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Wigner Functions versus WKB-Methods in Multivalued Geometrical Optics

Christof Sparber·Peter A. Markowich·Norbert J. Mauser
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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.

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Christof Sparber, Peter A. Markowich, Norbert J. Mauser. 2002-03-20. Wigner Functions versus WKB-Methods in Multivalued Geometrical Optics. https://arxiv.org/abs/math-ph/0109029

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