Nonlinear random waves

This book is mainly devoted to the dynamics of the one-dimensional nonlinear stochastic waves. It contains a description of the basic mathematical tools as well as the latest results in the following fields: exactly integrable nonlinear stochastic equations, dynamics of the nonlinear waves in random media, evolution of the random waves in nonlinear media and the basic concepts of the numerical simulations in nonlinear random wave dynamics. A brief outline of the localization phenomenon in the nonlinear medium is also given. The approach is interdisciplinary describing the general methods with application to specific examples. The results presented may be useful for those who work in the areas of solid state physics, hydrodynamics, nonlinear optics, plasma physics, mathematical models of micromolecules and biological structures, etc. Since many results are based on the inverse scattering technique, perturbation theory for solitons and the methods of the statistical radiophysics, the terminology of the respective fields is used.

1. Introduction -- 2. Linear random waves. Some basic results. 2.1. Mathematical preliminaries. 2.2. Mean field method. 2.3. Localization. 2.4. Parabolic equation -- 3. Exactly solvable models. 3.1. Inverse scattering technique. 3.2. Exactly integrable stochastic equations -- 4. Direct perturbation methods. 4.1. Born approximation in the nonlinear wave theory. 4.2. Mean field method in nonlinear theory -- 5. From inverse scattering to perturbative approach. 5.1. The perturbed Korteweg-de Vries equation. One soliton dynamics. 5.2. Perturbation theory for nonlinear Schrödinger solitons. 5.3. Perturbation theory for kinks of the Sine-Gordon equation. 5.4. General remarks -- 6. Dynamical solitons under random perturbations. 6.1. Stochastic dynamics of the Korteweg-de Vries soliton. 6.2. Solitons on inhomogeneous lattices. 6.3. Perturbed dynamics of the nonlinear Schrödinger soliton -- 7. Sine-Gordon kinks under random perturbations. 7.1. The forced dynamics of a kink. 7.2. Multisoliton effects in randomly perturbed systems -- 8. Random wave packets in nonlinear media. 8.1. Direct analytical methods. 8.2. Intensity of a nonsoliton random wave packet. 8.3. Weakly nonlinear incoherent wa…