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Supercritical geometric optics for nonlinear Schrödinger equations. (English) Zbl 1179.35302
Summary: We consider the small time semi-classical limit for nonlinear Schrödinger 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 [Commun. Partial Differ. Equations 25, No. 3–4, 737–754 (2000; Zbl 0970.35110)]. The system we obtain is hyperbolic symmetric, and the justification of WKB analysis follows.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
78A05 Geometric optics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35B65 Smoothness and regularity of solutions to PDEs
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