Some problems of asymptotic theory of nonlinear waves

Asymptotic methods developed recently in the nonlinear wave theory and applicable to the analysis of nonsinusoidal processes are discussed. These methods are based on the assumption of sufficiently slow space-time modulation of wave parameters (shortwave approximation). The intention of this paper is neither to review all problems of asymptotic wave theory nor to give a complete bibliography. Its purpose is to show the main ideas of the methods and to present concrete procedures for the construction of the approximate solutions that correspond to the "nonlinear geometrical optics" for various types of waves. A brief classification of these processes is also given in this paper. The following questions are considered: a) the asymptotic method for quasi-stationary waves; b) a modification of this method describing slowly varying aperiodic waves (shocks and solitary waves); c) the derivation of "averaged variational principle" and its generalization for dissipative systems; d) the method of reduction of a system of equations close to the linear hyperbolic ones which is valid for nonstationary waves propagating along the characteristics; and e) the concrete problem of cylindrical finite-amplitude electromagnetic wave propagation in isotropic dielectrics.