by the recent developments in quan- Beatrice Pelloni tum hydrodynamic instability and turbulence in atomic University of Reading, UK Bose-Einstein condensates. [email protected] Makoto Tsubota Department of Physics, Osaka City University Athanassios Fokas Japan University of Cambridge [email protected] [email protected] Jonatan Lenells MS20 Department of Mathematics Asymptotics for a Fredholm Determinant Involving Baylor University the Second Painleve Transcendent jonatan [email protected] This talk discusses the large s-asymptotics of the deter- minant det(I − αKs) of an integrable Fredholm operator MS20 acting on the interval (−s, s) with a real-valued parameter Automatic Contour Deformation α. The kernel of the operator Ks is constructed out of the Ψ-function associated with the Hastings-McLeod solution In many cases the standard contour of a Riemann Hilbert of the Painleve II equation. This kernel appears in Ran- problem is not the most useful one for numerical analysis. dom Matrix Theory and describes the critical behavior of For example to calculate the solution of a Riemann Hilbert the eigenvalue gap probablities of a random Hermitian ma- problem it is often necessary to deform the contour in order trix chosen from the Unitary Ensemble in the bulk double to reduce the condition of the problem. Currently these scaling limit near a quadratic zero of the limiting mean deformations are done by hand and this talk will give some eigenvalue density. The given integrable form of the Fred- details on how some of these deformations can also be done holm operator allows us to connect the resolvent kernel to by an algorithm. the solution of a Riemann-Hilbert problem which can be analysed rigorously via the Deift-Zhou nonlinear steepest Georg Wechslberger descent method. We will highlight certain technical fea- Mathematics Department tures in the implementation of the method related to dif- Technical University Munich ferent choices of the parameter α.Thisisjointworkwith [email protected] Alexander Its.

Thomas J. Bothner MS21 Indiana University and Purdue University Resonant Dynamics of Fourier Modes [email protected] We analyze the resonant transfer of energy between Fourier modes for the 1D nonlinear Schr¨odinger equation in a finite MS20 interval with homogeneous Dirichlet or Neumann boundary Toeplitz-like Structure of RiemannHilbert Prob- conditions. For the cubic nonlinearity we show that there lems is no long term energy exchange between Fourier modes as opposed to higher nonlinearities. This slow dynamics Associated with a (matrix-valued) Riemann–Hilbert prob- is explained by simple amplitude equations. This method lem is a linear singular integral equation involving the can be applied to other systems to reveal their long term Cauchy transform and its left and right limits. By rep- dynamics. resenting this linear equation as an operator on Chebyshev coefficients, we find that it has the structure of a pertur- Jean-Guy Caputo bation of a block-Toeplitz operator. In suitably defined Universite Rouen spaces, this perturbation is compact. This allows us to France prove convergence and stability of a numerical method for [email protected] solving Riemann–Hilbert problems. Sheehan Olver NW12 Abstracts 65

MS21 Index (BFI) is a useful parameter to compute the proba- Singular Structure of Wave Modes in Rotating bility of occurrence of rogue waves in a wave field. This Shallow Water index depends on the dispersive and nonlinear coefficients of the NLS equation. We have shown that a shear cur- Abstract not available at time of publication. rent co-flowing with the waves increases the value of the BFI and so the number of steep wave events is expected to David J. Muraki increase. Department of Mathematics Simon Fraser University Christian Kharif, Roland Thomas [email protected] Institut de Recherche sur les Phenomenes Hors Equilibre [email protected], [email protected] MS21 Wave Train Defocusing over a Highly Disordered MS23 Bathymetry Numerical and Experimental Investigation for Spectral Evolution of Nonlinear Ocean Surface Different water wave regimes lead to different types of Waves and their Stability wave-topography interactions. For example when the bot- tom topography is considered to be random and rapidly Efficient and accurate computation of evolving nonlinear varying, pulse shaped waves are attenuated through this ocean surface waves is a challenging hydrodynamic prob- interaction. In some cases the attenuation is asymptot- lem. Resolution of various spatial and temporal scales and ically captured in a linear diffusive-like fashion. In the preservation of required coherent structures, is at heart of present work wave trains are considered. Then a multiple an efficient asymptotic model. In this study we examine scale, amplitude modulation analysis leads to a Nonlinear both short and long term spectral evolution of nonlinear Schrodinger equation. It is shown that large bottom varia- ocean surface waves, in the context of a third-order phase- tions can modify the effective nonlinearity coefficient thus resolving asymptotic model. An array of randomly initial- leading to a defocusing effect. ized Monte-Carlo simulations, solved via Fourier pseudo- spectral method, and corresponding stability analysis of Andre Nachbin various models in question will be presented. Furthermore, IMPA,Rio,Brazil to give further validity to several new findings, a prelimi- [email protected] nary set of laboratory/wave-tank experiments will also be presented. In particular, the downshifting of the spectral Ana Maria Luz peak and a quasi steady-state of the ensemble averaged Math. Dept., UFF, spectra, in the absence of damping and wave-breaking have Niteroi, Brazil. been observed in both the experiments and numerical sim- [email protected] ulations. Matt Malej MS21 NJIT Spontaneous Breaking of the Spatial Homogeneity [email protected] Symmetry in Wave Turbulence and the Emergence of Coherent Structures Wooyoung Choi Dept of Mathematics Spatial homogeneity, the symmetry property that all sta- New Jersey Institute of Tech tistical moments are functions only of the relative geometry [email protected] of any configuration of points, can be spontaneously broken by the instability of the finite flux Kolmogorov-Zakharov spectrum in certain (usually one dimensional) situations. MS23 As a result, the nature of the statistical attractor changes Time Dependent Hydro-Elastic Waves dramatically, from a sea of resonantly interacting disper- sive waves to an ensemble of coherent radiating pulses. The problem of forced unsteady water waves under an elas- tic sheet is a model for waves under ice or under large float- Benno Rumpf ing structures. Even though small amplitude solitary waves Southern Methodist University are not predicted to exist by standard perturbation anal- [email protected] yses, we find large amplitude solitary waves, and explore their crucial role in the forced problem of a moving load on the surface. This is meant to represent a model of the MS23 use of extended ice sheets as roads and aircraft runways. Modulational Instability of Weakly Nonlinear Gravity Waves on a Current of Uniform Vorticity Paul A. Milewski in Arbitrary Depth Dept. of Mathematics Univ. of Wisconsin A nonlinear Schrodinger equation in one dimension is de- [email protected] rived from the fully nonlinear water equations in arbitrary depth in the presence of uniform shear current of constant vorticity by using a multiple-scale method. A stability MS23 analysis demonstrates that Stokes waves counter propagat- Stability of Traveling Wave Solutions to Euler’s ing over a shear current are generally stable under given Equations conditions of the value of the vorticity for any value of the dispersive parameter kh, where k and h are the carrier Euler’s equations describe the dynamics of gravity waves wavenumber and depth respectively. The Benjamin-Feir on the surface of an ideal fluid with arbitrary depth. In this 66 NW12 Abstracts

talk, we discuss the stability of periodic traveling wave so- a new pump-free configuration, which give rise to a much lutions for the full set of Euler’s equations with constant easy to implement promising device. vorticity via a generalization of a non-local formulation of the water wave problem due to Ablowitz, Fokas & Mus- Julien Fatome, Stephane Pitois, Philippe Morin slimani [Ablowitz et al 2006, J. Fluid Mech., Ashton & University of Bourgogne Fokas 2011, arxiv.org]. We determine the spectral stabil- [email protected], [email protected], ity for the periodic traveling wave solution by extending [email protected] Fourier-Floquet analysis to apply to the non-local problem. We will discuss some interesting and new relationships be- tween the stability of the traveling wave with respect to MS24 long-wave perturbations and the structure of the bifurca- Theory of Polarization Attraction in Randomly tion curve for small amplitude solutions. Birefringent Nonlinear Optical Fibers Katie Oliveras We derive the coupled stochastic nonlinear equations de- Seattle University scribing the interaction among different optical waves in Mathematics Department fibers with randomly varying birefringence. These equa- [email protected] tions describe polarization attraction in conservative loss- less polarizers based on cross-polarization modulation, as Vishal Vasan well as dissipative polarization attraction induced by either Department of Applied Mathematics stimulated Raman scattering or parametric four-wave mix- University of Washington ing. We present extensive numerical simulations that link [email protected] the stochastic properties of the fiber, such as polarization mode-dispersion, to the repolarization capabilities of the different fiber structures. MS23 Stefan Wabnitz On the Evolution of Ocean Swell Over Long Dis- University of Burgundy tances [email protected] Segur et al. (2005) showed mathematically that even weak dissipation can stabilize Benjamin-Feir instability. Their Victor V. Kozlov, Massimiliano Guasoni experiments, on deep-water waves in a wave tank, sup- University of Brescia ported this claim for small perturbations; for larger per- [email protected], [email protected] turbations, they observed frequency downshifting, which is not explained by any current model. This talk summa- rizes our work to apply these ideas to ocean swell, and to MS24 construct an accurate model of downshifting. Hamiltonian Relaxation Phenomena and Applica- tions to Nonlinear Optics Harvey Segur Department of Applied Mathematics In this talk, we provide a generalized understanding of the University of Colorado phenomenon of polarization attraction. More precisely, we [email protected] analyze the polarization dynamics of counterpropagating waves in two non standard optical fiber systems, i.e. the spun fiber and the randomly birefringent fiber, the latter MS24 being relevant to optical telecommunication systems. Our Four-wave-mixing and Optical Wave Turbulence in theoretical analysis is based on recently developed mathe- Fiber Lasers matical and geometrical techniques, such as the concept of singular torus. Experimental examples will be also given. An overview of research in optical wave turbulence in fiber lasers will be presented in this talk. Four-wave mixing A. Picozzi induced wave turbulence of a number of weakly interacting Universit´e de Bourgogne longitudinal modes generated in nonlinear and dispersive [email protected] cavity defines basis properties of fiber lasers. Our recent results on coherent structures generation in fiber lasers will Elie Ass´emat, Dominique Sugny, Hans Jauslin be presented. Laboratoire Interdisciplinaire Carnot de Bourgogne [email protected], Dmitry Churkin [email protected], [email protected] Aston University [email protected] MS24 Collapses in Optical Turbulence MS24 All-optical Control of Polarization State in Optical We consider turbulence in the framework of Nonlinear Fibers for Telecommunication Applications Schrodinger Equation with focusing nonlinearity, dissi- pation, and forcing. Dissipation saturates catastrophic In this work, we report last developments in all-optical non- growth of collapses, causing their break down into almost linear control of polarization state in optical fibers based linear waves. These waves form a nearly-Gaussian random on a counter-propagating four-wave mixing interaction. In field which seeds new collapses. We analyze statistics of the particular, we will show that it is possible, using a unique amplitude fluctuations in the turbulent filed and connect it segment of fiber, to all-optically manipulate both the po- to the evolution and structure of individual collapses and larization state and intensity profile of telecommunication signals. We will also report significant results dealing with NW12 Abstracts 67

the statistics of collapses in the field. MS25 The Effects of Rotation on Internal Solitary Waves Natalia Vladimirova Department of Mathematics and Statistics, In the weakly nonlinear long wave regime, internal solitary University of New Mexico waves are often modeled by the Korteweg-de Vries equa- [email protected] tion, which is well-known to support an exact solitary wave solution. However, when the effect of background rotation Pavel M. Lushnikov is taken into account, an additional term is needed and the Department of Mathematics and Statistics outcome is the Ostrovsky equation. The addition of this University of New Mexico term has the drastic effect of destroying the solitary wave [email protected] solution. Instead an initial solitary-like disturbance decays into radiating oscillatory waves, with the eventual forma- tion of a nonlinear envelope solitary wave, whose carrier MS25 wavenumber is determined by an extremum in the group On the Dynamics of Turbulence Near a Free Sur- velocity. This process is addressed through a combination face of theoretical analyses, numerical simulations and labora- tory experiments. It is becoming increasingly clear that stably-stratified flows can support a stratified turbulence inertial range, differ- Karl Helfrich ent from Kolmogorov’s (e.g., Riley and Lindborg, J. At- Woods Hole Oceanographic Institution mos. Sci., 2008). Stratification inhibits vertical motions [email protected] at larger scales, but the largescale, quasi-horizontal mo- tions produce a strong vertical shearing and small-scale Roger Grimshaw −5/3 instabilities. The result is a k horizontal spectrum University of Loughborough, UK for the horizontal velocities and density at scales larger [email protected] than the Ozmidov scale, the largest scale that can over- turn. For smaller scales, the classical Kolmogorov k−5/3 spectrum applies. Inspired by data taken near the water MS25 surface in a tidal river (Chickadel, Talke, Horner-Devine How Nonlinear are the Internal Tides in the Luzon and Jessup, IEEE Geos. Remote Sens. Ltrs., 2011), we Strait? explore here to what extent the dynamics of the nonlinear spectral energy transfer of near-surface turbulence with no In recent years, the Luzon Strait has been the focus of mean shear (i.e., turbulence bounded by a free-slip sur- extensive field studies, remote observations and numeri- face) is analogous to stablystratified turbulence. To that cal simulations to gain more insights on the propagation end we perform DNS of decaying, non-sheared turbulence and generation mechanism(s) of some of the largest inter- with Reλ ∼ 100, but bounded by a free-slip surface. We nal solitary waves observed worldwide. To complement the find that, indeed, the behavior of the flow near the free- previous studies, we performed a laboratory experiment at slip surface is similar to stratified turbulence, with a tenta- the Coriolis facility in Grenoble, France, site of the world tive k−5/3 range. Recent field measurements by Thomson largest rotating table. We modeled the generation of in- and Polagye (private communication, 2011) and by Dugan ternal tides using realistic three-dimensional topography, and Piotrowski (JGR, 2012) indicate a similar behavior for realistic density stratification and barotropic tidal forcing. larger-scale quasi-horizontal flows limited by depth. We Particular care was taken to achieve dynamical similarity propose that the mechanism exhibited by strong stratified with the ocean problem. We present here the influence flows is a more general feature of turbulent flows in which of the tidal characteristics (amplitude, frequency) and of one component of the velocity has been strongly inhibited. the background rotation on the nonlinear aspects of the internal tide generation. Oscar Flores Universidad Carlos III Matthieu Mercier Massachusetts Institute of Technology [email protected] James Riley University of Washington Louis Gostiaux Laboratoire des Ecoulements Geophysique et Industriels, MS25 CNRS Transient Interfacial Waves - Nonlinear Calcula- [email protected] tions in Three Dimensions Sasan Saidi We investigate nonlinear internal wave generation by tidal Massachusetts Institute of Technology flow over realistic topography which is three-dimensional, [email protected] where few or no calculations exist, particularly for highly nonlinear wave amplitudes. Method is fully dispersive and Joel Sommeria, Samuel Viboud fully nonlinear. Energy transfer processes in the ocean are LEGI/Coriolis investigated as well as propagation in two horizontal direc- INPG-UJF-CNRS, Grenoble tions. Successive Fourier transform is extensively used for [email protected], rapid calculations. [email protected] John Grue University of Oslo Henri Didelle [email protected] Laboratoire des Ecoulements Geophysique et Industriels CNRS [email protected] 68 NW12 Abstracts

