A method for obtaining evolution equations for nonlinear waves in a random medium

Abstract

Most of the methods used to derive deterministic equations governing the evolution of linear waves in random media are based on the mean field approach. For a given linear system of equations with random coefficients this approach results in an approximate deterministic equation (or system) for the mean (averaged over the set of all realizations) field. The coefficients of such equations are associated with the statistical moments of the random coefficients of the initial system. Sometimes similar approaches are also applied to nonlinear problems. However, recently, in a number of examples, it was shown that the mean field approach in nonlinear problems may give the wrong results. The error is related to the infinite growth of the root-mean-square phase fluctuations due to fluctuations of the wave velocity in an inhomogeneous medium. To overcome this effect the idea of eliminating the unbounded growth of phase fluctuations by using an appropriate coordinate transformation was proposed resulting in a method called the “mean waveform method”. In the present paper we extend the idea of the mean waveform method in order to develop a systematic approach which enables the construction of an approximate deterministic evolution equation for a given quasi-hyperbolic and quasi-linear system of equations with weak nonlinearity and stationary random coefficients. The applicability of the proposed approach is demonstrated by means of an example.