Soliton Evolution in the Presence of Perturbation

A perturbation theory for nonlinear waves based on the inverse scattering method is presented. The theory is applied to the description of soliton evolution in the presence of permanent perturbation. It is shown that small perturbation leads to three main effects: (i) a slow change of soliton parameters; (ii) a deformation of its shape (iii) formation of a soliton "tail" which is a small amplitude wave packet with growing length. All these effects are investigated in detail for the Korteweg-de Vries, modified Korteweg-de Vries and nonlinear Schrödinger equations to which perturbation terms of general form are added. It is show, in particular, that for the last equation, in contrast to the previous two, the tails do not appear for perturbations of a very broad type.

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