Asymptotic Theory of Solitons Generated from an Intense Wave Pulse

Abstract

A theory of conversion of an intense initial wave pulse into solitons for asymptotically long evolution times has been developed using the approach based on the fact that such a transformation occurs via an intermediate stage of formation and evolution of dispersion shock waves. The number of nonlinear oscillations in such waves turns out to be equal to the number of solitons in the asymptotic state. Using the Poincaré–Cartan integral invariant theory, it is shown that the number of oscillations equal to the classical action of a particle associated with the wave packet in the vicinity of the small-amplitude edge of a dispersion shock wave remains unchanged upon a transfer by a flow described by a nondispersive limit of the nonlinear wave equations considered here. This makes it possible to formulate a generalized Bohr–Sommerfeld quantization rule that determines the set of “eigenvalues” associated with soliton physical parameters in the asymptotic state (in particular, with their velocities). In the theory, the properties of full integrability of nonlinear wave equations are not used, but the corresponding results are reproduced in this case also. The analytical results are confirmed by numerical solutions to nonlinear wave equations.