WKB-Methods in Multivalued Geometrical Optics

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. This work was supported by the Austrian START award (FWF Y-137-TEC) of N.J.M., the ”Wittgenstein Award” of P.A.M. and by the FWF Wissenschaftskolleg ”Differential Equations”. ∗ e-mail: [email protected] † e-mail: [email protected] or http://mailbox.univie.ac.at/peter.markowich ‡ e-mail: [email protected]