Wigner functions versus WKB-methods in multivalued geometrical optics

Christof Sparber, Peter Markowich, Norbert J. Mauser

Research output: Contribution to journalArticlepeer-review

56 Scopus citations

Abstract

We consider the Cauchy problem for a class of scalar linear dispersive equations with rapidly oscillating initial data. The problem of the high-frequency asymptotics of such models is reviewed, in particular we highlight the difficulties in crossing caustics when using (time-dependent) WKB-methods. Using Wigner measures we present an alternative approach to such asymptotic problems. We first discuss the connection of the naive WKB solutions to transport equations of Liouville type (with mono-kinetic solutions) in the pre-breaking regime. Further we show how the Wigner measure approach can be used to analyze high-frequency limits in the post-breaking regime, in comparison with the traditional Fourier integral operator method. Finally we present some illustrating examples.

Original languageEnglish (US)
Pages (from-to)153-187
Number of pages35
JournalAsymptotic Analysis
Volume33
Issue number2
StatePublished - Feb 1 2003

ASJC Scopus subject areas

  • Mathematics(all)

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