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58 NW14 Abstracts 58 NW14 Abstracts IP1 Mathematics and Engineering Physics Stability and Synchrony in the Kuramoto Model University of Wisconsin, Madison [email protected] The phenomenon of the synchronization of weakly coupled oscillators is a old one, first been described by Huygens in his “Horoloquium Oscilatorium.’ Some other exam- IP4 ples from science and engineering include the cardiac pace- Coherent Structures in 2D Turbulence: Identifica- maker, the instability in the Millenium Bridge, and the tion and their Use synchronous flashing of fireflies. A canonical model is the Kuramoto model Ideas from dynamical systems have recently provided fresh insight into transitional fluid flows. Viewing such flows as dθ i = ω + γ sin(θ − θ ) a trajectory through a phase space littered with coherent dt i j i j structures (meaning ’exact’ solutions here) and their sta- ble and unstable manifolds has proved a fruitful way of We describe the fully synchronized states of this model understanding such flows. Motivated by the challenge of together with the dimensions of their unstable manifolds. extending this success to turbulent flows, I will discuss how Along the way we will encounter a high dimensional poly- coherent structures can be extracted directly from long- tope, a Coxeter group, and a curious combinatorial iden- time simulations of body-forced turbulence on a 2D torus. tity. This leads to a proof of the existence of a phase tran- These will then be used to ’postdict’ certain statistics of sition in the case where the frequencies are chosen from as the turbulence. iid Random variables. Rich Kerswell Jared Bronski Department of Mathematics University of Illinois Urbana-Champaign University of Bristol, U.K. Department of Mathematics [email protected] [email protected] IP5 IP2 A Neural Field Model of Binocular Rivalry Waves Integrability: Initial- Boundary Value Problems in 1+1, and Solitons in 3+1 Binocular rivalry is the phenomenon where perception switches back and forth between different images presented A review will be presented of the two most challenging to the two eyes. The resulting fluctuations in perceptual problems in the analysis of integrable nonlinear evolution dominance and suppression provide a basis for non-invasive equations: (a) initial- boundary value problems in 1+1, i.e. studies of the human visual system and the identification evolution equations in one spatial variable, and (b) the con- of possible neural mechanisms underlying conscious visual struction and solution of equations in 3+1. Regarding (a), awareness. In this talk we present a neural field model of it will be shown that the so- called unified transform, which binocular rivalry waves in visual cortex continuum neural provides the proper generalization of the inverse scattering fields are integro-differential equations that describe the transform, yields effective formulae for the large t- asymp- large-scale spatiotemporal dynamics of neuronal popula- totics of a variety of physically significant problems; this tions. We derive an analytical expression for the speed of includes problems on the half-line with t-periodic boundary a binocular rivalry wave as a function of various neurophys- conditions. Regarding (b), an integrable generalization of iological parameters, and show how properties of the wave the Davey-Stewardson equation 4+2, i.e. four spatial and are consistent with the wave-like propagation of perceptual two time variables, will be presented. A reduction of this dominance observed in recent psychophysical experiments. equation to an equation in 3+1 will also be discussed. Fur- We then analyze the effects of extrinsic noise on wave prop- thermore, one and two soliton solutions for both the 4+2 agation in a stochastic version of the neural field model. We and 3+1 Davey-Stewardson equations will be presented. end by describing recent work on rotating rivalry stimuli Thanasis Fokas and direction selectivity. Department of Applied Mathematics and Theoretical Physics Paul C. Bressloff University of Cambridge, UK University of Utah and University of Oxford, UK [email protected] Department of Mathematics bressloff@math.utah.edu IP3 IP6 Minimal Models for Precipitating Convection Engineering Extreme Materials with Defects and Despite the importance of organized convection (e.g. squall Nonlinearity lines and hurricanes), numerical simulations remain a chal- lenge. Comprehensive cloud resolving models would in- We study the fundamental dynamic response of discrete clude water vapor, liquid water and ice, and liquid wa- nonlinear systems and study the effects of defects in the ter would be separated into cloud water and rain water. energy localization and propagation. We exploit this un- Here we take a minimalist approach to cloud microphysics