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Semiclassical regularization of Vlasov equations and wavepackets for nonlinear Schrödinger equations

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posted on 2018-05-18, 09:04 authored by Agissilaos Athanassoulis
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.

History

Citation

Nonlinearity, 2018, 31 (3), pp. 1045-1072

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

Version

  • AM (Accepted Manuscript)

Published in

Nonlinearity

Publisher

IOP Publishing, London Mathematical Society

issn

0951-7715

eissn

1361-6544

Acceptance date

2017-11-14

Copyright date

2018

Available date

2019-02-12

Publisher version

http://iopscience.iop.org/article/10.1088/1361-6544/aa9a86/meta

Notes

The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.

Language

en

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