Wigner functions versus WKB‐methods in multivalued geometrical optics

Article type: Research Article

Authors: Sparber, Christof | Markowich, Peter A. | Mauser, Norbert J.

Affiliations: Institut für Mathematik, Universität Wien, Strudlhofg. 4, A‐1090 Vienna, Austria E‐mail: christof.sparber@univie.ac.at, peter.markowich@univie.ac.at, mauser@courant.nyu.edu

Abstract: 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 pre‐breaking regime. Further we show how 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.

Journal: Asymptotic Analysis, vol. 33, no. 2, pp. 153-187, 2003