Communications in Mathematical Sciences

Volume 15 (2017)

Number 7

Averaging of nonlinear Schrödinger equations with strong magnetic confinement

Pages: 1933 – 1945

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n7.a7

Authors

Rupert L. Frank (Mathematisches Institut, Ludwig-Maximilans Universität München, Germany; and Department of Mathematics, California Institute of Technology, Pasadena, Calif., U.S.A.)

Florian Méhats (IRMAR, Université de Rennes, France; and INRIA, IPSO Project, Campus de Beaulieu, Rennes, France)

Christof Sparber (Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, Il., U.S.A.)

Abstract

We consider the dynamics of nonlinear Schrödinger equations with strong constant magnetic fields. In an asymptotic scaling limit the system exhibits a purely magnetic confinement, based on the spectral properties of the Landau Hamiltonian. Using an averaging technique we derive an associated effective description via an averaged model of nonlinear Schrödinger type. In a special case this also yields a derivation of the LLL equation.

Keywords

nonlinear Schrödinger equation, magnetic confinement, Landau levels, averaging

2010 Mathematics Subject Classification

35B25, 35Q55

Full Text (PDF format)

R.L.F. has been supported by the U.S. National Science Foundation through grant no. DMS-1363432.

F.M. acknowledges support by the ANR project Moonrise ANR-14-CE23-0007-01.

C.S. has been supported by the U.S. National Science Foundation through grant no. DMS-1348092.

Received 15 March 2017

Accepted 30 June 2017

Published 16 October 2017