Karl Helfrich [email protected] Woods Hole Oceanographic Institution [email protected] Mark A. Hoefer North Carolina State University Thierry Dauxois [email protected] Ecole Normale Sup´erieure [email protected] MS26 Thomas Peacock The Semiclassical Modified Nonlinear Schrdinger Massachusetts Institute of Technology Equation [email protected] The modified nonlinear Schr¨odinger (MNLS) equation is a completely integrable system that appears to be a pertur- MS25 bation of the focusing nonlinear Schr¨odinger (NLS) equa- Generation and Propagation of Internal Waves in tion. However, the perturbation is singular and it turns a Model of the Deep Ocean out that one of its effects is that for certain initial data the problem behaves more like a perturbed defocusing NLS We present studies of internal gravity waves in a stratified equation than a perturbed focusing NLS equation. This fluid designed to model the deep ocean. King et al. recently effect is particularly dramatic in the semiclassical limit, in found that stratification in regions of the deep ocean be- which it can be seen that the modulational instability of low (previously unknown) turning depths is too weak to the unperturbed problem completely disappears in the per- support tidally generated internal waves. The present ex- turbed problem for certain initial conditions. This is joint periments and simulations examine internal wave reflection work with Jeffery DiFranco and Benson Muite. at turning depths. Further, we study internal wave gener- ation by tidal flow over bottom topography that is below Peter D. Miller a turning depth. University of Michigan, Ann Arbor [email protected] Matthew S. Paoletti, Harry Swinney University of Texas at Austin Center for Nonlinear Dynamics MS26 [email protected], [email protected] Thermodynamic Phase Transitions and Shock Sin- gularities

MS26 We show that, under rather general assumptions on the Dispersive Shock Wave Propagation in Weakly form of the entropy function, the energy balance equation Non-Uniform Media for a system in thermodynamic equilibrium is equivalent to a set of nonlinear equations of hydrodynamic type. This We consider the propagation of dispersive shock waves set of equations is integrable via the method of character- (DSWs) through weakly non-uniform media in the frame- istics and it provides the equation of state for the gas. The work of the KdV equation with a slowly varying disper- shock wave catastrophe set identifies the phase transition. sion coefficient. We show that the interaction of DSWs A family of explicitly solvable models of non-hydrodynamic with variable environments can result in a number of non- type such as the classical plasma and the ideal Bose gas is local and non-adiabatic responses including the genera- also discussed. tion of multi-phase regions and expanding soliton trains. In particular, our solutions describe the transformation of Antonio Moro shallow-water undular bores on a slope. SISSA-International School for Advanced Studies Trieste, Italy Gennady El [email protected] Loughborough University, United Kingdom [email protected] MS26 Semiclassical Dynamics and Rogue Waves Forma- MS26 tion in Quasi 1D Attractive BoseEinstein Conden- Dark Solitons, Dispersive Shock Waves and their sates: The Riemann-Hilbert Problem Approach Transverse Instabilities The strongly interacting regime for attractive BoseEin- Transverse instabilities of dark solitons for the (2+1)- stein condensates (BECs) tightly confined in an extended dimensional defocusing nonlinear Schrodinger / Gross- cylindrical trap is studied. For appropriately prepared, Pitaevskii equation are considered. Asymptotics and com- non-collapsing BECs, the ensuing dynamics are found to putations of the linearized equation yield the dispersion be governed by the one-dimensional focusing Nonlinear relation, which, in turn, yields the separatrix for the Schrdinger equation (NLS) in the semiclassical (small dis- transition between convective and absolute instabilities of persion) regime. In spite of the modulational instability of dark solitons. The implications for stationary and non- this regime, some mathematically rigorous results on the stationary oblique dispersive shock waves are elucidated. strong asymptotics of the semiclassical limiting solutions Our results have application to controlling nonlinear waves were obtained recently. Using these results, implosion- in dispersive media, such as dispersive shocks in Bose- like and explosion-like events are predicted whereby an Einstein condensates and nonlinear optics. initial hump focuses into a sharp spike which then ex- plodes into two rapidly oscillating radiative waves (rogue Boaz Ilan waves), whose initial amplitude is about 3 times the ampli- School of Natural Sciences tude of the background waves. Seemingly related behavior University of California, Merced has been observed in three-dimensional experiments and NW12 Abstracts 69

models, where a BEC with a sufficient number of atoms able via the numerical method of Sheehan Olver. This gives undergoes collapse. The dynamical regimes we consider, an efficient and spectrally accurate numerical method for however, are not predicted to undergo collapse. Instead, computing finite-genus solutions. distinct, ordered structures, appearing after the implosion, yield interesting new observables that may be experimen- Thomas D. Trogdon tally accessible. All the dynamical results follow from the Department of Applied Mathematics rigorous analysis of the Riemann-Hilbert problem that de- University of Washington scribes the inverse scattering transformation for the NLS. [email protected]

Bernard Deconinck Alexander Tovbis University of Washington University of Central Florida [email protected] Department of Mathematics [email protected] MS28 Wave Propagation in Anharmonic Discrete Sys- MS27 tems Painlev´e Functions and Critical Behavior in Non- linear Wave Equations We present some anharmonic nonlinear discrete models and study, numerically and asymptotically, the possibility The solution to a nonlinear wave equation associated to of propagation of coherent structures. In one of the cases a given initial condition will often display two or more we show the possibility of wave propagation of Toda’s solu- qualitatively different behaviors in different regions of tions type and in another case exponential cusp like travel- space-time, such as having an oscillatory zone and a non- ing wave solution is shown to exist for a cubic interaction. oscillatory zone. The boundaries between these regions These results are obtained in the fully discrete regime. may become well-defined in certain limits (such as small dispersion). In this case it is natural to consider the tran- Luis A Cisneros-Ake sition behavior between the two regions. In the past three ESFM-Instituto Politecnico Nacional years, it has been discovered that for several equations, [email protected] including the Korteweg–de Vries, nonlinear Schr¨odinger, and Camassa–Holm equations, that certain critical behav- ior can be described for wide classes of initial conditions MS28 in terms of Painlev´e functions. These functions, which are Stable, Conservative Solution Methods for Large, solutions of nonlinear ordinary differential equations, seem Stiff Hamiltonian Systems Modeling Coherent Phe- to play a role for nonlinear equations analagous to the role nomena in Nonlinear Optics and Biophysics played by the classical special functions for linear equa- tions. We will discuss recent work with P. Miller using the A variety of problems in biophysics and nonlinear optics Riemann–Hilbert approach on Painlev´e-type asymptotics lead to the need to solve large, stiff, mildly nonlinear sys- in solutions of the sine–Gordon equation, which in turn tems of ODEs that have Hamiltonian form, or perturba- has led to a better understanding of interesting behavior tions of such by slight dissipation or noise. This talk of the Painlev´e functions. presents a variant of the discrete gradient approach of Gon- zales and Simo, adapted to deal with stiffness in the com- Robert Buckingham mon scenario where this arises only from dominant linear University of Cincinnati terms, not the nonlinearities. Several applications will be [email protected] considered as time permits; in particular, some new co- herent phenomena in ODE systems of discrete nonlinear Schr¨odinger equation form relating to a novel continuum MS27 limit approximation, quite different from the usual NLS Large Size Riemann-Hilbert Problems in Random approximation. Matrix Theory Brenton J. LeMesurier We discuss some large size Riemann-Hilbert problems aris- College of Charleston ing from critical phenomena in random matrix theory and Department of Mathematics non-intersecting Brownian motion models. The size of the [email protected] Riemann-Hilbert problems is 3x3 or 4x4. The jump con- tour consists of 10 rays through the origin. MS28 Steven Delvaux Second Harmonic Generation in Negative Index Department of Computer Science Materials Katholieke Universiteit Leuven, Belgium [email protected] We present novel energy conversion from the fundamen- tal frequency to the second harmonic by use of negative index materials. One of the most fundamental properties MS27 of negative index materials is an opposite directionality Finite-Genus Solutions of Integrable Equations: A of the Poynting vector, thus energy transfer occurs while Numerical Riemann-Hilbert Approach the fields counterpropagate, instead of the traditional co- propagating format. By clever use of conservation laws we For many integrable equations, computing finite-genus so- can find input output solutions and show efficient energy lutions can be reduced to computing a certain hyperelliptic conversion even in the presence of phase mismacth Baker-Akhiezer function. We present a Riemann-Hilbert problem, on the plane, for these Baker-Akhiezer functions. Alejandro Aceves After deformation, this Riemann-Hilbert problem is solv- Southern Methodist University 70 NW12 Abstracts

[email protected] University of Massachusetts panayotis kevrekidis ¡[email protected]¿ Zhaxylyk Kudyshev Department of Physics, Al-Farabi Kazakh National Jesus Cuevas University, Almaty, Kazakhstan Departamento de Fisica Aplicada I [email protected] Escuela Politecnica Superior. Universidad de Sevilla jess cuevas maraver ¡[email protected]¿ Ildar Gabitov, Alyssa Pampell Department of Mathematics Southern Methodist University MS29 [email protected], [email protected] Nonlinear Frequency Conversion Through Driven Granular Media

MS28 The frequency spectrum of vibrational based sources is im- Instabilities of Breathers in a Discrete Nls portant in a variety of applications. We examine methods to control the spectra of these sources through driven gran- We review some numerical and analytical results on the ular crystals. The nonlinear Hertzian interaction allows continuation of breathers in the cubic discrete NLS equa- granular systems to be highly tunable and exhibit three dis- tion in finite one dimensional lattices. Breathers can be tinct regimes, linear, weakly nonlinear, and strongly non- viewed as fixed points in a reduced system and we use linear. By varying the precompression and length of the the stability properties of breathers to obtain information granular chain, we leverage nonlinear phenomena to shift on the topology of the energy hypersurface in a system of energy between modes of different frequencies. three sites. The change to a connected energy hypersurface corresponds to elliptic-hyperbolic breathers, and we study Joseph Lydon numerically Lyapunov periodic orbits, and their stable and California Institute of Technology unstable manifolds. We see evidence of homoclinic orbits [email protected] and we also discuss heteroclinic orbits and the question of transport of energy. MS29 Panayotis Panayotaros Dispersion Estimation for Weakly and Strongly Depto. Matematicas y Mecanica Nonlinear Periodic Systems IIMAS-UNAM [email protected] Techniques developed for the estimation of the disper- sion properties of nonlinear periodic media are presented. Weekly nonlinear systems are first analyzed through a MS28 perturbation approach implemented in a commercial FE Light Propagation in Two Dimensional Plasmonic tool. Next, strongly nonlinear one-dimensional and two- Arrays dimensional granular lattices are studied as examples of strongly nonlinear systems through a harmonic balance We present results on the dynamics of light beams propa- method. Both methodologies illustrate the amplitude- gating in two dimensional waveguides. We show how differ- dependent dispersion properties of nonlinear systems, and ent configurations provide a rich dynamics of localization, suggest their potentials as tunable waveguides. solitary wave formation and stability. Massimo Ruzzene Danhua Wang, Alejandro Aceves Georgia Institute of Technology Southern Methodist University [email protected] [email protected], [email protected] MS29 MS29 Resonances, Solitary Waves, and Passive Wave From Newton’s Cradle to the p-Schr¨odinger Equa- Redirection in Granular Media tion We discuss rich dynamics of ordered heterogeneous granu- We study localized waves in chains of oscillators coupled lar media with no precompression, including countable in- by Hertzian interactions, a problem originally motivated by finities of nonlinear resonances in dimer chains leading to the Newton’s cradle under the effect of gravity. We con- strong attenuation of propagating pulses; countable infini- sider an unusual setting where local oscillations and binary ties of nonlinear anti-resonances in the same dimers leading collisions occur on similar time scales. Using both direct to families of solitary waves; and capacity for passive wave numerical computations and an asymptotic model (the dis- redirection in weakly coupled homogeneous chains through crete p-Schr¨odinger equation), we obtain static and travel- Landau-Zenner tunneling in space. The nonlinear dynam- ing breathers with unusual properties, including enhanced ical mechanisms governing these phenomena are studied localization, almost vanishing Peierls-Nabarro barrier and leading to fully predictive material designs. direction-reversing motion. Alexander Vakakis Guillaume James Department of Applied Mathematical and Physical Laboratoire Jean Kuntzmann, Universit´e de Grenoble Sciences and CNRS National Technical University of Athens [email protected] [email protected]

Panayotis Kevrekidis Yuli Starosvetsky Department of Mathematics and Statistics University of Illinois at Urbana Champaign NW12 Abstracts 71

1416 Mechanical Engineering Laboratory MS30 [email protected] Traveling Water Waves in Three Dimensions

KR Jayaprakash, Md. Arif Hasan In this talk I shall present a new single equation for the Mechanical Science and Engineering surface elevation of a traveling water-wave in an incom- University of Illinois pressible, inviscid, irrotational fluid. This new equation is [email protected], [email protected] derived from the full set of Eulers Equations and is valid for both a one- and two- dimensional traveling wave surfaces, with and without surface tension. Some sample solutions, MS30 obtained through Stokes expansions as well as numerical Variational Methods for Solitary Water Waves computations, will also be presented.

