The short-wave limit for nonlinear, symmetric, hyperbolic systems

Abstract

For general semilinear or quasilinear symmetric, hyperbolic systems, the short-wave approximation is studied by constructing approximate solutions to the initial-value problem associated with the hyperbolic operator by means of a multiscale WKB expansion. In the diffractive optics regime, the components of the leading-order terms of the approximate solutions are shown to satisfy differential equations of KP type. Whether these equations are scalar or not depends on the polarization of the initial datum and on the asymptotic behavior of the branches of the characteristic variety of the hyperbolic operator. The asymptotic stability of the approximate solution is proved. Following J.-L. Joly, G. Métivier, and J. Rauch, the cornerstone of this study is the characteristic variety of the hyperbolic operator. It is examined here in terms of perturbation theory, which yields new proofs of the so-called algebraic lemmas of geometric optics.