NONLINEAR-WAVES TRAVELING UPON A FRONT OF SOLITONS

The equation for perturbations with finite amplitude and finite wavelength traveling on a front of exponential solitons has been derived by means of an asymptotic procedure. The shock waves traveling on a front of exponentially decaying solitons, like the "shock-shocks" of Whitham, have been described. Stability conditions of solitons relative to the multidimensional perturbations have also been obtained. The spectrum of the soliton vibrations in the stable case, the stability threshold, and the growth rate in the unstable case have been calculated. The resulting equation is a generalization of the nonlinear quasioptics for the modulation of solitons with finite front width, in this case, the nonlinearity and diffusion of the perturbations are not independent.