I will speak on variational methods for solitary waves on Vishal Vasan the free surface of a two-dimensional steady, irrotational Department of Applied Mathematics flow of water, acted upon by gravity. I will begin by formu- University of Washington lating the steady water wave problem as a nonlinear pseu- [email protected] dodifferential equation via conformal mappings and com- pare to Babenko’s equation for Stokes waves. I will argue Katie Oliveras non-existence in the infinite-depth case by Pohozaev iden- Seattle University tity techniques. After commenting how the Korteweg-de Mathematics Department Vries equation arises in the finite-depth case as the leading- [email protected] order approximation in a certain weakly-nonlinear long- wave regime, I will explain existence in the finite-depth case of solitary waves as minimizers. MS30 Global Continuation for Solitary Waves with Vor- Vera Mikyoung Hur ticity University of Illinois at Urbana-Champaign, USA [email protected] We consider the solitary water wave problem with an ar- bitrary distribution of vorticity. Small amplitude solutions have been constructed in [Hur, 2008] and later in [Groves MS30 and Wahl´en, 2008]. We use degree theory to prove a con- Small Divisors Made Visible tinuation result, constructing a global connected set of so- lutions. Wavefront propagation in a multiply periodic medium ex- poses in a very tangible way the small divisor problem of Miles Wheeler classical mechanics. Consider the speed of a macroscopi- Department of Mathematics cally planar wavefront as a function of direction angle in Brown University R2 . Naive perturbation analysis in the limit of small varia- [email protected] tions from a uniform medium leads to a formal asymptotic expansion, whose terms are in one-to-one correspondence with rational di- rections: A line from the origin has a ra- MS30 tional direction if it intersects a prime integer lattice point Breakdown of Self-similarity at the Crests of Large- k∗ = m∗, n∗), where m∗ and n∗ are integers with no com- amplitude Standing Water Waves mon factor. The sum diverges in any rational direction, but standard diophantine analysis establishes convergence We study the limiting behavior of large-amplitude standing in almost all of the remaining irrational directions. Ob- waves on deep water using high-resolution numerical simu- viously the small divisor series begs the question what is lations in double and quadruple precision. While traveling the actual structure of the speed function? Formal singular waves are known to approach Stokes’s 120 degree corner perturbation analysis provides an intriguing clue: Let e be wave in an asymptotically self-similar manner, standing the gauge parameter measuring the deviation from a uni- waves do not approach Penney and Price’s conjectured 90 form medium: The partial sum of the small divisor series degree corner solution. Instead, a variety of oscillatory −p with |k∗| < ,(p = positive exponent) is regarded as an structures form near the crest tip, causing the bifurca- outer expansion and we examine its asymptotic matching tion curve to fragment into disjoint branches. Additional to inner expansions around all the rational directions with branches of solutions nucleate and merge as fluid depth −p |k∗| < . The inner expansions are done by Arnold av- decreases. For comparison, we consider the effect of small eraging. We use a nonstandard variation of the usual Dio- divisors in a model KAM system. phantine analysis to show that the outer expansion matches with all the inner expansions if 0

Boulder metrology, spectroscopy, and ultrafast applications. [email protected] Alexander Gaeta Ian Coddington Applied and Engineering Physics NIST Boulder Cornell University, Ithaca, NY 14853 [email protected] [email protected]

MS31 MS31 Ultralow Noise Mode-locked Semiconductor Diode Carrier-envelope Phase Locking of Multi-pulse Based Fiber Lasers Lasers

The development of ultrastable mode-locked lasers based We propose the use of an intra-cavity Mach Zehnder in- on harmonic mode-locking have recently produced wide terferometer (MZI), for increasing the repetition rate at spaced frequency combs (∼10 GHz) with long term sta- which carrier-envelope phase-locked pulses are generated bility and narrow comb tooth linewidth (∼500 Hz). The in passively mode-locked fiber lasers. The attractive fea- overall phase noise (timing jitter) integrated to Nyquist can ture of the proposed scheme is that light escaping through be less that 2 femtoseconds. Key to achieving this level of the open output ports of the MZI can be used as a monitor performance requires a fundamental understanding of the signal feeding a servo loop that allows multiple pulses to underlying nonlinear optical mechanisms of pulse gener- co-exist in the cavity, while rigidly controlling their sepa- ation, as well as the environmental effect that drive can ration. The proposed scheme enables in principle a signifi- perturb the systems overall stability. This talk will review cant increase in the pulse-rate with no deterioration in the the recent progress in stabilized combs from harmonically properties of the generated pulses. mode-locked diode lasers and highlight key mechanisms Mark Shtaif that play critical roles in the overall output pulse train Physical Electronics characteristics. Tel Aviv University Peter Delfyett [email protected] Center for Research in Optoelectronics and Lasers University of Central Florida Curtis R. Menyuk [email protected] UMBC Baltimore, MD [email protected] MS31 Coherent Combs: Optimized Supercontinuum Generation for Carrier-envelope Phase-locked MS32 Lasers Shock-driven Jamming and Periodic Fracture at Particulate Interfaces The optical frequency comb has become an indispensable tool in many areas of pure and applied science, opening When a monolayer of hydrophobic particles at the air- new frontiers in fields as diverse as astrophysics, atomic water interface is disturbed by a surfactant droplet, a ra- physics and sensing. However, the generation of broad- dially divergent surfactant shock emanates and packs the band phase-stable combs for the most demanding applica- particles into a jammed annulus that grows with time. This tions still often remains a matter of trial and error, limiting 2D, disordered, elastic solid then fractures to form periodic the potential uptake of this technology. The use of optical triangular cracks with robust geometrical features. We de- fiber supercontinuum generation for comb stabilization is scribe a simple experiment complemented by minimal sim- of course a well-known technique, but surprisingly, the dy- ulation that studies the formation and failure of a disor- namics of this process and the impact on comb stability re- dered solid at the air-water interface. main poorly understood. In this talk we review work in this field, with the overall aim of providing clear guidelines for Mahesh Bandi the generation of stable phase-stabilised frequency combs Okinawa Institute of Science and Technology from mode-locked lasers. If time permits, links with other Collective Interactions Unit areas of wave instability physics will also be addressed. [email protected]

John Dudley Univeriste de Franche Comte MS32 [email protected] Localized Structures in Complex Fluids In the Faraday system a thin layer of fluid is vertically vi- MS31 brated. The primary instability is a transition from a flat Chip-Based Parametric Frequency Combs to a wavy interface. While much is known about extended states in this system, far less is known about localized We describe recent work on the generation of ultrabroad- structures. Recently, a new class of localized structures band frequency combs based on parametric four-wave mix- kinks and persistent holes were discovered in the Faraday ing in silicon-based micro resonators. This system exhibits system with particulate suspensions as the working fluid. extreme nonlinear optical effects at modest power levels, These structures are markedly different from other exam- and many issues related to the dynamics, noise, and mode ples: they oscillate about an unstable state and are inac- locking remain not well understood. Ultimately, this sys- cessible via infinitesimal perturbation from the weakly non- tem offer the promise of highly compact, robust sources for linear states. I will present experimental results on these structures in particulate suspensions, worm-like micellar NW12 Abstracts 73

solutions, and emulsions. MS33 A Numerical Study of Stability of Periodic Robert Deegan Kuramoto-Sivashinsky Waves University of Michigan Department of Physics We consider the spectral and nonlinear stability of peri- [email protected] odic traveling wave solutions of a generalized Kuramoto- Sivashinsky equation. In particular, we resolve the long- standing question of nonlinear modulational stability by MS32 demonstrating that spectrally stable waves are nonlinearly Plants in Motion: Mechanics and Function in stable when subject to small localized (integrable) per- Winding Tendrils turbations. We carry out a numerical Evans function study of the spectral problem and find bands of spec- The helical coiling of plant tendrils has fascinated scien- trally stable periodic travelling waves, in close agreement tists for centuries, yet the mechanism of coiling remains with previous numerical studies of Frisch-She-Thual, Bar- elusive. Moreover, despite Darwin’s widely accepted inter- Nepomnyashchy, Chang-Demekhin-Kopelevich, and others pretation of coiled tendrils as soft springs, their mechanical carried out by other techniques. We also compare pre- behavior is unknown. Our experiments on cucumber ten- dictions of the associated Whitham modulation equations, drils demonstrate that tendril coiling occurs via asymmet- which formally describe the dynamics of weak large scale ric contraction of an internal ‘fiber ribbon’ of specialized perturbations of a periodic wavetrain, with numerical time cells. The mechanical behavior of tendrils under tension evolution studies, demonstrating their effectiveness at a adds a new twist to the story and quantifies Darwins orig- practical level. inal interpretation. Blake Barker Sharon Gerbode Indiana University Harvey Mudd College BYU Department of Physics [email protected] [email protected] Mathew Johnson MS32 University of Kansas Helical Swimming in Viscoelastic Fluids [email protected]

Many bacteria swim by rotating helical flagella. These Pascal Noble, Miguel Rodrigues cells often live in polymer suspensions, which are viscoelas- University of Lyon tic. To explore the effect of viscoelasticity on the bacterial [email protected], [email protected] motility, the helical swimmer is simulated by a model sys- tem - a rotating helical coil. When immersed in a vis- Kevin Zumbrun coelastic fluid, the helix swims faster as compared with the Indiana University Newtonian case. The enhancement is maximized when the USA rotation rate of the helix matches the relaxation time of [email protected] the fluid.

Bin Liu MS33 Brown University Stability of Periodic Traveling Waves in a Department of Physics KdV/Kuramoto-Sivashinsky Equation Equation bin [email protected] In this talk, we consider the spectral and nonlinear stability of periodic traveling wave solutions of a KdV/Kuramoto- MS32 Sivashinsky equation modeling thin film flow down an in- Linking Structure to Dynamics in Weak Turbu- cline. In special cases it has been known since 1976 that, lence when subject to small localized perturbations, spectrally stable solutions of this form exist. Although numerical Despite an enormous range of applications and centuries of time-evolution studies indicate that these waves should also scientific study, understanding and predicting the flow of be nonlinearly stable to such perturbations, an analytical fluids remains a tremendous challenge, particularly when verification of this result has only recently been provided. the flow is chaotic or turbulent. Turbulent flows tend to Here, I will discuss this recent result and, time permit- be characterized by violent fluctuations, broad ranges of ting, I will discuss numerical and analytical verifications strongly coupled degrees of freedom, and significant vari- of the required spectral stability and structural hypoth- ability in space and time. But despite all this, turbulent is esis of this theorem in particular canonical limits of dis- not random. Rather, it tends to self-organize into striking persion/dissipation. This is joint work with Blake Barker, patterns and features. Some of these “coherent structures,’ Pascal Noble, L. Miguel Rodrigues, and Kevin Zumbrun. such as strong vortices, are readily apparent; others are more subtle. But how much can we learn about the flow Mat Johnson from studying coherent structures? To begin to answer this University of Kansas question, I will discuss experiments that suggest deep links [email protected] between flow structure (that is, localized spatiotemporally coherent regions) and dynamics (that is, the flow of energy in the system). MS33 (In)stability of Traveling Waves for the Sine- Nicholas T. Ouellette Gordon Equation: How Bands Balloon Yale University [email protected] Traveling waves of the Sine-Gordon equation exist as kinks 74 NW12 Abstracts

and oscillations in both the subluminal and superluminal stationary localized states, both oscillating localized states regimes. It has been folklore that only subluminal kinks and traveling pulses. are stable in the spectral sense and all of the other waves are unstable. The spectral problem has, however, not been John Burke properly analyzed because of its non self-adjoint structure. Boston University We show that spectral bands can become balloons, and [email protected] do so except in the subluminal kink case, which is indeed stable. MS34 Christopher Jones Interacting Invasion Fronts University of North Carolina and University of Warwick [email protected] We consider invasion fronts in systems of reaction-diffusion equations representing the displacement of an unstable ho- Robert Marangell mogeneous state by a stable one. In particular, we are in- University of Sydney terested in systems for which this transition takes place via [email protected] a third, intermediate state. In some cases, the invasion pro- cess splits into a pair of propagating fronts traveling with different speeds while in other situations a single front is Peter D. Miller formed. We discuss mechanisms leading to both behaviors University of Michigan, Ann Arbor and consider applications to pattern forming systems using [email protected] amplitude equations. Ramon G. Plaza Matt Holzer Institute of Applied Mathematics (IIMAS) School of Mathematics National University of Mexico (UNAM) University of Minnesota [email protected] [email protected]

Arnd Scheel MS33 University of Minnesota On the Spectral Stability of Nonlinear Waves in [email protected] Continuum Mechanics

We discuss some non-standard spectral problems associ- MS34 ated to the stability of nonlinear waves in continuum me- Colliding Convectons chanics. Applications to elasto-chemical waves and waves in thermoelasticity, among others, will be discussed. Convectons are strongly nonlinear spatially localized states found in thermally driven fluid systems. In binary fluid Ramon G. Plaza convection with midplane reflection symmetry convectons Institute of Applied Mathematics (IIMAS) of odd and even parity lie on a pair of intertwined branches National University of Mexico (UNAM) (J. Fluid Mech. 667 (2011) 586) that form the backbone [email protected] of the snakes-and-ladders structure of the so-called pin- ning region. These branches are connected by branches of MS33 asymmetric localized states that drift. When the midplane reflection symmetry is broken, the odd parity convectons Breather Stability in Klein-Gordon Equations also drift, greatly modifying the snakes-and-ladders struc- We will present results on breather stability in Klein- ture of the pinning region. The resulting speed depends Gordon equations. We will show results for discrete Klein- on the magnitude of the symmetry-breaking and the con- Gordon chains that relate the type of multibreathers – vecton length. Head-on and follow-on collisions between where oscillators are at rest, in or out of phase – to the odd parity drifting convectons of different lengths are de- sign of eigenvalues in the perturbation matrix. The weak scribed and the results compared with corresponding dy- coupling perturbation is with respect to the case without namics in a Swift-Hohenberg model studied by Houghton coupling between the oscillators. Next, we will show results and Knobloch (PRE 84 (2011) 016204). for continuous Klein-Gordon equations that approximate Edgar Knobloch the discrete ones. University of California at Berkeley Zoi Rapti Dept of Physics University of Illinois at Urbana-Champaign [email protected] [email protected] Isabel Mercader, Oriol Batiste UPC, Barcelona, Spain MS34 [email protected], [email protected] Localized Oscillations in a Nonvariational Swift - Hohenberg Equation Arantxa Alonso Fisica Aplicada Stationary spatially localized states occur in many systems UPC, Barcelona of physical interest, and are often organized in a so-called [email protected] ’snakes-and- ladders’ structure. In recent years the Swift- Hohenberg equation has garnered much attention as the standard model exhibiting this behavior. In this talk I con- MS34 sider a generalized version of the Swift- Hohenberg equa- Gluing Localized Structures in Reaction-diffusion tion and show that it exhibits, in addition to the usual NW12 Abstracts 75

Systems Darmstadt.

