Supercritical geometric optics for nonlinear Schrodinger equations
Analysis of PDEs
2009-10-06 v2 Mathematical Physics
math.MP
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.
Keywords
Cite
@article{arxiv.0704.2488,
title = {Supercritical geometric optics for nonlinear Schrodinger equations},
author = {Thomas Alazard and Rémi Carles},
journal= {arXiv preprint arXiv:0704.2488},
year = {2009}
}