Semiclassical regularization of Vlasov equations and wavepackets for nonlinear Schr\"odinger equations

TL;DR

This paper develops a semiclassical regularization method for Vlasov equations derived from nonlinear Schrödinger equations, resolving non-uniqueness issues and accurately capturing the semiclassical limit for wavepacket initial data.

Contribution

It introduces a regularization approach that selects the correct measure-valued solution of Vlasov equations in the semiclassical limit, addressing non-uniqueness problems.

Findings

Successfully recovers the Wigner measure for nonlinear Schrödinger equations.

Provides a method to select physically relevant solutions for Vlasov-type equations.

Extends results to the Vlasov-Dirac-Benney equation with wavepacket initial data.

Abstract

We consider the semiclassical limit of nonlinear Schr\"odinger equations with wavepacket initial data. We recover the Wigner measure of the problem, a macroscopic phase-space density which controls the propagation of the physical observables such as mass, energy and momentum. Wigner measures have been used to create effective models for wave propagation in random media, quantum molecular dynamics, mean field limits, and the propagation of electrons in graphene. In nonlinear settings, the Vlasov-type equations obtained for the Wigner measure are often ill-posed on the physically interesting spaces of initial data. In this paper we are able to select the measure-valued solution of the 1+1 dimensional Vlasov-Poisson equation which correctly captures the semiclassical limit, thus finally resolving the non-uniqueness in the seminal result of [Zhang, Zheng \& Mauser, Comm. Pure Appl. Math. (2002) 55, doi:10.1002/cpa.3017]. The same approach is also applied to the Vlasov-Dirac-Benney equation with small wavepacket initial data, extending several known results.