The Swift–Hohenberg equation serves as a model system Alexei F. Cheviakov to study pattern formation in reaction-diffusion equations. Department of Mathematics and Statistics An infinite family of stationary multi-pulse spot solutions University of Saskatchewan to the radially symmetric Swift–Hohenberg equation are [email protected] constructed for dimensions one through three. This re- quires matching a spot and ring solution, which both scale identically with the bifurcation parameter. The multi- MS35 pulses exhibit an anomalous scaling that differs from those Symbolic Computation of Perturbation-Iteration for each of the glued solutions. Solutions for Nonlinear Differential Equations Scott McCalla An University of California Los Angeles algorithm for the symbolic computation of perturbation- [email protected] iteration solutions of nonlinear differential equations will be presented. In the algorithm, the number of correction terms in the perturbation expansion and the number of MS34 terms in the Taylor expansion can be arbitrary. The steps Heterogeneity-induced Spot Dynamics in Dissipa- of the algorithm will be illustrated on a Bratu type initial tive Systems value problem. The algorithm has been implemented in Mathematica. The package PerturbationIteration.m will We discuss the behaviors of traveling 2D spots arising in be discussed and demonstrated. a three component reaction diffusion system when the me- dia has a jump heterogeneity along the line. The travel- Unal Goktas ing spot responds in various ways depending on the the Dept. Computer Engineering height of the jump and the incident angle when it encoun- Turgut Ozal University, Turkey ters the jump line. Refraction and reflection are commonly [email protected] observed. Two issues are discussed here: One is the rela- tion between the incident angle and the refraction angle when the spot crosses the jump. In a scaling limit near MS35 a drift bifurcation, a Snell’s-like law holds for the refrac- Symbolic Computation of Scaling Invariant Lax tion case. The other is the transition from refraction to Pairs in Operator Form for Integrable Systems reflection. Such a transition occurs as the incident angle is increased for a fixed height or the height is increased for a Based on scaling symmetry properties, a direct method to fixed incident angle. As the angle (or height) approaches compute Lax pairs in operator form for completely inte- the critical one, the spot spends much longer time in the grable systems will be presented. By splitting the deter- right half plane after crossing the jump line and it even- mining equations for the Lax pair into kinematic and dy- tually converges to the one traveling parallel to the jump namical constraints, the problem can be reduced to solving line but infinitely far from it, namely it is a traveling spot nonlinear algebraic equations. The method will be illus- in the homogeneous space located at right infinity. We call trated with well-known examples from soliton theory and such a special solution ”scattor” located at infinity. applied to a three-parameter class of fifth-order KdV-like evolution equations. A Mathematica implementation will Takashi Teramoto be demonstrated. Chitose Institute of Science and Technology [email protected] Willy A. Hereman Department of Mathematical and Computer Sciences Yasumasa Nishiura Colorado School of Mines RIES, Hokkaido University [email protected] Japan [email protected] Mark Hickman University of Canterbury Christchurch, New Zealand MS35 [email protected] Symbolic Computation of Point Symmetries of an Infinite Family of Multi-point Correlation Equa- Unal Goktas tionsin Turbulence Theory Dept. Computer Engineering Turgut Ozal University, Turkey The nonlinear Euler or Navier-Stokes equations governing [email protected] the incompressible fluid motions can be re-cast through ensemble averaging into an infinite countable system of Jennifer Larue multi-point correlation equations for the unknown corre- Dept. Applied Mathematics & Statistics lation tensors of increasing order. In particular, the first Colorado School of Mines such tensor is the mean flow velocity. The system consists [email protected] of coupled linear partial differential equations of increas- ing complexity. We demonstrate the use of a modification of GeM symbolic software for Maple to consistently seek MS35 symmetries of such a system of equations, in particular, to Construction of Lax Pairs in Matrix Form and the discover symmetries of multi-point correlation equations Drinfel’d-Sokolov Method for Conservation Laws inherited from the original nonlinear model. This is a joint work with Andreas Rosteck and Martin Oberlack from TU We will present an algorithmic method for the construction of a class of Lax pairs in matrix form for completely inte- 76 NW12 Abstracts

grable nonlinear PDEs. Using a method by Drinfel’d and Flow Instabilities Sokolov, the Lax pairs allow one to construct conservation laws for the given PDE. The method can be implemented A particular interaction-diffusion plant-surface water in any computer algebra system. Prototype Mathemat- model system for the development of spontaneous station- ica software will be demonstrated for nonlinear integrable ary vegetative patterns in an arid flat environment is in- evolution equations, including the Korteweg-de Vries, sine- vestigated by means of a weakly nonlinear diffusive insta- Gordon, and nonlinear Schrodinger equations. bility analysis. The main results of this analysis can be represented by closed-form plots in the rate of precipita- Jacob Rezac tion versus the specific rate of plant loss parameter space. Dept. Applied Mathematics & Statistics From these plots, regions corresponding to bare ground and Colorado School of Mines vegetative patterns consisting of tiger bush, labyrinth-like [email protected] mazes, pearled bush, irregular mosaics, and homogeneous distributions of vegetation may be identified in this param- Willy A. Hereman eter space. Then those Turing diffusive instability predic- Department of Mathematical and Computer Sciences tions are compared with both relevant observational evi- Colorado School of Mines dence and existing numerical simulations involving differ- [email protected] ential flow migrating stripe instabilities for the associated interaction-dispersion-advection plant-surface water model system. MS35 Traveling Waves and Conservation Laws for Com- Bonni Kealy plex mKdV-type Equations Department of Mathematics Washington State University Traveling waves and conservation laws are presented for a [email protected] class of U(1)-invariant complex mKdV equations contain- ing the known integrable generalizations of the real mKdV David J. Wollkind equation. We derive new complex solitary waves and kinks Washington State University that generalize the well-known mKdV sech and tanh solu- Department of Mathematics tions. With respect to conservation laws, we explicitly find [email protected] all first-order conserved densities that yield phase-invariant counterparts of the mKdV conserved densities for momen- tum, energy, and Galilean energy, and a new conserved MS36 density describing the angular twist of complex kink solu- The Stability of Hot-Spots in a Reaction-Diffusion tions. Model of Urban Crime

Thomas Wolf, Stephen Anco Over the past five years, agent-based stochastic models Department of Mathematics have been developed by various researchers to predict Brock University spatio-temporal concentrations of criminal activity in ur- [email protected], [email protected] ban settings. The continuum of these models leads to reaction-diffusion systems with chemotactic terms, that Mohammad Mohiuddin have some common features with more well-known prey- Department of Mathematics Brock University taxis type of predator-prey interactions. In this cotext, [email protected] and in a particular singularly perturbed limit, we analyti- cally construct localized equilibrium and quasi-equilibrium solutions characterizing hot-spots of criminal activity for MS36 the reaction-diffusion system of Short et al. (Mathematical The Brownian Rachet Revisited: Multiple Fila- Models and Methods in Applied Science, Vol. 18, Suppl. mentous Bundle Growth 2008, pp. 1249-1267). Explicit thresholds for the diffusiv- ity of the criminal activity density determining the stability We present a model based on a diffusion formalism for the of these localized patterns are obtained for 1-D and 2-D do- Brownian Rachet (BR), which we extend to incorporate mains by first deriving and then analyzing a novel class of a bundle of N identical filaments. In the absence of a nonlocal eigenvalue problems (NLEP). The implications of load, the bundle growth rate is similar to that of a single these results are discussed together with some open prob- filament. However, under the stalling condition, the bundle lems related to the dynamics of hot-spot patterns. can oppose N times the external force. We derive a set of relationships describing the velocity of the BR movement Michael Ward (Vz) and its apparent diffusivity (Dz) as functions of the Department of Mathematics resistant force (F ) and the number of filaments in a bundle University of British Columbia (N). [email protected]

Christine L. Cole,HongQian Department of Applied Mathematics MS36 University of Washington Mussel Pattern Formation Model Systems: Com- [email protected], parison of Turing Diffusive and Differential Flow [email protected] Instabilities A particular interaction-diffusion mussel-algae model sys- MS36 tem for the development of spontaneous stationary young Vegetative Pattern Formation Model Systems: mussel bed patterning on a homogeneous substrate covered Comparison of Turing Diffusive and Differential by a static marine layer containing algae as a food source is investigated employing a weakly nonlinear diffusive in- NW12 Abstracts 77

stability analysis. The main results of this analysis can be represented by plots in the ratio of mussel motility to algae lateral diffusion versus the rate of mussel growth parameter Benjamin Akers space. From these plots, regions corresponding to bare sed- Air Force Institute of Technology iment and mussel patterns consisting of bands, labyrinth- benjamin.akers@afit.edu like mazes, hexagonal arrays of clumps or gaps, irregu- lar mosaics, and homogeneous distributions of low to high density may be identified in this parameter space. Then MS37 those Turing diffusive instabiltiy predictions are compared Time-periodic Vortex Sheets with Surface Tension with both relevant laboratory experimental evidence and existing numerical simulations involving differential flow We present results from joint work with Jon Wilkening migrating band instabilities for the associated interaction- on computation of time-periodic solutions of the vortex dispersion-advection mussel-algae model system. sheet with surface tension. These are found by defining a functional which is zero for time-periodic solutions and David J. Wollkind positive otherwise; the functional is then minimized by a Washington State University gradient descent algorithm. Differences between solutions Department of Mathematics with zero and nonzero mean vortex sheet strength will be [email protected] discussed. If time allows, a discussion of analytical progress on the question of existence of these solutions (which is Richard Cangelosi joint work with C. Eugene Wayne) will be included. Department of Mathematics David Ambrose WashingtonState University Drexel University [email protected] [email protected]

MS36 MS37 Row Formation in Lepidopteran Wings by Origin- On the Existence and Stability of Solitary-wave Dependent Cell Adhesion: An Individual Cell In- Solutions to a Class of Evolution Equations of teraction Model Whitham Type

Regardless of the obvious differences in the color patterns We consider a class of pseudodifferential evolution equa- exhibited by lepidopteran wings in nature, the scale cells tions of the form that display colors have a remarkable similarity in spatial patterning. Precursors of scale cells develop and migrate ut +(n(u)+Lu)x =0, into rows roughly parallel to the proximodistal axis. Exper- iments have revealed that a gradient of activity along the L n wings, as well as the maintenance of cell adhesivity within in which is a linear smoothing operator and is at least each cell, may have been reasons for the formation of rows. quadratic near the origin; this class includes in particular In addition, cell polarization along the proximodistal axis the Whitham equation. A family of solitary-wave solutions in some species, and the extension of filopodia in response is found using a constrained minimisation principle and to protein expression, are observed. Based on mechanisms concentration-compactness methods for noncoercive func- observed in experiments, a model of individual precursor tionals. The solitary waves are approximated by (scalings interaction and migration is developed to simulate the for- of) the corresponding solutions to partial differential equa- mation of parallel rows. The results show that the ori- tions arising as weakly nonlinear approximations; in the gin dependent cell adhesion is crucial in producing parallel case of the Whitham equation the approximation is the rows, although short-range and long-range cell interactions Korteweg-deVries equation. We also demonstrate that the are important in generating stable spatial patterns. In con- family of solitary-wave solutions is conditionally energeti- trast, the effect of polarization is less significant. This indi- cally stable. vidual cell interaction model is compared with the existing Mark D. Groves nonlinear differential integral origin-dependent cell adhe- Universit¨at des Saarlandes sion model using cell density as a model variable. [email protected] Mei Zhu Pacific Lutheran University MS37 Department of Mathematics [email protected] Stability of Gravity Waves with Surface Tension Recently, a reformulation of the surface water wave prob- MS37 lem was presented, allowing improved computational ef- ficiency [Ablowitz et al 2006, J. Fluid Mech.]. Building Wilton Ripples in Weakly Nonlinear Models on previous work [Akers & Nicholls 2010, SIAM J. Appl. A method for computing Wilton ripples, traveling waves Math., Deconinck & Oliveras 2011, J. Fluid Mech.], we supported at two resonant harmonics, is presented for a compute stationary periodic solutions of the problem, in class of weakly nonlinear model equations. This method the presence of small surface tension. The aim of the is perturbative in nature; all perturbation orders are com- project is to examine the perturbative effect of surface ten- puted exactly. The method is used to numerically compute sion on the slow-growing oscillatory instabilities in water solution profiles as well as the disc of analyticity of branches waves. of solutions. The procedure provides a natural framework Olga Trichtchenko to prove existence and analyticity of branches of solutions. Department of Applied Mathematics University of Washington [email protected] 78 NW12 Abstracts

Bernard Deconinck Feng Li University of Washington Dept. of Electronic and Information Engineering [email protected] Hong Kong Polytechnic University [email protected] Katie Oliveras Seattle University Edwin Ding Mathematics Department Azusa Pacific University [email protected] [email protected]

Nathan Kutz MS38 Dept. of Applied Mathematics Dependence of Mode-locked Laser Linewidth on University of Washington Pulse Parameter Jitter [email protected] We study the spectral linewidth of a finite sequence of near-identical pulses produced by a model for mode-locked MS38 lasers. In particular, we show how dispersive radiation can Linearized Stability Analysis of the Haus Mode- have a significant impact on the random dynamics of the Locking Equation pulse phases, translating to an increased linewidth relative to studies neglecting this phenomenon. Using simulations, Haus mode-locking equation (HME) is the simplest scalar we compare the true probability of critical line broadening model for mode-locking, in which the fast saturable ab- with the approximate value obtained by using the com- sorber is modeled as a loss term proportional to the in- puted variance with a Gaussian assumption. tensity of the electrical field. There are several models introduced for the fast saturable absorption that claimed Richard O. Moore, Daniel Cargill would be a better match to the physical saturable absorp- New Jersey Institute of Technology tion process. We perform a parameter study of the HME [email protected], [email protected] with these different saturable absorption models. Tobias Schaefer Shaokang Wang Department of Mathematics UMBC The College of Staten Island, City University of New York [email protected] [email protected] Andrew Docherty University of Maryland Baltimore County MS38 [email protected] Effect of the Birefringent Beat Length on Variabil- ity in Passively Modelocked Fiber Lasers BrianMarks,CurtisR.Menyuk Birefringence leads to sensitivity of the power transfer to UMBC polarization controller settings, loop length, and birefrin- Baltimore, MD gent beat length in passively modelocked fiber lasers that [email protected], [email protected] use nonlinear polarization rotation for fast saturable ab- sorption. We discuss the parameters that lead to the great- MS39 est sensitivity and suggest good operating regimes for these lasers. Swimming and Propulsion in Complex Fluids Brian Marks,CurtisR.Menyuk Abstract not available at time of publication. UMBC Paulo E. Arratia Baltimore, MD Mechanical Engineering and Applied Mechanics [email protected], [email protected] University of Pennsylvania, Philadelphia. parratia@seas.upenn.edu MS38 Engineering the Laser Cavity Transmission for En- MS39 hanced Energy in Mode-locked Fiber Lasers Microbial Flow Fields, Noise, and Cell-cell Interac- The multi-pulsing instability of mode-locked fiber lasers tions must be avoided in applications where high energy pulses It is currently believed that deterministic long-range fluid are required. Recently by using a simple geometric model, dynamical effects govern bacterial cell-cell and cell-surface we showed that it is possible to engineer the laser cavity scattering - the elementary events that lead to swarming dynamics by modifying the nonlinear loss curve. In this and collective swimming in active suspensions and to the paper, we theoretically demonstrated that the energy per- formation of biofilms. I will present direct measurements of formance can be increased by including a second set of the bacterial flow field, generated by individual swimming waveplates and polarizer to a laser cavity mode-locked by Escherichia coli, and use these measurements to show that a set of waveplates and polarizer. rotational diffusion drowns out long-range hydrodynamic P. K. Alex Wai effects in cell-cell and cell-surface interactions. Hong Kong Polytechnic University Knut Drescher Dept. of Electronic and Information Engineering Princeton University [email protected] [email protected] NW12 Abstracts 79

