Supercritical geometric optics for nonlinear Schrodinger equations

Abstract

We consider the small time semi-classical limit for nonlinear Schrodinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To overcome this issue, we introduce new unknown functions, which are defined nonlinearly in terms of the wave function itself. This approach provides a local version of the modulated energy functional introduced by Y.Brenier. The system we obtain is hyperbolic symmetric, and the justification of WKB analysis follows.

29 pages. Some typos fixed

References in corpus (5)

• Low-dimensional Bose liquids: beyond the Gross-Pitaevskii approximation
• Low Mach number limit of the full Navier-Stokes equations
• Geometric optics and instability for semi-classical Schrodinger equations
• WKB analysis for nonlinear Schrödinger equations with potential
• On the instability for the cubic nonlinear Schrodinger equation

Cited by in corpus (6)

• Loss of regularity for supercritical nonlinear Schrodinger equations
• On Fourier time-splitting methods for nonlinear Schrodinger equations in the semi-classical limit II. Analytic regularity
• An asymptotic preserving approach for nonlinear Schrodinger equation in the semiclassical limit
• WKB analysis of the Logarithmic Nonlinear Schrodinger Equation in an analytic framework
• WKB analysis of generalized derivative nonlinear Schrodinger equations without hyperbolicity
• On nonlinear effects in multiphase WKB analysis for the nonlinear Schrodinger equation