Abstract

The Inverse Scattering Method for the Equation Describing the High-Frequency Waves in a Relaxing Medium

V.O.Vakhnenko (Institute for Geophysics, Kiev, Ukraine); E.J.Parkes (Department of Mathematics, University of Strathclyde, Glasgow, UK)

e-mail: vakhnenko@bitp.kiev.ua

Describing real media under the action of intense waves is often unsuccessful in the framework of equilibrium models of continuum mechanics. To develop physical models for wave propagation through media with complicated inner kinetics, the notions based on the relaxational nature of a phenomenon are regarded to be promising. We consider the nonlinear evolution equation (u_t+uu_x)_x+u=0 (Vakhnenko equation - VE) that arose as a result describing the high-frequency perturbations in a relaxing medium. The study of the VE has scientific interest both from the viewpoint of the existence of stable wave formations and from the viewpoint of the general problem of integrability of nonlinear equations. The VE has two families of travelling wave solutions, both of which are stable to long wavelength perturbations. In particular, the VE has a loop-like soliton solution. The interaction of two solitons by both Hirota's method and the IST method are considered. The IST method has a third order eigenvalue problem. This has been achieved by finding a Backlund transformation. A procedure for finding the exact N-soliton solution to the VE via the IST method is described. Under the interaction of solitons there are features that are not typical for the KdV equation.

Section : 1