MS39 Schr¨odinger equations, etc.; however, the theory is not as Title Not Available at Time of Publication well established for quadratic pencils. In this talk I will discuss the extension of the linear theory to the quadratic Abstract not available at time of publication. theory, and apply the theoretical results to the study of the spectral stability of spatially periodic waves. William Irvine University of Chicago Todd Kapitula [email protected] Calvin College [email protected] MS39 Fluidic Computation: Dynamics of Drops and Bub- MS40 bles in Geometrical Fluid Networks Evans-Krein Function and Eigenvalue Counts Abstract not available at time of publication. The key question in spectral stability of nonlinear waves is the existence of eigenvalues with positive real part of a Manu Prakash linearized operator. Two concepts very different in nature Stanford University proved to be useful in search for such an unstable spec- Department of Bioengineering trum: the Evans function, an analytic function with zeros [email protected] at isolated unstable eigenvalues, and the Krein signature, an algebraic quantity capturing the ability of an eigenvalue to be or to become unstable under a change of a parameter MS39 in a system. Although the Evans function does not pro- Dynamics of Passive Flexible Wings vide full information about the Krein signature, we show that its simple extension, the Evans-Krein function, allows We investigate the dynamics of passive flexible wings freely to calculate the Krein signature of an eigenvalue at almost falling under the influence of gravity. Particular attention no additional computational cost. The method used also is given to elucidating the role of flexibility in passive flight. enables us to give very elegant proofs of eigenvalue counts The effect of bending on the dynamics of fluttering wings for linearized Hamiltonians: the Grillakis-Shatah-Strauss is examined through an experimental investigation of de- criterion, its generalization for systems with broken Hamil- formable rectangular wings falling in water. Elastic defor- tonian symmetry, and a count of real eigenvalues for diag- mations induced by the flow strongly affect the flight char- onazible Hamiltonians originally obtained by Jones. acteristics and suggest the existence of an optimal bending rigidity minimizing the descent velocity. Richard Kollar Comenius University Daniel Tam,JohnBush Department of Applied Mathematics and Statistics MIT [email protected] dan [email protected], Peter D. Miller MS40 University of Michigan, Ann Arbor Two Parameter Methods for Symmetrizable Non- [email protected] self-adjoint Eigenproblems

Many apparently non-self-adjoint eigenproblems can be re- MS40 cast in the form Ay − λBy =0whereA and B are self- Stability of Solitary Waves of a Sixth-Order Boussi- adjoint operators in a suitable Hilbert space. They may nesq Equation however be indefinite, and the use of a two parameter em- bedding Ay − λBy = μy will be explored. Under fairly We consider the stability of solitary waves of a sixth-order Boussinesq equation. For a class of homogeneous nonlin- general conditions this allows one to “see” certain aspects d c of the spectrum in terms of the (λ, μ) eigencurves. earities, we determine the nodal set of the function ( ) that determines the stability of solitary waves. Paul Binding Department of Mathematics Steve Levandosky University of Calgary College of the Holy Cross [email protected] [email protected]

MS40 MS40 Index Theorems for Quadratic Pencils with Appli- Spectral Pollution and Boundary Conditions for cations PDEs on Singular Domains

In dispersive wave equations which are second-order in PDE spectral problems on singular domains (waveguides; time, e.g., sine-Gordon and the “good” Boussinesq, the exterior domains; domains in which coefficients blow up or linearized eigenvalue problem associated with the spec- strong ellipticity fails) often have essential spectra. As a tral stability of a wave can be realized as a quadratic consequence, variational methods and domain truncation pencil, where each coefficient of the pencil is either a methods may suffer from spectral pollution: eigenvalues self-adjoint or skew-symmetric operator. There is a well- are generated which converge to points not in the spec- developed unstable eigenvalue index theory for linear pen- trum of the original problem. This talk will review some cils (going back to Grillakis, Jones, etc., and continu- approaches for avoiding this undesirable phenomenon, in- ing to Pelinovsky, K/Kevrekidis/Sandstede, etc.), which cluding an approach proposed by the author involving the arise when discussing generalized KdV, coupled systems of 80 NW12 Abstracts

use of relatively compact dissipative perturbations. MS41 Stability of Concatenated Waves Marco Marletta Cardiff university UK Doug Ward (Drexel) and Sabrina Selle (Bielefeld) have [email protected]ff.ac.uk studied stability of concatenated traveling waves, moving with different speeds, in dissipative systems. Their ap- proach treats concatenated waves as a sum of waves. I will MS41 describe an alternate approach that respects the concate- Patterns on Sea Urchin Embryos nated wave structure and uses Laplace transforms to solve the linearized problem. Patterns of bone morphogen proteins (BMP) on embryos are important for development. BMP patterns have been Xiao-Biao Lin, Stephen Schecter studied extensively on fly (drosophila melanogaster) em- North Carolina State University bryos. This joint project was motivated by experimental Department of Mathematics work of Prof. C. Bradham on sea urchin embryos. We en- [email protected], [email protected] deavor to explain a number of experimental observations using ARD modeling, and we are in the process of conduct- ing new epxeriments suggested by the analysis. MS41 Transonic Evaporation Waves in a Spherical Sym- Tasso J. Kaper metric Nozzle Boston University Department of Mathematics We study the liquid to vapor phase transition in a cone [email protected] shaped nozzle. Using geometric singular perturbation the- ory, we extend results on subsonic and supersonic evapora- Cyndi Bradham tion waves by Fan and Lin (2011) to transonic waves. We BIology, Boston University are able to show the existence of evaporation waves that [email protected] cross from supersonic to subsonic regions and evaporation waves that connect from the subsonic regions to the sonic Heather Hardway surface and then continue onto the supersonic branch via Mathematics, Boston University the slow flow. [email protected] Martin Wechselberger University of Sydney Peter van Heijster [email protected] Division of Applied Mathematics Brown University, Providence RI [email protected] Xiao-Biao Lin North Carolina State University [email protected] MS41 Poiseuille Flow and Apparent Viscosity of Nematic MS43 Liquid-crystals Modeling Landslides and Landslide-generated Abstract not available at time of publication. Tsunamis Weishi Liu Landslide-generated tsunamis pose a significant threat, yet Department of Mathematics assessing their potential is challenging due to an uncon- University of Kansas strained source. Dynamically modeling landslides can elu- [email protected] cidate the tsunamigenic nature of offshore geomorphology, but this remains a difficult endeavor due to the compli- cated physics of granular-fluid flows. I will describe a two- MS41 phased depth-averaged model for tsunamigenic landslides. Fundamental Chemotactic Traveling Wave Phe- The model is a nonconservative system of hyperbolic equa- nomena tions similar to the shallow water equations.

Advection-Reaction-Diffusion systems are commonplace in David George the mathematical biology literature on pattern formation U.S. Geological Survey and structure in cell populations. Where directed cell Cascades Volcano Observatory motion in response to a biochemical gradient occurs - a [email protected] process termed chemotaxis - simple advection terms in the governing continuum model are typically used. In contrast to the classic models of bacteria chemotaxis by MS43 Keller and Segel for highly diffusive populations we con- A Probabilistic Tsunami Hazard Assessment Study sider advection-dominated travelling waves as models of of Crescent City, Ca dense motile cell populations. A probabilistic tsunami hazard assessment (PTHA) of Graeme Pettet Crescent City, CA will be presented. The study is based Queensland University of Technology on nonlinear tsunami inundation simulations with initial School of Mathematical Sciences conditions corresponding to near- and far-field earthquake [email protected] sources in the Cascadia, Alaska, Japan-Kamchatka-Kurile and Chile Subduction Zones. The methodology produces maps that provide an estimate of the maximum flood level NW12 Abstracts 81

that will be exceeded with a specific annual probability at tics. each cell of the inundation computational grid. Elena Tolkova Frank I. Gonzalez JISAO / NOAA Center for tsunami research U. Washington, Dept. of Earth and Space Sciences [email protected] fi[email protected] William L. Power Randall LeVeque, Jonathan Varkovitzky GNS Science U. Washington, Applied Mathematics [email protected] [email protected], [email protected] MS43 MS43 Three-dimensional Tsunami Runup with GPUSPH Simulation of Landslide-Generated Tsunamis with the Hysea Platform: the Lituya Bay 1958 Event Abstract not available at time of publication. We present a multi GPU implementation of the IFCP FV Robert Weiss scheme of the two-layer Savage-Hutter type model devel- Virginia Tech oped by E. D. Fern´andez-Nieto et al (JCP, 2008) to study Department of Geosciences submarine avalanches. In this model, a layer composed [email protected] of fluidized granular material is assumed to flow within a layer composed of an inviscid fluid. A new web-based platform named HySEA is used as interface for simulating MS44 landslide-generated tsunamis focusing in the Lituya Bay Laser Light Condensation Phenomena 1958 mega-tsunami event. We present in a theoretical and experimental study vari- Jose Manuel Gonzalez-Vida ous condensation phenomena in cw and mode-locked lasers. University of Malaga, Spain They are based on weighting the laser modes in a noisy en- [email protected] vironment (spontaneous emission, etc.) by a loss-gain scale rather than in photon energy. They are characterized by M. De La Asunci´on, M. J. Castro a sharp transition from multi- to single-mode oscillation. Dpt. An´alisis Matem´atico The study uses a simple linear multivariate Langevin for- University of M´alaga mulation with a global constraint which gives a mode oc- [email protected], [email protected] cupation hierarchy that functions like Bose-Einstein statis- tics. We also discuss how and when condensation occurs in photon systems, how it relates to lasing, and the difficul- E. D. Fern´andez-Nieto ties to observe regular photon Bose-Einstein condensation Dpt. Matem´atica Aplicada (BEC) in laser cavities. University of Seville [email protected] Baruch Fischer Department of Electrical Engineering J. Mac´ıas Technion Dpt. An´alisis Matem´atico fi[email protected] University of M´alaga [email protected] MS44 T. Morales Phase-coherent Repetition Rate Multiplication of Dpt. Matem´aticas a Mode-locked Laser by Injection Locking University of C´ordoba [email protected] We have used injection locking to multiply the repetition rate of a passively mode-locked femtosecond fiber laser from 40 MHz to 1 GHz while preserving optical phase co- C. Par´es,C.S´anchez-Linares herence between the master laser and the slave output. Dpt. An´alisis Matem´atico The slave system is implemented almost completely in fiber University of M´alaga and incorporates gain and passive saturable absorption. carlos [email protected], [email protected] The slave repetition rate is set to a rational harmonic of the master repetition rate, inducing pulse formation at the MS43 least common multiple of the master and slave repetition rates. Detecting Standing Waves in Modeled Tsunami Wave-Fields David Kielpinski Australian Attosecond Science Facility Continental shelves can serve as a barrier protecting from Griffith University, Brisbane, Australia tsunami, or as a resonator trapping tsunami wave energy [email protected] in standing waves. Bathymetric features in an adjacent ocean can also contribute to standing wave formation by acting as waveguides. Knowing how tsunami wave energy MS44 is transferred from the open ocean into oscillations next The Sinusoidal Ginzburg-Landau Equation for to the coast, and the spatial distribution of those oscilla- High-Energy Mode-Locking tions, helps identify areas likely to experience the greatest amplitudes. We demonstrate techniques that use tsunami The sinusoidal Ginzburg-Landau Equation (SGLE) is pre- simulations to determine an area’s resonance characteris- sented to characterize the pulse evolution in a passively 82 NW12 Abstracts

mode-locked ring cavity laser. The model gives a better [email protected] description of the cavity dynamics by accounting explicitly for the full periodic transmission generated by the wave- Arnaud Goullet plates and polarizer. The SGLE agrees well with the full NJIT governing model, and it supports high energy pulses that [email protected] are not predicted by the master mode-locking theory, thus providing a platform for optimizing the laser performance. Wooyoung Choi Dept of Mathematics Edwin Ding New Jersey Institute of Tech Azusa Pacific University [email protected] [email protected] MS45 J. Nathan Kutz Stable Solitary Waves in Two-layer Flows University of Washington Dept of Applied Mathematics The system of equations describing nonlinear and weakly [email protected] dispersive interfacial waves between two layers of inviscid fluids of different densities and bounded by top and bot- Eli Shlizerman tom walls is known to be mathematically ill-posed despite University of Washington, Seattle the fact that physically stable internal waves have been ob- [email protected] served. We obtain a stable nonhydrostatic model for this system and illustrate our results with solitary waves.

MS44 Anakewit Boonkasame Ultra-short Pulse Dynamics Towards Modeling University of Wisconsin at Madison, USA Atto-second Physics [email protected] A new theoretical model is proposed for characterizing the ultrashort (few femtoseconds and below) propagation dy- MS45 namics in a laser cavity that is mode-locked with a sat- Strongly Nonlinear Interaction of Crossing Large urable absorbers. The theory circumvents the standard, Amplitude Internal Waves and problematic, center frequency expansion methods that typically result in the nonlinear Schrodinger based master A regularized model was introduced recently to describe mode-locking equation. The resulting short-pulse equa- strongly nonlinear internal waves in a two-layer system tion framework, which is the equivalent of the nonlin- to eliminate the shear instability at the interface. The ear Schrodinger equation for ultrafast pulses, provides an two-dimensional time evolution equations for the interface asymptotically valid description of the electric field ampli- and velocity fields are solved numerically using a pseudo- tude even as pulses are shortened below a single-cyle of spectral method to study the strongly nonlinear interac- the electric field. Given the lack of theory in the ultra- tion of crossing internal solitary waves. Our simulations fast regime, the model provides the beginning theoretical are compared with the solution of the weakly nonlinear framework for quantifying the pulse dynamics and stability KP equation. as pulsewidths approach the attosecond regime. Arnaud Goullet J. Nathan Kutz NJIT University of Washington [email protected] Dept of Applied Mathematics [email protected] MS45 Modeling Three Dimensional Gravity-capillary MS45 Waves in Deep Water Two-dimensional Nonlinear Internal Waves Gravity-Capillary waves enjoy a range of applications. In Large amplitude internal solitary waves excited typically order to accurately compute complex time dependent so- by the interaction of tidal currents with bottom topography lutions, we simplify the full Euler equations by taking a have been observed frequently in coastal oceans through cubic truncation of the Dirichlet-to-Neumann operator for in-situ measurements and satellite images. Although there the velocity potential on the free surface with the full sur- has been a considerably intensive research on nonlinear in- face tension. This equation agrees remarkably well with ternal waves, most of the existing work is devoted strictly the full equations of solitary waves. In 3d, fully localised to the one-dimensional case. To study the evolution of two- solitary waves are computed and the stability, interaction dimensional large amplitude internal waves in a two-layer and focussing phenomena of both line and lump solitary system with variable bottom topography, we derive a fully waves are investigated via numerical time evolution, and two-dimensional strongly nonlinear model. This is a gen- some interesting dynamical phenomena are observed. eralization of the one-dimensional model of Choi, Barros & Jo (2009) that is known to be free from shear instability Zhan Wang for a wide range of physical parameters. After investigating University of Wisconsin-Madison, USA shear instability of the regularized model for ?at bottom, [email protected] weakly two-dimensional and weakly nonlinear limits are discussed. MS45 Ricardo Barros Exact Solution of Linear Wave Motion Over Pe- Instituto Nacional de Matematica Pura e Aplicada, Brazil riodic Topography of Large Amplitude and Arbi- NW12 Abstracts 83

trary Shape the onset of instability in shallow water.

