Supercritical geometric optics for nonlinear Schrodinger equations

Thomas Alazard (1) , Rémi Carles (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier

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.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Thomas Alazard, Rémi Carles. Supercritical geometric optics for nonlinear Schrodinger equations. Archive for Rational Mechanics and Analysis, 2009, 194 (1), pp.315-347. ⟨10.1007/s00205-008-0176-7⟩. ⟨hal-00142467v2⟩