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.Comment: 38 page
