MODELLING OF THE KdV-BURGERS EQUATION SOLITARY WAVES IN DISSIPATIVE NONHOMOGENEOUS MEDIA

Abstract

The work is a continuation of the research begun in previous works of the authors. At present, the theory of nonlinear waves is experiencing rapid development, and its results find numerous practical applications. One can mention the direction associated with the study of the origin and evolution of shock waves, solitary waves, kinks, periodic and quasiperiodic oscillations (for example, cnoidal waves) and many others. In this series, problems with the motion of solitons in a nonhomogeneous medium remain insufficiently studied; in this paper we consider the simplest model of such a medium: layered-inhomogeneous. The behavior of solutions of the single-wave type for the KdV-Burgers equation at various dissipative medium nonhomogeneities is considered. Various kinds of finite obstacles, as well as the transition from a dissipative medium to a free one are scrutinized. Numerical models of the solution behavior are obtained. The simulation was carried out using the Maple mathematical program through the PDETools package. The tasks considered in the paper are computationally-intensive and require a great deal of computer time. Of particular interest is the case of increasing the height of the obstacle while maintaining its width. When analyzing numerical experiments, the unexpected effect of increasing the wave height with increasing obstacle height is observed, and this may be the subject of further research. Along with this, as the height of the obstacle increases, ripples run ahead of the wave. It should be noted that in the previous work of the authors, another situation related to the appearance of a ripple was described. If, however, when the height of the obstacle remains constant, we again increase the width, then we observe an appreciable decrease in the wave amplitude, as demonstrated in the model charts. Thus, by this work of an experimental nature some new interesting properties of quasi-soliton motion are demonstrated on the basis of numerical simulation; they depend on the type and size of the dissipative obstacles.

Keywords

• korteweg-de vries-burgers equation
• soliton
• nonhomogeneous dissipation medium