derstanding to create experimentally novel materials and by incorporating fast auto-conversion from cloud to rain devices at different scales (for example, for application in water, and fast condensation and evaporation. Numeri- energy absorption, acoustic imaging and energy harvest- cal and analytical results will be discussed, e.g., saturated ing). We use granular systems as a basic platform for test- regions have a linear stability boundary associated with ing, and control the constitutive behavior of the new ma- parcel (finite-amplitude) stability in unsaturated regions. terials selecting the particles geometry, their arrangement and materials properties. Ordered arrangements of par- Leslie Smith ticles exhibit a highly nonlinear dynamic response, which NW14 Abstracts 59 has opened the door to exciting fundamental physical ob- [email protected] servations (i.e., compact solitary waves, energy trapping phenomena, and acoustic rectification). This talk will fo- cus on energy localization and redirection in one- and two- CP1 dimensional systems. The Rogue Wave Solutions of the Kpi Chiara Daraio Rogue wave solutions of integrable partial differential equa- ETH Z¨urich, Switzerland and California Institute of Tech tions are one kind of large-amplitude doubly localized ratio- [email protected] nal solutions. Recently, the first-order rogue wave solution of the NLS has been observed in water tanks and optical fibers. In this talk, we shall provide new rogue wave so- lutions of a well-known (2+1)- dimensional equation: the IP7 KPI, from the solutions two (1+1)-dimensional equations: Momentum Maps, Shape Analysis and Solitons NLS and complex mKdV. The main technique is the Dar- boux transformation. Much of this talk is based on work done with Jerry Mars- Jingsong He den (1942 - 2010) on shared geometric properties in the Ningbo University analysis of fluid flow and shape transformations. The talk [email protected] will discuss uses of geometric mechanics in the problem of registration of images, primarily in the example of planar closed curves. Many types of mathematics apply in this CP1 problem, including soliton theory and momentum maps. Mcc Type Strongly Nonlinear Internal Wave Mod- Some trade secrets will be revealed. els DarrylD.Holm We discuss MCC type strongly nonlinear internal long Imperial College London wave models in two- and three-layer fluid systems in de- [email protected] tail. Their dispersion relations and stability properties comparing to coupled Euler equations will be presented. In addition, generalized strongly nonlinear models are also IP8 considered. Non-stationary time evolution for their perfor- mance tests will be presented as well. Symmetry, Modulation and Nonlinear Waves Tae-Chang Jo Modulation underpins many facets of the analysis of non- Inha University linear waves and patterns. A new perspective on modula- [email protected] tion combined with symmetry and conservation laws will be presented, which leads to a mechanism for the emer- gence of model wave PDEs such as KdV, KP, Boussinesq CP1 and others. This combination results in simple geomet- Laboratory Experiments and Simulations for Soli- ric formulae for the coefficients. Generalizations to multi- tary Internal Waves with Trapped Cores phase wavetrains, systems of conservation laws, Whitham equations with dispersion, and a mechanism for the forma- Under appropriate conditions, solitary internal waves in tion of multi-pulse planforms in pattern formation will also the ocean and atmosphere can develop recirculating re- be presented. gions, known as “trapped cores’. Fundamental properties, essential for constructing and testing theoretical models, remain to be quantified experimentally. These include core Tom J. Bridges circulation and density, as well as mass and energy trans- University of Surrey port, and Richardson number. By using a new technique, [email protected] we measure these properties in detail. To examine larger waves, we perform simulations. The corresponding theory agrees closely with our experiments. SP1 Paolo Luzzatto-Fegiz Martin D. Kruskal Prize Lecture - Nonlinear Waves University of Cambridge from Beaches to Photonics [email protected] Based on work of Kruskal et al the Korteweg-deVries equa- Karl Helfrich tion was found to have many remarkable properties. These Woods Hole Oceanographic Institution properties are shared by many other equations including [email protected] the 2D Kadomtsev-Petvishvili (KP) equation, which arises in the study of shallow water waves. Some solutions of KP and their relation to waves on shallow beaches will be dis- CP1 cussed. Nonlinear waves in photonic periodic lattices will InternalWavesIncidentUponAnInterface also be described; photonic graphene is a remarkable case. Since this is an
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