A review of dispersive limits of (non)linear schrödinger-type equations

Ingenuin Gasser*, Chi Kun Lin, Peter Markowich

*Corresponding author for this work

Research output: Contribution to journalReview articlepeer-review

65 Scopus citations

Abstract

In this review paper we present the most important mathematical properties of dispersive limits of (non)linear Schrödinger type equations. Different formulations are used to study these singular limits, e.g., the kinetic formulation of the linear Schrödinger equation based on the Wigner transform is well suited for global-in-time analysis without using WKB-(expansion) techniques, while the modified Madelung transformation reformulating Schrödinger equations in terms of a dispersive perturbation of a quasilinear symmetric hyperbolic system usually only gives local-in-time results due to the hyperbolic nature of the limit equations. Deterministic analogues of turbulence are also discussed. There, turbulent diffusion appears naturally in the zero dispersion limit.

Original languageEnglish (US)
Pages (from-to)501-529
Number of pages29
JournalTaiwanese Journal of Mathematics
Volume4
Issue number4
DOIs
StatePublished - Jan 1 2000

Keywords

  • Compressible Euler equation
  • Dispersive limit
  • KdV equation
  • Quantum hydrodynamics
  • Wigner transform

ASJC Scopus subject areas

  • Mathematics(all)

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