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The nonlinear Schrödinger equation with a random potential: results and puzzles. (English) Zbl 1236.35166

Summary: The nonlinear Schrödinger 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 describe the main recent results obtained in this field and propose a list of specific problems to focus on, which we hope will enable to resolve these outstanding questions.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35R60 PDEs with randomness, stochastic partial differential equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
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