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42 NW12 Abstracts ∑ 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.
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