Wigner functions versus WKB-methods in multivalued geometrical optics, Asymptot

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 prebreaking regime. Further we show how the Wigner measure approach can be used to analyze high-frequency limits in the post-breaking regime, in comparson with the traditional Fourier integral operator method. Finally we present some illustrating examples.