Wigner Functions versus WKB-Methods in Multivalued Geometrical Optics

TL;DR

This paper compares Wigner measure methods with traditional WKB approaches for analyzing high-frequency asymptotics in dispersive equations, especially near caustics, providing new insights into post-breaking regimes.

Contribution

It introduces the use of Wigner measures as an alternative to WKB methods for high-frequency asymptotics, particularly in crossing caustics and post-breaking regimes.

Findings

Wigner measures effectively analyze post-breaking regimes.

WKB solutions relate to Liouville transport equations pre-breaking.

Wigner approach offers advantages over Fourier integral methods near caustics.

Abstract

We consider the Cauchy-problem for a class of scalar linear dispersive equations with rapidly oscillating initial data. The problem of 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 prebreaking regime. Further we show that 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.