Lev Ostrovsky, Colloquium

Here, some modern results of perturbation theory for solitons and fronts (kinks) are briefly described. The method is based on expansion of the solution in powers of a small parameter (e.g., small dissipation, slow variations of a medium in space and time, interaction with neighboring solitons, etc.) and preventing systematic increase of the perturbation by the slow variation of parameters of the basic solution (such as its amplitude). Among the physically sound results is the effect of earth rotation on a soliton, and dynamics of interacting solitons with close amplitudes in integrable and non-integrable systems. The latter is reduced to interaction of classical particles creating a field potential decreasing with distance. This interaction can be repulsing, attracting, or result in a multisoliton state. Also interaction of kinks in flat-top solitons in described. Under this approach, the soliton ensembles form “envelope solitons” of Toda type, and ensembles of kinks, by a double Toda system.

The theory is illustrated by examples of nonlinear waves in electromagnetic systems and nonlinear internal waves in the ocean.