Semiclassical regularization of Vlasov equations and wavepackets for nonlinear Schrödinger equations

TY - JOUR

T1 - Semiclassical regularization of Vlasov equations and wavepackets for nonlinear Schrödinger equations

AU - Athanassoulis, Agissilaos

PY - 2018/2/12

Y1 - 2018/2/12

N2 - 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.

AB - 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.

KW - Wigner measure

KW - nonlinear Schrödinger equation

KW - semiclassical asymptotics

KW - wavepackets

UR - https://www.scopus.com/pages/publications/85042545096

U2 - 10.1088/1361-6544/aa9a86

DO - 10.1088/1361-6544/aa9a86

M3 - Article

SN - 0951-7715

VL - 31

SP - 1045

EP - 1072

JO - Nonlinearity

JF - Nonlinearity

IS - 3

ER -