Semiclassical regularization of Vlasov equations and wavepackets for nonlinear Schrödinger equations

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Abstract

We consider the semiclassical limit of nonlinear Schrödinger equations with initial data that are well localized in both position and momentum (non-parametric wavepackets). We recover the Wigner measure (WM) of the problem, a macroscopic phase-space density which controls the propagation of the physical observables such as mass, energy and momentum. WMs 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 WM 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 et al (2012 Comm. Pure Appl. Math. 55 582-632). The same approach is also applied to the Vlasov-Dirac-Benney equation with small wavepacket initial data, extending several known results.

Original languageEnglish
Pages (from-to)1045-1072
Number of pages28
JournalNonlinearity
Volume31
Issue number3
DOIs
Publication statusPublished - 12 Feb 2018

Keywords

  • Wigner measure
  • nonlinear Schrödinger equation
  • semiclassical asymptotics
  • wavepackets

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