Wigner Functions versus WKB-Methods

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, leading to ”multi-valued” solutions of Hamilton-Jacobi equations. Using Wigner measures we present an alternative approach to such asymptotic problems. We discuss the connection of the WKB solutions to transport equations of Liouville type with mono-kinetic solutions in the prebreaking regime and we further show that the Wigner measure approach is, to some extent, more adapted to analyzing high-frequency limits in the post-breaking regime, where the semi-classical measure in general is no longer mono-kinetic. Finally we present some illustrating examples as well as further results and comments. 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]