Cascade of Phase Shifts and Creation of Nonlinear Focal Points for Supercritical Semiclassical Hartree Equation

TL;DR

This paper analyzes the semiclassical limit of the supercritical Hartree equation, revealing phase shifts and nonlinear focal points, and demonstrates the breakdown of WKB approximation due to singularity formation.

Contribution

It extends WKB approximation validity for larger initial data in supercritical regimes and analyzes the formation of singularities before focusing.

Findings

WKB approximation holds for larger data than previously known

Phase shifts occur near focusing points in supercritical regimes

Singularities form in the phase function before reaching the focal point

Abstract

We consider the semiclassical limit of the Hartree equation with a data causing a focusing at a point. We study the asymptotic behavior of phase function associated with the WKB approximation near the caustic when a nonlinearity is supercritical. In this case, it is known that a phase shift occurs in a neighborhood of focusing time in the case of focusing cubic nonlinear Schr\"odinger equation. Thanks to the smoothness of the nonlocal nonlinearities, we justify the WKB-type approximation of the solution for a data which is larger than in the previous results and is not necessarily well-prepared. We also show by an analysis of the limit hydrodynamical equaiton that, however, this WKB-type approximation breaks down before reaching the focal point: Nonlinear effects lead to the formation of singularity of the leading term of the phase function.