The Nonlinear Schroedinger Equation with a random potential: Results and Puzzles

Abstract

The Nonlinear Schroedinger Equation (NLSE) with a random potential is motivated by experiments in optics and in atom optics and is a paradigm for the competition between the randomness and nonlinearity. The analysis of the NLSE with a random (Anderson like) potential has been done at various levels of control: numerical, analytical and rigorous. Yet, this model equation presents us with a highly inconclusive and often contradictory picture. We will describe the main recent results obtained in this field and propose a list of specific problems to focus on, that we hope will enable to resolve these outstanding questions.

21 pages, 4 figures

References in corpus (12)

• Destruction of Anderson localization by a weak nonlinearity
• Universal spreading of wavepackets in disordered nonlinear systems
• Anderson Localization of Expanding Bose-Einstein Condensates in Random Potentials
• Absence of Wavepacket Diffusion in Disordered Nonlinear Systems
• Rigorous Derivation of the Gross-Pitaevskii Equation
• The crossover from strong to weak chaos for nonlinear waves in disordered systems
• Possible experimental manifestations of the many-body localization
• Superfluidity versus Anderson localization in a dilute Bose gas
• Nonlinear waves in disordered chains: probing the limits of chaos and spreading
• KAM tori in 1D random discrete nonlinear Schrödinger model?
• Anderson localization or nonlinear waves? A matter of probability
• Eigenvalue repulsion estimates and some applications for the one-dimensional Anderson model

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• Slow dissipation and spreading in disordered classical systems: A direct comparison between numerics and mathematical bounds
• Statistical Properties of the one dimensional Anderson model relevant for the Nonlinear Schrödinger Equation in a random potential
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