WKB analysis for nonlinear Schr\"{o}dinger equations with potential

TL;DR

This paper rigorously justifies the WKB approximation for the semiclassical nonlinear Schrödinger equation with subquadratic potentials across various critical regimes, extending previous results and handling unbounded initial phases.

Contribution

It extends WKB analysis to supercritical cases with defocusing cubic nonlinearities and unbounded initial phases, including new solution constructions for related Euler equations.

Findings

WKB approximation is justified in subcritical, critical, and supercritical regimes.

Extension of previous results to supercritical cases with defocusing cubic nonlinearities.

Construction of solutions for Euler equations with unbounded sources and initial velocities.

Abstract

We justify the WKB analysis for the semiclassical nonlinear Schr\"{o}dinger equation with a subquadratic potential. This concerns subcritical, critical, and supercritical cases as far as the geometrical optics method is concerned. In the supercritical case, this extends a previous result by E. Grenier; we also have to restrict to nonlinearities which are defocusing and cubic at the origin, but besides subquadratic potentials, we consider initial phases which may be unbounded. For this, we construct solutions for some compressible Euler equations with unbounded source term and unbounded initial velocity.