Nonlinear waves and their interaction

Abstract

The review is devoted to a number of problems connected with the propagation of strongly nonlinear periodic waves in the presence of various types of perturbing factors. The problems considered can be divided into three groups. The first includes questions connected with the employed formalism. These include the characteristic properties of nonlinear waves, perturbation-theory methods, canonical variables, and the Hamiltonian formalism. The second group of questions is devoted to the propagation of nonlinear waves in the presence of external perturbations. A description is given of the resonant interaction between the wave and an external force, the stochastic instability of a nonlinear wave, the change of the adiabatic invariant of a linear wave in a weakly-inhomogeneous medium, and the propagation of a nonlinear wave in the presence of random perturbations, particularly in a medium with random inhomogeneities. Finally, the third group of considered questions include problems connected with weak interaction of strongly nonlinear waves. The conditions under which the interaction of the waves is weak are determined, and the interaction of two waves and resonant interaction of three waves is considered. This group includes an investigation of an ensemble of a large number of nonlinear waves and its description with the aid of the kinetic equation. The Appendix discusses problems connected with the energy-momentum tensor of the nonlinear wave equation.