MODELLING SOLUTIONS TO THE KdV-BURGERS EQUATION IN THE CASE OF NONHOMOGENEOUS DISSIPATIVE MEDIA

Abstract

The behavior of the soliton type solutions to the KdV-Burgers equation is studied numerically in the case of non- homogeneous dissipative media. A soliton moves from left to right and it does not change its form. The solitons with great- er amplitude are narrower and move faster. The aim of the presented research is to study the behavior of the soliton that, while moving in nondissipative medium encounters a barrier (finite or infinite) with finite constant dissipation; one may imagine an impulse of light meeting on its way a partially absorbing layer. The modelling included the case of a finite dis- sipative layer similar to a wave passing through the air-glass-air as well as a wave passing from a nondissipative layer into a dissipative one (similar to the passage of light from air to water). The present paper is a continuation of the authors’ pub- lications. New results include a numerical model of the wave’s behavior for different types of the media non-homogeneity. The dissipation predictably results in reducing the soliton’s amplitude, but some new effects occur in the case of finite piecewise constant barrier on the soliton path: after the wave leaves the dissipative barrier it retains, on the whole, a soliton form yet some small and rapidly decreasing oscillations arises in front of the soliton. These oscillations are getting larger and spread as the soliton is moving of the barrier; the distance between the soliton and the oscillation grows. That is, the oscillations are faster than the soliton. The modelling used the Maple software PDETools packet; these activities were time and resources consuming.