Linear vs. nonlinear effects for nonlinear Schrodinger equations with potential

Abstract

We review some recent results on nonlinear Schrodinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and the nonlinear effects, can be described quite precisely. This includes semi-classical regimes, as well as finite time blow-up and scattering issues. We present the tools used for these problems, as well as their limitations, and outline the arguments of the proofs.

20 pages; survey of previous results

References in corpus (5)

• Low-dimensional Bose liquids: beyond the Gross-Pitaevskii approximation
• Remarks on nonlinear Schroedinger equations with harmonic potential
• Critical nonlinear Schroedinger equations with and without harmonic potential
• Similarity Solutions and Collapse in the Attractive Gross-Pitaevskii Equation
• Semi-classical Schrodinger equations with harmonic potential and nonlinear perturbation

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• Niederer's transformation, time-dependent oscillators and polarized gravitational waves
• An effective mass theorem for the bidimensional electron gas in a strong magnetic field
• Generalized Niederer's transformation for quantum Pais-Uhlenbeck oscillator
• Fractional Fourier Transform and Geometric Quantization
• Schrödinger-Lohe type models of quantum synchronization with nonidentical oscillators