We consider linear waves propagating over periodic to- David P. Nicholls pographies of arbitrary amplitude and wave form, general- University of Illinois at Chicago ising the method in Howard and Yu (2007). By a judicious [email protected] construction of a conformal map from the flow domain to a uniform strip, exact solutions of Floquet type can be developed in the mapped plane. These Floquet solutions MS46 are analogous to the complete set of flat-bottom wave and Stability and Bifurcations of Rotating Vortices in evanescent modes, therefore can be used to construct the the Gross-Pitaevskii Equation with a Harmonic Po- solutions to boundary value problems involving a wavy to- tential pography with a constant mean water depth. Various con- crete examples are given and quantitative results are dis- We show that the rotating symmetric vortices located at cussed. Comparisons with experimental data are made, the center of a two-dimensional harmonic potential under- and qualitative agreement is achieved. This is a collabora- take a pitchfork bifurcation with radial symmetry. This tion with Louis N Howard at MIT. Support of JY by NSF bifurcation leads to the family of vortices, which precess (Grants CBET-0756271 and CBET-0845957) is acknowl- constantly along an orbit enclosing the center of symme- edged. try. The radius of the orbit depends on the precessional frequency, or equivalently, on the chemical potential. We Jie Yu show that both symmetric and asymmetric vortices are North Carolina State University spectrally stable with respect to small time-dependent per- jie [email protected] turbations. At the same time, the symmetric vortex be- comes a local minimizer of energy in the parameter region where the asymmetric vortex exists, the latter corresponds MS46 to a saddle point of energy. Modulational Stability for a General Class of Dis- persive Waves Dmitry Pelinovsky McMaster University Abstract: We consider a very general class of dispersive Department of Mathematics wave equations having a variational structure of the fol- [email protected] lowing form: there are two conserved quantities, a Hamil- tonian and a Momentum, and a conserved Casimir, a Mass, along with some scaling relations. This includes a number MS46 of models describing water waves. Under these assump- Asymptotic Stability of Semi-Strong Interactions tions the stability of long waves is governed by a common in Weakly Damped Systems: Attack of the Point normal form, and the modulational stability of such waves Spectrum can be determined rather generally. We demonstrate the asymptotic stability of semi-strong N- Jared Bronski pulse solutions for a class of singularly perturbed reaction University of Illinois Urbana-Champain diffusion equations. The key step to both the existence Department of Mathematics and stability is the analysis of a family of non-self adjoint [email protected] eigenvalue problems, particularly the control of the point spectrum, which we show breaks into a controllable part Vera Hur, Nanjundamurthy Venkatasubbu and a ’semi-strong’ portion, which is amenable to an ana- University of Illinois lytic reduction. This is joint work with Tom Bellsky. [email protected], [email protected] Keith Promislow Michigan State University MS46 [email protected] Spectral Stability of Water Waves: Stable, High- Order Computation in the Presence of Resonance Tom Bellsky Arizona State University We study spectral stability of periodic traveling waves in [email protected] fluids in two dimensions. Rather than simply substituting a computed traveling wave into the linearized water wave problem and appealing to a numerical solver, we use the MS47 fact that waves come in analytic branches to show that, Bifurcations of Front Solutions for a Three- generically, the spectral data can also be parametrized an- component Reaction Diffusion System alytically. We followed the behavior of the spectrum in the complex plane as a wave height/steepness parameter was By means of a center manifold type reduction we derive increased until divergence of the method. The singularities an ODE system describing the evolution of front solutions in the expansions (resulting in the divergence of the numer- in a three-component reaction-diffusion system. The re- ical scheme) are mandated by the form of the expansions: duced systems exhibits a surprisingly complicated bifur- only purely imaginary eigenvalues can be produced so that cation structure including a butterfly catastrophe and a the algorithm cannot compute spectrum with a non-zero Bogdanov-Takens type scenario. These results shed light real part. In the light of this, we pose a conjecture that on numerically observed accelerations and oscillations and not only is the presence of a singularity necessary for the pave the way for the analysis of front interactions in a pa- onset of (spectral) instability, but it is also sufficient. After rameter regime where the essential spectrum of a single careful numerical investigation, we find that the conjecture front approaches the imaginary axis asymptotically. is largery justified in the case of deep water but unpredicts Martina Chirilus-Bruckner Boston University 84 NW12 Abstracts

martina [email protected] of dark breathers in a one-dimensional uniform chain of beads under precompression. We derive a defocusing non- linear Schr¨odinger equation (NLS) for frequencies that are MS47 close to the edge of the linear spectrum and use this to Blowup of Solitary Waves of the Generalised construct targeted initial conditions to numerically find the Korteweg-de Vries Equation dark breather solutions. The range of validity of the NLS approximation is also explored and we discuss experimental We study blowup solutions of the generalized Korteweg-de implications of our results. Vries equation (GKdV). These solutions arise when a soli- ton turns unstable and then becomes infinite in finite time; Christopher Chong in other words blow up. Through a dynamical rescaling we University of Massachusetts, Amherst reduce the GKdV to an ODE. Then, we use asymptotic [email protected] methods and matching techniques to construct bounded solutions of the ODE. Moreover, with the asymptotic anal- ysis we determine the parameter range over which these MS48 solutions may exist. Topological and Spectral Features of Nonlinear Wave Motion in Periodic Chains Vivi Rottschafer Leiden University Wave motion in periodic chains with cubic nonlinearities is Dept of Mathematics investigated with the objective of providing a comprehen- [email protected] sive account of the distinctive topological and spectral fea- tures of one-dimensional nonlinear wave propagation. The investigation focuses on analogies and differences between MS47 nonlinear chains and their linear dispersive counterparts, Pulse Stability in a Gierer-Meinhardt System with as well as on the interplay and competing roles of disper- a Slow Nonlinearity sive and nonlinear mechanisms in the formation and clas- sification of wave distortion. The signatures of salient be- The existence and stability of localized pulses in a Gierer- havior associated with nonlinear mechanisms are collected Meinhardt equation with an additional ‘slow’ nonlinearity both in the physical (space-time) and spectral (frequency- is studied. This system is an explicit example of a gen- wavenumber) space, and the special case of waves with eral class of singularly perturbed , two-component reaction- identical spectral content and markedly different topolo- diffusion equations that goes beyond model systems such as gies in the physical space is investigated. The analysis is Gray-Scott and Gierer-Meinhardt. The additional nonlin- carried out using full-scale simulations in conjunction with earity influences the stability analysis and stability proper- a variety of global and local signal processing techniques, ties of the pulse. Moreover, unlike in GS/GM type models, and the results are checked against existing unit-cell based pulse solutions exhibit complex behaviour near Hopf bifur- perturbation methods. cations. Stefano Gonella Frits Veerman Department of Civil Engineering Mathematical Institute University of Minnesota Leiden University [email protected] [email protected] Ganesh Ramakrishnan MS47 University of Minnesota [email protected] Destabilizing of Stationary Spots in a Planar Three-component FitzHugh-Nagumo Equation MS48 In this talk, I will analyze the destabilization of a stable stationary radially symmetric spot arising in a specific pla- Ill-posedness of a Linearized Compacton Equation nar three-component FitzHugh-Nagumo equation. In par- After linearizing the K(2, 2) equation around a known peri- ticular, I am interested in the bifurcation of the stationary odic solution (whose spacial truncation is the famous com- spot to a traveling spot. As it turns out, there is a compe- pacton solution), we are able do demonstrate that the spec- tition between this drift bifurcation and several other Hopf trum of the linear operator, as an operator on H 3([−π,π]), bifurcations. I will formally determine asymptotic condi- is the entire complex plane. This is accomplished by turn- tions for these bifurcations and also check the asymptotic ing the eigenvalue value problem into a discrete dynamical results with AUTO and a direct solver. system which we then solve. We will also briefly consider K n, m Peter van Heijster the operators that arise as linearizations of the ( ) Division of Applied Mathematics equation. Brown University, Providence RI Daniel Jordon [email protected] Department of Mathematics Drexel University Bjorn Sandstede [email protected] Brown University bjorn [email protected] MS48 Ill-Posedness Due to Degenerate Dispersion MS48 Dark Breathers in Granular Crystals We study the behavior of solutions for PDEs with de- generate dispersions. Using the degenerate Airy equation We study the existence, stability and bifurcation structure NW12 Abstracts 85

ut =2uuxxx as an example, we show that unlike uniform neously start cooperating or developing shared behaviors, dispersive effects, degenerate dispersion even just by it- norms, and culture. In this connection, it is important to self may lead to ill-posedness of the corresponding initial study the role of social mechanisms such as repeated inter- value problem. In particular, we establish the existence of actions, group selection, network formation, costly punish- a compactly supported self-similar solution for the degen- ment and group pressure, and how they allow to transform erate Airy equation. When combined with certain scaling social dilemmas into interactive situations that promote invariances, such a self-similar solution implies that the de- the social system. Furthermore, it is interesting to study generate Airy equation is ill-posed for initial data in H 2, the role that social inequality, the protection of private i.e., its solutions do not depend smoothly on initial condi- property, or the on-going globalization play for the result- tions. ing ”character” of a social system (cooperative or not). It is well-known that social cooperation can suddenly break Dennis Guang Yang down, giving rise to poverty or conflict. The decline of high Department of Mathematics cultures and the outbreak of civil wars or revolutions are Drexel University well-known examples. The more suprising is it that one [email protected] can develop an integrated game-theoretical description of phenomena as different as the outbreak and breakdown of cooperation, the formation of norms or subcultures, and MS49 the occurence of conflicts. Threshold Models and Abrupt Change in Social Norms in Complex Social Networks Dirk Helbing ETH Zurich In our work we seek for description of extensive simula- [email protected] tions and mathematical analysis of simple systems of peo- ple affected by both personal preferences and observation of their neighbors behaviour. We show that the influential MS49 Schelling-Granovetter model of threshold behavior change Social Interactions on Networks: Self-excitation, is equivalent to a zero-temperature, mean-field limit of a Third-party inhibition, and the Link with Game very well studied model in physics the Ising model. We Theory also show that we can study the effects of locality and mix- ing rate on behavior change by studying these models on We introduce a point process model for social interactions lattices and networks. on a network, including self-excitation and third-party in- hibition. Here, a coupled system of state-dependent jump Andrei Akhmetzhanov stochastic differential equations is used to model the con- McMaster University ditional intensities of the directed network of interactions. [email protected] The model produces a wide variety of transient or station- ary weighted network configurations and we investigate un- Lee Worden der what conditions each type of network forms in the con- UC Berkeley tinuum limit. We also explore the link between this model [email protected] and recent work on repeated games.

Jonathan Dushoff Martin Short McMaster University UCLA Department of Biology [email protected] dushoff@mcmaster.ca PP1 MS49 Some Exact Solutions Of Two 5th Order KdV-Type An Adversarial Evolutionary Game for Criminal Nonlinear Evolution Equations Behavior We consider the generalized 5th order nonlinear KdV equa- We consider an adversarial evolutionary game developed tion which is invariant under a scaling transformation with for criminal activity where players may or may not com- suitable weighting scheme. An extension of tanh method mit crimes and cooperate with authorities. Among the four has been used rigorously for solving this fifth order non- possible player strategies, the so called “informant” who linear evolutionary partial differential equation. The gen- cooperates while still committing crimes. We find two pos- eral solutions of the parameters are formed considering an sible equilibration regimes, a defection-dominated and an ansazt of the solution in terms of tanh. Then, using these ideal, cooperation-dominated one and show that the num- results some exact solutions are found for the two 5th order ber of informants is crucial in determining which of these KdV-type equations which also include soliton solutions. two regimes is achieved. Tania Akter Maria Rita D’Orsogna World University California State University, Northridge [email protected] [email protected] Abdus Sattar Mia Brock University, Canada MS49 sattar [email protected] Towards Simulating Societies: Simple Models with Complex Dynamics In order to understand social systems, it is essential to identify the circumstances under which individuals sponta- 86 NW12 Abstracts

PP1 New Jersey Institute of Technology Compressed Sensing in Retinal Image Processing [email protected] Retinal image processing transforms photons into mem- brane potentials via several nonlinear transformations. PP1 This process begins in a large network of photoreceptors A Numerical Method For Solving Nonlinear and ends in a relatively small ganglion cell network. We Schrodinger Equation posit the loss of visual information despite the decrease in network size is minimized via compressed sensing. Using The Finite-Difference Time-Domain (FDTD) method is a an idealized mathematical model of the retina and a mean- well-known technique for the analysis of quantum devices. field analytical reduction, we demonstrate firing patterns It solves a discretized Schrodinger equation in an explicitly among ganglion cells can be used to reconstruct input im- iterative process. In this research, we apply the idea of the ages. FDTD method to the development of a numerical method for solving the time dependent nonlinear Schrodinger equa- Victor Barranca tion. The scheme is shown to satisfy the discrete energy Rensselaer Polytechnic Institute conservation laws. Finally, the scheme is tested by simu- [email protected] lating the propagation and formation of solitons.

