Supercritical geometric optics for nonlinear Schrodinger equations

Details Supercritical geometric optics for nonlinear Schrodinger equations Supercritical geometric optics for nonlinear Schrodinger equations

2007-04-19

arxiv.org/abs/0704.2488v2

Archive for Rational Mechanics and Analysis 194, 1 (2009) 315-347

Mathematics

Analysis of PDEs

29 pages. Some typos fixed

Scientific paper

10.1007/s00205-008-0176-7

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.

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