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Summary: Supercritical geometric optics for nonlinear
Schr¨odinger equations
Thomas Alazard and R´emi Carles
Abstract. We consider the small time semi-classical limit for nonlin-
ear Schr¨odinger 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.
1. Introduction
1.1. Presentation. We study the behavior of the solution u to
(1.1) itu
+
2
2
u
= |u
|2
|