Gregor Kovacic Weizhong Dai Rensselaer Polytechnic Inst College of Engineering and Science Dept of Mathematical Sciences Louisiana Tech University [email protected] [email protected]

David Cai Frederick Moxley New York University Louisiana tech University Courant institute [email protected] [email protected] PP1 PP1 FifthOrderEvolutionEquationforLongWaveDis- Computational Reductions and Dynamics of Neu- sipative Solitons ronal Models with Adaptation Current Solitary waves may arise in systems with dissipation and The stiffness and complexities associated with the nonlin- instability if a balance between these effects and nonlinear- earities of the Hodgkin-Huxley neuron model have mo- ities exists. We search for a consistent model equation to tivated a wide array of computational reductions. The describe long wave dissipative solitons including fifth or- dynamical systems underlying these reductions, however, der dispersion. The equation found includes quadratic and rarely generalize to neurons containing realistic adaptive cubic nonlinearities. For certain parameter values the bi- ionic currents. Using several numerical simulations and furcation from the basic state is subcritical. In this case associated measures of robustness, we demonstrate that we show that periodic solutions in a small box do not blow our novel modeling algorithm accomplishes the goals of at- up in finite time. taining accuracy and efficiency while retaining enough gen- Cristina Depassier, Juvenal Letelier erality to replicate the nontrivial dynamics of specialized Departamento de Fisica neurons. Universidad Catolica de Chile Victor Barranca [email protected], [email protected] Rensselaer Polytechnic Institute [email protected] PP1 Feigenbaum Universality, Lyapunov Exponents and Gregor Kovacic Fractal Dimensions in Two Dimensional Chaotic Rensselaer Polytechnic Inst Models Dept of Mathematical Sciences [email protected] This paper highlights the following objectives in some two dimensional chaotic models: (i) Sophisticated numerical David Cai methods have been developed to determine Feigenbaum New York University bifurcation tree leading to chaos, (ii)Determination of Lya- Courant institute punov exponents, Correlation, box- counting and informa- [email protected] tion dimensions as measure of chaos (iii) Statistical tools are employed to confirm the results .

PP1 Tarini K. Dutta Simultaneous Frequency Conversion, Regeneration PROFESSOR OF MATHEMATICS, GAUHATI and Reshaping of Optical Signals UNIVERSITY, GUWAHATI, ASSAM STATE: 781014, INDIA Nondegenerate four-wave mixing in fibers enables the tun- [email protected] able and low-noise frequency conversion of optical signals. This poster shows that four-wave mixing driven by pulsed pumps can also regenerate and reshape optical signal pulses PP1 arbitrarily. Confinement of the Eigenvalues of the Ablowitz- Daniel Cargill NW12 Abstracts 87

Ladik Eigenvalue Problem. PP1 Random and Regular Dynamics of Stochastically Klaus and Shaw (2001) proved that if initial conditions for Driven Neuronal Networks the Nonlinear Schr¨odinger equation were real and single- lobe, the eigenvalues of the Zakarov-Shabat system should Dynamical properties of and uncertainty quantification in be imaginary. To the knowledge of the authors, there is no Integrate-and-Fire neuronal networks with multiple time analogous to discrete problems. The natural extension for scales of excitatory and inhibitory neuronal conductances this result is that if the single-lobe property was satisfied driven by random Poisson trains of external spikes will be by the potentials of the Ablowitz-Ladik (AL) eigenvalue discussed. Both the asynchronous regime in which the net- problem, then the eigenvalues should be real. We show an work spikes arrive at completely random times and the extension of Klaus-Shaw potential for the AL lattice. synchronous regime in which network spikes arrive within periodically repeating, well-separated time periods, even J. Adrian Espinola-Rocha though individual neurons spike randomly will be pre- Dept. of Mathematics sented. Centro de Investigaci´on en Matem´aticas, A.C. [email protected] Pamela B. Fuller Rensselaer Polytechnic Institute Patrick Shipman [email protected] Department of Mathematics Colorado State University [email protected] PP1 Vaccinating Against HPV in Dynamic Network Stephen P. Shipman Department of Mathematics We develop a dynamical network model to examine the rel- Louisiana State University ative merits of strategies for vaccinating women against the [email protected] sexually transmitted Human Papillomavirus, which can in- duce cervical cancer. The model community is represented as a sexual network of individuals with links dynamically PP1 created and destroyed through statistical rules based on On the Nonlinear Schrodinger Equation with Em- the node characteristics. Various strategies for distribut- bedded Eigenvalues ing an allotted number of doses of vaccine are tested for effectiveness in reducing the incidence of cervical cancer. We revisit the inverse scattering transform for the one- dimensional focusing nonlinear Schrodinger equation with Pamela B. Fuller, Toni Wagner, Mark Yuhas zero boundary conditions at infinity. Explicitly, we study Rensselaer Polytechnic Institute cases in which the analytic scattering coefficients admit ze- [email protected], [email protected], ros on the real axis. We present specific examples and we [email protected] discuss the appropriate formulation of the inverse prob- lem, issues of existence and uniqueness and the long-time Peter R. Kramer behavior of the solutions. Rensselaer Polytechnic Institute Department of Mathematical Sciences Emily Fagerstrom [email protected] State University of New York at Buffalo emilyrf@buffalo.edu PP1 Gino Biondini A Hamiltonian Water Wave Model with Horizon- State University of New York at Buffalo tally Sheared Currents Department of Mathematics biondini@buffalo.edu We will show the Hamiltonian dynamics of a new, vari- ational water wave model of Cotter and Bokhove (2010). This model has a threedimensional velocity ?eld consisting PP1 of the full threedimensional potential ?eld plus horizontal Travelling Waves of a Reaction-Diffusion Model for velocity components, such that the vertical component of the Acidic Nitrate-Ferroin Reaction vorticity is nonzero. We aim to augment the new model locally with bores and will discuss how our existing varia- We consider a system of two reaction-diffusion equations, tional, potential-flow ?nite element model can be extended which was derived to model the acidic nitrate-ferroin re- to include bores. action. The existence and stability of travelling waves for this system are investigated. The proofs for the obtained Elena Gagarina, Onno Bokhove results are rigorous. University of Twente [email protected], [email protected] Sheng-Chen Fu Department of Mathematical Sciences National ChengChi University PP1 [email protected] Many-Body Interaction in Fast Collisions of Nls Solitons

Je-Chiang Tsai We study the effects of fast collisions between NLS solitons Department of Mathematics in the presence of weak nonlinear loss − |ψ|2mψ.Weshow National Chung Cheng University (Taiwan) that for m ≥ 2, n-body interaction with m +1 ≥ n ≥ [email protected] 3 gives a significant contribution to the collision-induced 88 NW12 Abstracts

amplitude shift. We characterize the dependence of this from geometric singular perturbation theory and geometric contribution on m, n, the group velocity difference, and desingularization. Second, we show that this front is a the initial soliton positions. pushed front despite the fact that it propagates slower than the linear spreading speed. Avner Peleg State University of New York at Buffalo Matt Holzer apeleg@buffalo.edu School of Mathematics University of Minnesota Yeojin Chung [email protected] Southern Methodist University [email protected] Arnd Scheel University of Minnesota Paul Glenn [email protected] State University of New York at Buffalo paulglen@buffalo.edu PP1 Quan Nguyen Stability and Dynamics of Solitary Waves in Bec Vietnam National University at Ho Chi Minh city Spinor Lattices [email protected] The work presented in this talk focuses on understanding solitary waves in a spinor BEC lattice system. This sys- PP1 tem is motivated by the spinor BEC which can be described by a quasi-one dimensional model. Here, we discuss two- Dynamics of a Mean Field Cortex Model and three-component dynamical lattice which contains a We study a PDE model of the membrane potential and mean field nonlinearity. Our analysis of solitary waves in- synaptic interaction of excitatory and inhibitory neuron volves (i) an examination of the anti-continuum limit for populations in a two-dimensional slab of cortical tissue. our model of interest, (ii) the existence and stability of Considering the equations as an autonomous dynamical these solitary waves via a perturbative approach and (iii) system, we aim to parse the model using bifurcation anal- understanding the structure of these waves in excited sites ysis. We compute equilibria, waves, and periodic patterns, of the lattice. and study their dependence on external input, connectiv- Rudy L. Horne ity parameters and the domain size. Our computations are Department of Mathematics done using open source code built on PETSc. Morehouse College Kevin R. Green, Lennaert van Veen [email protected] UOIT [email protected], [email protected] Hadi Susanto The University of Nottingham United Kingdom PP1 [email protected] Pattern Competition in Spatially-Forced Systems Nathan Whitaker Spatial periodic forcing of pattern-forming systems is an University of Massachusetts Amherst important, but lightly studied, method of pattern control. [email protected] Forcing can be used to control the wavenumber of one- dimensional periodic patterns, to increase pattern ampli- Panos Kevrekidis tudes, to stabilize unstable patterns, or to induce patterns University of Massachussets below instability onset. We show how in one spatial di- [email protected] mension forcing acts to reinforce patterns, while in two dimensions it acts to destabilize or displace them by pro- ducing two-dimensional rectangular and oblique patterns. PP1 Weakly Subcritical Stationary Patterns: Eckhaus Aric Hagberg Instability and Homoclinic Snaking Los Alamos National Laboratory The transition from subcritical to supercritical station- [email protected] ary periodic patterns is described by the one-dimensional cubic-quintic Ginzburg-Landau equation At = μA + Axx + 2 2 ∗ 2 4 Yair Mau, Ehud Meron i(a1|A| Ax + a2A Ax)+b|A| A −|A| A,whereA(x, t)rep- Ben-Gurion University resents the pattern amplitude and the coefficients μ, a1, [email protected], [email protected] a2 and b are real. The conditions for Eckhaus instabil- ity of periodic solutions are determined, and the resulting spatially modulated states are computed. Some of these PP1 evolve into spatially localized structures in the vicinity of A Slow Pushed Front in a Lotka-Volterra Compe- a Maxwell point, while others resemble defect states. The tition Model results are used to shed light on the behavior of localized structures in systems exhibiting homoclinic snaking during We study the existence and stability of a traveling front the transition from subcriticality to supercriticality. in the Lotka-Volterra competition model when the rate of diffusion of one species is small. This front is noteworthy Hsien-Ching Kao in two respects. First, we show that it is the selected, University of California at Berkeley, Department of or critical, front for this system. We utilize techniques Physics NW12 Abstracts 89

[email protected] Queen’s University Belfast Centre for Plasma Physics (CPP) Edgar Knobloch [email protected] University of California at Berkeley Dept of Physics Vikrant Saxena [email protected] Centre for Plasma Physics (CPP) Queen’s University Belfast, UK [email protected] PP1 Localized Structures in Rotating Convection Amita Das Institute for Plasma Research Convection in a horizontal layer heated from below and Bhat, Gandhinagar, India rotating about the vertical is studied. The system admits [email protected] 2D spatially localized structures. Such structures are orga- nized in the form of slanted snaking and are present over a large range of Rayleigh numbers, regardless of the di- Abhijit Sen rection of branching of periodic convection. This behavior Institute for Plasma Research is a consequence of a conserved quantity. The results are [email protected] compared with predictions based on a fifth order amplitude equation. This is joint work with C Beaume (IMFT), A Predhiman Kaw Bergeon(IMFT)andH-CKao(UCBerkeley). Institute for Plasma Research Bhat, Gandhinagar, India Edgar Knobloch [email protected] University of California at Berkeley Dept of Physics [email protected] PP1 Synchronous and Asynchronous Dynamics of Com- plex Neuronal Networks PP1 Gain-Controlled Soliton Routing in Dissipative For integrate-and-fire neuronal networks with complex con- Lattices. nectivity topology, we study the dependence of their pulse rate on the underlying architectural connectivity statistics. We demonstrate an effective mechanism for gain-controlled We derive the distribution of the firing rate from this de- soliton routing in dissipative lattices. An effective particle pendence and determine when the underlying scale-free ar- model reveals the essential features of soliton dynamics in chitectural connectivity gives rise to a scale-free pulse-rate such structures. The results presented are directly applica- distribution. We identify the scaling of the pairwise cou- ble to general inhomogeneous systems where interplay be- pling between the dynamical units in this network class tween gain and loss mechanisms occurs. Focusing on the that keeps their pulse rates bounded in the infinite-network complexity of soliton dynamics rather than on the com- limit. plexity of their profiles we have analyzed scenarios that are promising for dynamical soliton control concepts and Gregor Kovacic applications. Rensselaer Polytechnic Inst Dept of Mathematical Sciences Yannis Kominis [email protected] School of Electrical and Computer Engineering National Technical University of Athens [email protected] PP1 Spatiotemporal Dynamics in Three Species Model Sotiris Droulias, Panagiotis Papagiannis Systems with Self Diffusion School of Electrical and Computer Engineering, National Technical University of Athens Input your abstract, including TeX commands, here. In [email protected], [email protected] this paper, the complex dynamics of two types of tri- trophic food chain model systems modeling two real sit- uations of marine ecosystem have been investigated in the Kyriakos Hizanidis presence of diffusion, both analytically and through numer- National Technical University, Greece ical simulations. These models have been studied earlier, [email protected] but in the absence of diffusion. The numerical simula- tion leads to spontaneous and interesting pattern forma- PP1 tion. Diffusion driven analysis is carried out and effect of diffusion on the chaotic dynamics of the model systems are Mutual Interactions Between Electromagnetic Soli- studied. The existence of chaotic attractor and long term tons in Plasmas chaotic behavior demonstrate the effect of diffusion on the We numerically investigate the mutual interactions be- dynamics of the model systems tween two electromagnetic solitons in a cold unmagnetized Nitu Kumari plasma, both for finite and vanishing group speeds of the Indian Institute of Technology Mandi, Himachal Pradesh interacting structures. The interaction of two spatially India, 175001 overlapping standing electromagnetic solitons is studied for [email protected], [email protected] partial as well as complete overlap and the corresponding end states are compared. Ioannis Kourakis 90 NW12 Abstracts

