On nonlinear effects in multiphase WKB analysis for the nonlinear Schrodinger equation

TL;DR

This paper investigates the nonlinear Schrödinger equation in the semiclassical limit, demonstrating superposition principles and interference behaviors depending on initial data size, with implications for understanding wave interactions in nonlinear quantum systems.

Contribution

It extends WKB analysis to nonlinear regimes, showing superposition and interference effects, and connects these phenomena to properties of Euler equations and explicit phase computations.

Findings

Superposition principle holds for certain time intervals in nonlinear Schrödinger equations.

No nonlinear interference occurs for small initial data within specific time frames.

Explicit phase computations reveal timing of interference effects.

Abstract

We consider the Schrodinger equation with an external potential and a cubic nonlinearity, in the semiclassical limit. The initial data are sums of WKB states, with smooth phases and smooth, compactly supported initial amplitudes, with disjoint supports. We show that like in the linear case, a superposition principle holds on some time interval independent of the semiclassical parameter, in several regimes in term of the size of initial data with respect to the semiclassical parameter. For large data, we invoke properties of compressible Euler equations. For smaller data, we show that there may be no nonlinear interferences on some time interval independent of the semiclassical parameter, and interferences for later time, thanks to explicit computations available for particular phases.