a Mathematics Department, Yuba College, Marysville, CA 95901-7699, USA
b Department of Mathematics, University of California, Davis, CA 95616-8633, USA
Abstract:
We use an asymptotic expansion introduced by Benilov and Pelinovski
to study the propagation of a weakly nonlinear hyperbolic wave pulse through a stationary random medium in one space dimension. We also study the scattering of such a wave by a background scattering wave. The leading-order solution is non-random with respect to a realization-dependent reference frame, as in the linear theory of O’Doherty and Anstey. The wave profile satisfies an inviscid Burgers equation with a nonlocal, lower-order dissipative and dispersive term that describes the effects of double scattering of waves on the pulse. We apply the asymptotic expansion to gas dynamics, nonlinear elasticity, and magnetohydrodynamics.