PP1 PP1 Localized Patterns on the Surface of a Magnetic Bistable Fronts in Discrete Inhomogeneous Media Fluid Bistable differential-difference equations with inhomoge- The discovery of two-dimensional solitons on the surface of neous diffusion are considered using McKean’s caricature a ferrofluid under a vertical magnetic field by Richter and of the cubic. Front solutions are constructed for essentially Barashenkov (PRL, 2005) has raised many new theoretical arbitrary inhomogeneous discrete diffusion and these solu- questions. Due to the complicated nature of the equations tions correspond, in the case of homogeneous diffusion, to modelling ferrofluids, Richter and Barashenkov proposed a monotone traveling front solutions, or stationary front solu- conservative analogue of the Swift-Hohenberg equation tions in the case of propagation failure. Explicit conditions reveal relationships between zero wave speed and defects in the medium, and changes in wave speed and shape are 2 2 3 utt = −(1 + Δ) u − νu + μu − u , (1) analyzed as fronts propagate. Brian E. Moore University of Central Florida as a phenomenological model that could provide an un- u USA derstanding of localised axisymmetric solitons. Here, = [email protected] u(x, y, t) describes the surface height, ν is related to the strength of the magnetic field (and is the usual control parameter) and μ is related to the permeability of the Tony R. Humphries ferrofluid. Here, we review what is known about station- McGill University ary (time-independent) solutions of the equation above and Mathematics and Statistics show where this phenomenological model is (un)successful [email protected] in predicting the qualitative nature of localised patterns on the surface of a ferrofluid. Furthermore, we show nu- Erik Van Vleck merically that homoclinic snaking does exist in the full Department of Mathematics Ferrofluid equations. University of Kansas [email protected] David Lloyd University of Surrey [email protected] PP1 Numerical Simulation of Internal Wave Generation from a Collapsing Mixed Region in Stratified Fluid PP1 Transition Between Slow and Fast Wavefronts from The dynamics of a density perturbed fluid into an ambient the Point of View of Speed Selection fluid with complex stratification is examined by numeri- cal simulations. Algorithms based on the Navier-Stokes On an example of a system of coupled reaction-diffusion equations in the primitive variables and on the Eulerian- equations, we discuss a sudden transition between slow and Lagrangian coordinate system and moving grid are utilized. fast wavefronts. We relate the mechanism of such transi- High-order, monotone approximation of convective terms tion to the known facts about speed selection for fronts. are used. Parametric calculations demonstrated the ap- plicability of the numerical models. The analysis focuses Peter Mclarnan upon the generation and propagation of internal waves and Miami University on the dynamics of collapsing region. [email protected] Nikolay Moshkin Anna Ghazaryan Suranaree University of Technology Department of Mathematics [email protected], [email protected] Miami University and University of Kansas [email protected] Gennadii Chernykh, Andrey Zudin Institute of Computational Technologies, Russia Christopher Jones [email protected], [email protected] University of North Carolina and University of Warwick [email protected] PP1 Synchrony in Stochastic Pulse-Coupled Neuronal PP1 Network Models Oscillons Near Forced Hopf Bifurcations We are interested in the synchronous dynamics of simple Oscillons are planar, spatially localized, temporally oscil- pulse-coupled models for neuron dynamics. These time lating, radially symmetric structures. We present a proof correlations in firing times are seen experimentally. In our of the existence of oscillons in the forced planar complex model, the size of synchronous firing events depends on the Ginzburg-Landau equation through a geometric blow-up probabilistic dynamics between such events as well as the analysis. Our analysis is complemented by a numerical con- network structure representing the neuron connections. We tinuation study of oscillons in the forced Ginzburg-Landau presents both analytical results and numerical simulations equation using Matlab and AUTO. of these global dynamics. Kelly Mcquighan, Bjorn Sandstede Katherine Newhall Brown University Courant Institute of Mathematical Science kelly [email protected], bjorn [email protected] New York University [email protected] NW12 Abstracts 91

Aaditya Rangan lutions of the EBBM-equation are not known analytically, Courant Institute of Mathematical Sciences they are generated and investigated numerically. It tran- New York University spires that there can be as many as three stability regimes [email protected] for the EBBM-solitary waves. We present numerical simu- lations of the formation and long-time evolution of solitary waves, the behavior of solitary waves under amplitude per- PP1 turbations and the interaction of solitary waves. Minimal An Improved Local Well-Posedness Result for the perturbations necessary to force a solitary wave to change Periodic ”Good” Boussinesq Equation. stability regimes are also determined. We prove that the ”good” Boussinesq model with the peri- Sanja Pantic odic boundary condition is locally well-posed in the space University of Illinois at Chicago H s × H s−2 for s ≥−3/8. In the proof, we employ the [email protected] normal form approach, which allows us to explicitly ex- tract the rougher part of the solution. This also leads to the conclusion that the remainder is in a smoother space PP1 C([0; T ]; H s+a), where 0 ≤ a−1/4. Soliton dynamics in a large variety of longitudinally mod- ulated lattices are studied in terms of phase-space analy- Seungly Oh, Atanas Stefanov sis for an effective particle approach and direct numerical University of Kansas simulations. A rich set of qualitatively distinct dynami- [email protected], [email protected] cal features of soliton propagation that have no counter- part in longitudinally uniform lattices is illustrated. This set includes cases of enhanced soliton mobility, dynamical PP1 switching, extended trapping in several transverse lattice Effects of Stochastic Perturbations on the Atlantic periods, and quasiperiodic trapping, which are promising Thermohaline Circulation for soliton control applications. The Atlantic thermohaline circulation (THC) transports Panagiotis Papagiannis large amounts of heat from the equatorial region north- School of Electrical and Computer Engineering, ward toward the polar regions and is responsible for the National Technical University of Athens relatively mild climate in the north Atlantic. Various stud- [email protected] ies have shown the Atlantic THC can have multiple stable equilibria each representing very different climatic states. Yannis Kominis Stochastic ODEs are derived from simple box models and School of Electrical and Computer Engineering we will show how stochastic perturbations such as freshwa- National Technical University of Athens ter influx can produce transitions between these equilibria. [email protected]

Alejandro Aceves Kyriakos Hizanidis Southern Methodist University National Technical University, Greece [email protected] [email protected]

Alyssa Pampell Sotiris Droulias Department of Mathematics School of Electrical and Computer Engineering, Southern Methodist University National Technical University of Athens [email protected] [email protected]

PP1 PP1 The Extension of the Generalized Regularized Point vortex equilibria and Calogero-Sutherland Long Wave Equation equation The Regularized Long-Wave equation, also known as the We consider linear evolution flows on the space of polyno- Benjamin-Bona-Mahony (BBM)-equation was first studied mials and compute the associated dynamics of the polyno- as a model for small-amplitude long waves that propagate mial roots . This leads to a set of equations for the root on the free surface of a perfect fluid. As an alternative to dynamics, whose equilibria are used to compute relative the Korteweg-de Vries equation, it features a balance be- equilibria for the point vortex problem in 2-D fluids. We tween nonlinearity and a frequency dispersion term that make use of properties of the Calogero-Sutherland model allows the existence of traveling waves that are smooth in two variables to construct novel point vortex confingu- and symmetric about their maximum. Such waves decay rations. rapidly to zero on their outskirts and, because of their Ronald Perline tendency to travel alone, are known as ‘solitary waves’. Drexel University We investigate the behavior of solitary-wave solutions for [email protected] the Extended BBM (EBBM)-equation which is the BBM- equation, but with two power nonlinearities in general, gra- dient form. We prove that this model is globally well-posed, which provides a rigorous foundation for the stability the- ory of their solitary-wave solutions. Since solitary-wave so- 92 NW12 Abstracts

PP1 School of Applied Mathematical and Physical Sciences Snakes and Ladders: Stability of Fronts and Pulses National Technical University of Athens [email protected] The subject of this poster are localized patches composed of spatially periodic structures. These patterns often exhibit Dimitri Frantzeskakis snaking: in parameter space, the localized states lie on a University of Athens vertical sine-shaped bifurcation curve so that the width of [email protected] the underlying periodic pattern increases as one moves up along the bifurcation curve. The issue addressed here is the Floyd Williams stability of these structures as a function of the bifurcation Department of Mathematics and Statistics parameter. University of Massachusetts, Amherst Bjorn Sandstede [email protected] Brown University bjorn [email protected] Avadh Saxena Los Alamos Natl. Lab. [email protected] PP1 On the Spectral Stability of Soltion Solutions of the Atanas Stefanov Vector Nonlinear Schr¨odinger Equation University of Kansas [email protected] We consider a system of coupled cubic nonlinear Schr¨odinger (NLS) equations Theodoros Horikis n ∂ψ ∂2ψ  Department of Mathematics i j − j α |ψ |2ψ j , ,...,n University of Ioannina ∂t = ∂x2 + jk k j =1 2 k=1 [email protected] α α where jk = kj are real. The stability of solitary wave Nathaniel Whitaker solutions is examined numerically using Hill’s method (De- Department of Mathematics and Statistics coninck, B. and J. N. Kutz. J. Comput. Phys., 219: 296- University of Massachusetts, Amherst 321, 2006). Our results extend the analysis of Nguyen [email protected] (Commun. Math. Sci., 9: 997-1012, 2001.) to incorpo- rate solitons with nontrivial phase profile. Panos Kevrekidis Natalie Sheils University of Massachussets Department of Applied Mathematics [email protected] University of Washington [email protected] PP1 Bernard Deconinck Global Existence for a System of Schr¨odinger Equa- University of Washington tions with Power-Type Nonlinearities [email protected] We consider the Cauchy problem for a Schr¨odinger system with power-type nonlinearities Nghiem Nguyen, Rushun Tian Utah State University [email protected], [email protected]   i ∂ u u m a |u |p|u |p−2u , ∂t j + j + k=1 jk k j j =0 uj (x, 0) = ψj0(x), PP1 On Model of Short Pulse Type in Continuous Me- dia N N where uj : R × R → C,ψj0 : R → C for j =1, 2, ..., m We consider short pulse propagation in nonlinear metama- and ajk = akj are real numbers. Global existence is first 3 ≤ p< 2 terials characterized by a weak Kerr-type nonlinearity in established for 2 1+ N . Then we deduce a sharp their dielectric response. We derive two short-pulse equa- form of vector-valued Gagliardo-Nirenberg inequality and tions (SPEs) for the high- and low-frequency band for 1 di- use it to prove global existence in the critical case for small mensional case. Then we generalize this into 2 dimensional initial data. At the end, we consider the stability of solu- case and give a 2D version of the short pulse equation. For tions in the critical case. 1D soutions, we will discuss the connection with the soli- ton solution of the nonlinear Schr¨odinger equation, also we Rushun Tian, Nghiem Nguyen will discuss the robustness of various solutions emanating Utah State University from the sine-Gordon equation and their periodic general- [email protected], [email protected] izations. For the 2D short pulse equation, we will discuss several possible forms and their solutions. Bernard Deconinck University of Washington Yannan Shen [email protected] Department of Mathematics and Statistics University of Massachusetts, Amherst Natalie Sheils [email protected] Department of Applied Mathematics University of Washington Nikolaos Tsitsas [email protected] NW12 Abstracts 93

PP1 different from the FitzHugh-Nagumo equations. A Single Equation Describing Free Surface Water Waves. Je-Chiang Tsai Department of Mathematics Zakharov (1968) introduced canonical coordinates for the National Chung Cheng University (Taiwan) free surface water wave equations. Using the same canoni- [email protected] cal coordinates on the formulation of the water wave prob- lem due to Ablowitz, Fokas and Musslimani (2006), we Wenjun Zhang derive a single nonlinear complex equation describing the Department of Mathematics dynamics of the free surface. In contrast to Zakharov’s University of Auckland equation, our equation is not restricted to waves of small [email protected] amplitude. The properties of this equation are examined. Vivien Kirk, James Sneyd Olga Trichtchenko University of Auckland Department of Applied Mathematics [email protected], [email protected] University of Washington [email protected] PP1 Bernard Deconinck Behaviour of Torsional Surface Wave in a Homoge- University of Washington neous Substratum over a Dissipative Half Space [email protected] The propagation behaviour of Torsional surface wave in a homogeneous isotropic layer lying over a viscoelastic half PP1 space has been taken into account. Numerical result has Numerical Inverse Scattering: Uniform Approxi- been obtained to show the effect of internal friction, rigidity mation of Solutions of Integrable Pdes and wave number on phase velocity of Torsional surface wave. Dispersion equation reduced in classical form when Riemann–Hilbert problems arise in many different applica- derived as a particular case. Graphical user interface (GUI) tions, ranging from integrable systems and special function has been developed using MATLAB to generalize the effect theory to card-shuffling and random matrix theory. A re- of various parameter discussed. cent collocation method allows for the numerical solution of Riemann–Hilbert problems. In this poster I will lay out Sumit K. Vishwakarma, Shishir Gupta how to use this method to, in effect, solve integrable equa- Indian School of Mines, Dhanbad, Jharkhand, tions on the whole line for arbitrary spectral data. Unifor- India-826004 mity statements about the approximation can be proved. [email protected], shishir [email protected] Additionally, this can be used to examine the validity of the known asymptotic formulas. PP1 Thomas D. Trogdon Propagation Failure in Discrete Periodic Media Department of Applied Mathematics University of Washington We study the periodic lattice dynamical systems with [email protected] bistable nonlinearity. We use Moser’s Theorem to show that there exist infinitely many stationary solutions when Sheehan Olver one of the migration coefficients is sufficiently small. More- University of Oxford over, we prove the propagation failure occurs when both UK migration coefficients are sufficiently small. [email protected] Chin-Chin Wu Department of Applied Math., National Chung Hsing Bernard Deconinck University University of Washington [email protected] [email protected] Shih-Gang Liao PP1 National Chung Hsing University Traveling Waves in a Simplified Model of Calcium [email protected] Dynamics We analyze traveling wave propagation in a simplified PP1 model of intracellular calcium dynamics. Despite its sim- Grain Boundaries in Swift-Hohenberg Equations plicity, the model is thought to capture fundamental fea- tures of wave propagation in calcium models. We explore We investigate grain boundaries in the Swift-Hohenberg aspects of the dynamics of traveling front, pulse and pe- equation posed in the plane. In dimension 2, we show the riodic wave solutions as J, a parameter in our model, is existence of grain boundaries—stationary solutions which varied. We focus on the closed-cell version of the model, are even in x, periodic in y and approach rotated members which corresponds to a singular limit of the full (open-cell) of the family of stationary periodic patterns as x goes to in- model. We use our results about the closed-cell model to finity. Grain boundaries in this spatially extended system make conjectures about the nature of wave solutions in resembles defects commonly seen in physical experiments. the open-cell version of the model. A comparison between We employ center manifold reduction and normal form the- the properties of wave solutions of the calcium model and ory to reduce the PDE to a finite-but high-dimensional wave solutions of the FitzHugh-Nagumo equations reveals ODE, in which we can locate heteroclinic orbits. Those cor- that the calcium model is an excitable system essentially respond to grain boundaries in the Swift-Hohenberg equa- 94 NW12 Abstracts

tion.

Qiliang Wu, Arnd Scheel University of Minnesota [email protected], [email protected]

PP1 Stationary Nonlinear Modes and Beam Amplifica- tion in PT -Symmetric Harmonic Potential We report a number of unusual properties of stationary modes in the nonlinear Schroedinger equation with PT - symmetric harmonic potential x2 − 2iαx.Inparticular, the modes, bifurcating from different eigenstates of the un- derlying linear problem, can actually belong to the same family of nonlinear modes. By proper adjustment of the coefficient α it is possible to enhance stability of nonlinear modes comparing to the case of real harmonic potential. Linear and nonlinear dynamics of the modes is also dis- cussed. Dmitry Zezyulin Centro de Fsica Te´orica e Computacional Universidade de Lisboa [email protected]

PP1 Multiple Solutions of Poisson-Nernst-Planck(pnp) Systems for Ion Channels we consider a one-dimensional PNP model for ionic flow through ion channels for two ion species with permanent charges. The PNP model problem can be viewed as a boundary value problem of a singularly perturbed system and the existence of solutions is reduced to that of an alge- braic system. Multiple solutions are shown to exist, under some conditions, through bifurcation analysis and numer- ical computations are consistent with our analysis. Exis- tence of multiple solutions in such or similar models might be relevant to some complex behaviors of ion channels. Mingji Zhang Department of Mathematics, University of Kansas [email protected]

Weishi Liu Department of Mathematics University of Kansas [email protected]