Conference Non- linear optical and atomic systems: deterministic and stochastic aspects. January 21-25, 2013

Transcription

1 January 21-25, 2013 An introduction to numerical methods for Schrödinger equations. Xavier ANTOINE (Institut Elie Cartan Nancy (IECN), Université de Lorraine) The aim of this course is to give an introduction to a few numerical methods for solving Schrödinger-type equations. Most particularly, I will consider the following topics computation of stationary solutions to nonlinear Schrödinger equations. Applications to the Gross-Pitaevskii equation with rotating terms. numerical methods for the time dependent Schrödinger equation. Linear and nonlinear equations. The special case of Gross-Pitaevskii equations. domain truncation for Schrödinger equations: absorbing boundary conditions, perfectly matched layers. Linear and nonlinear equations. All along the course, I will provide some examples of code implementation for different problems. Most of the codes are freely available. Stochastic models in fiber optics Anne DE BOUARD (Centre de Mathématiques Appliquées, CNRS et Ecole Polytechnique), We will describe in this talk mathematical and numerical results concerning stochastic PDEs based on nonlinear Schrödinger equations with a white noise dispersion which models the evolution of the complex envelope of a light beam propagating in an optical fiber, in the presence of so called dispersion management. We will in particular show how the presence of the noise affects the mathematical properties of the model, and show that the conservative Crank-Nicolson scheme gives a better approximation than is usual for the numerical simulation of this stochastic equation. Solitonization of the Anderson localization. Claudio CONTI (Department of Physics, University Sapienza), We will review some results on the effect of nonlinearity on disorder induced localization, with special emphasis on nonlinear optics and photonics. We will report on a simple model for the theoretical description of Anderson states in the presence of nonlinearity, their affinities with bright solitary waves and the links with statistical mechanics of disordered systems. Cold atomic gases in 2D: From Kosterlitz-Thouless to Quantum Hall Physics. Jean DALIBARD (Laboratoire Kastler Brossel) In his world-famous novel "Flatland" published in 1884, the English writer Edwin Abbott imagined a social life in a two-dimensional world. With a very original use of geometrical notions, E. Abbott produced a unique satire of his own society. Long after Abbott's visionary allegory, microscopic physics has provided a practical path for the

2 exploration of low-dimensional worlds. With the realization of quantum wells for example, it has been possible to produce two-dimensional gases of electrons. The properties of these fluids dramatically differ from the standard three-dimensional case, and some of them are still lacking a full understanding. During the last decade, a novel environment has been developed for the study of low-dimensional phenomena. It consists of cold atomic gases that are confined in tailor-made electromagnetic traps. With these gases, one hopes to simulate and understand complex condensed-matter phenomena, such as the Kosterlitz-Thouless superfluid transition. The talk will discuss some aspects of this research, both from an experimental and a theoretical perspective. Connections with other domains of 2D many-body physics, such as the Quantum Hall phenomenon, will also be addressed. Numerical methods for computing vortex states in rotating Bose- Einstein condensates Ionut DANAILA (Laboratoire de mathématiques Raphaël Salem Université de Rouen) Vortex states in a rotating Bose-Einstein condensate are computed by numerically solving the stationary Gross-Pitaevskii (GP) equation in 2D and 3D. Different types of methods are used: (i) direct minimization of the GP energy functional using Newton-type methods or steepest descent methods based on Sobolev gradients and (ii) imaginary time propagation of the wave function. Advantages and drawbacks of each method are summarized and convergence properties are presented. Numerical setups using 6th order finite difference schemes and finite elements with mesh adaptivity are proved to be effective in computing difficult cases with quantized vortices. A rich variety of vortex arrangements (single-line vortex, Abrikosov lattice, giant vortex) is obtained using different trapping potentials, corresponding to real laboratory experiments performed at ENS Paris in the group of J. Dalibard. New configurations with arrays of condensates in 1D rotating optical lattices are also presented. Splitting scheme for a stochastic Gross-Pitaevskii equation Romain DUBOSCQ (Institut Elie Cartan Nancy (IECN), Université de Lorraine), In this talk, we investigate a time-splitting scheme to numerically solve a dynamic Gross-Pitaesvkii equation with a randomly perturbed potential. The random perturbation is modeled using a Wiener process. We provide the analysis of the convergence of the splitting scheme and show that, in our case, the order of the scheme is bounded below by 1/2. Furthermore, using numerical simulations, we will see that the order is in fact equal to 1. Numerical examples will be given for two-dimensional Gross-Pitaevskii equations. Extreme events in nature, randomness and rogue wave in optics. John DUDLEY (Department of Optics, Institut FEMTO-ST, University of Franche-Comté), Recent work in nonlinear fiber optics has demonstrated qualitative and quantitative links between instabilities in optical propagation and the giant destructive rogue or freak waves on the surface of the ocean. The analogy between the appearance of instabilities in optics and the rogue waves on the ocean's surface is both intriguing and attractive, as it opens up possibilities to explore the extreme value dynamics in a convenient benchtop optical environment. The purpose of this talk will be to discuss the results that have been obtained in optics, and to consider both the similarities and the differences with oceanic

3 rogue wave counterparts. The talk will review experimental work in this field and will cover rogue waves in supercontinuum generation and the formation of localized new classes of soliton on finite background. New applications relating to the generation of random numbers at arbitrary optical wavelengths will also be discussed. Dispersive shock waves in nonlinear Schrödinger flows. Gennady El (Department of Mathematical Sciences, Loughborough University), Dispersive shock waves (DSWs) are coherent unsteady nonlinear wave structures which provide dispersive regularization of hydrodynamic singularities in conservative media. The fundamental role of DSWs in such media is similar to that of viscous shock waves in classical gas and fluid dynamics. At the same time, DSWs are sharply distinct from their well-studied dissipative counterpart both in terms of physical significance and mathematical description. DSWs have recently attracted significant attention due to ground-breaking experiments in Bose-Einstein condensates and nonlinear optical media. The mathematical description of DSWs involves a synthesis of methods from hyperbolic quasi-linear systems, asymptotics, and soliton theory. The principal tool is nonlinear wave modulation theory, often referred to as Whitham averaging. In my talk I outline main mathematical ideas of the DSW theory and present some recent applications of this theory to superfluid flows and nonlinear optics systems described by the defocusing nonlinear Schrödinger equations. Nonlinear Waves in Localizing Systems. Sergej FLACH (New Zealand Institute for Advanced Study, Centre for Theoretical Physics and Chemistry, Massey University), Various extended media are capable of localizing waves, in particular in low dimensions. Examples are random potentials (Anderson localization), quasiperiodic potentials (Aubry-Andre localization), Wannier-Stark ladders (Bloch oscillations). Going from real to momentum space adds also the kicked quantum rotor (dynamical localization). Wave localization relies on the phase coherence of the waves. Experiments with ultracold atomic clouds in optical potentials, and light propagating through structured media, have confirmed that. When atom-atom interaction is added and is treated on a mean field level, or when light intensity is increased, the wave equations turn nonlinear. This has far reaching consequences, since nonlinearity may annihilate integrability, lead to deterministic chaos, and ultimately destroy phase coherence. Insulators turn into conductors, wave packet localization is destroyed, and a number of very intricate mathematical problems on how to connect to the linear wave theory pop up. I will give an introduction into this fascinating field, discuss and explain the main results, extend to quantum aspects, and discuss open problems. Introduction to stochastic processes and applications to Schrodinger equations Josselin GARNIER (Laboratoire de Probabilités et Modèles Aléatoires et Laboratoire Jacques-Louis Lions)

4 Shock waves in disorder media Silvia GENTILINI (ISC-CNR c/o Physics Department of Univerity of Rome La Sapienza), Dispersive shock waves (DSWs), or undular bores, are observed in nonlinear optics in systems described by universal models, such as the nonlinear Schrödinger equation, when the hydrodynamical approximation holds true. I will report on the experimental investigation of the effect of disorder on the formation and propagation of optical DSWs. Our experimental technique allows the visualization of a propagating Gaussian laser beam in a thermal defocusing medium in the presence of controllable disorder obtained by a colloidal dispersion of low index contrast dielectric particles. We show that, by increasing the strength of nonlinearity, the shock formation is enhanced, while, on the other hand, random light scatterers hamper and eventually inhibit the wave breaking phenomenon. We quantify such a competition by analyzing images of the laser beam along the propagation direction, and the far field distribution intensity: these allows to measure the relevant scaling laws relating the shock position with the input power and strength of disorder [1], and the wavevector spectrum generated by the shock and its relation to the underlying physics of the nonlocal nonlinearity [2]. References [1] N.Ghofraniha, S. Gentilini, V.Folli, E.DelRe, and C.Conti, Phys.Rev.Lett., to be published. [2] S.Gentilini, N.Ghofraniha, E.DelRe, and C.Conti, Opt.Exp., 20, (2012). Stability for the solitons of the one-dimensional Gross-Pitaevskii equation Philippe GRAVEJAT (Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique), We present two results in collaboration with F. Béthuel and D. Smets concerning the orbital stability of multi-solitons and the asymptotic stability of single solitons for the onedimensional Gross-Pitaevskii equation An overview of the KAM theorem for nonlinear PDEs Benoît GREBERT (Laboratoire de Mathématiques Jean Leray, Université de Nantes) I will present a quick overview of the KAM results proved in the context of nonlinear PDEs. In particular I will detail the recent result that I have obtained in collaboration with H. Eliasson and S. Kuksin for multidimensional PDEs. Dispersive blow-up for Schrödinger type equations Jean-Claude SAUT (Université Paris Sud) We review old and recent results on dispersive blow-up (by focusing on short or long waves) for various Schrödinger type equations (based on joint work with Jerry Bona and Christof Sparber).

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