Stability analysis of traveling wave solutions of a generalized Korteweg–de Vries–Burgers equation with variable dissipation parameter

Abstract

We consider traveling wave solutions of a generalized Korteweg–de Vries–Burgers equation. The dissipation coefficient depends on coordinate and time and has a smoothed step-like form at every instant of time. Small-scale processes of dissipation and dispersion determine the solution of the traveling wave problem in the high-gradient region. The flux function is non-convex and has two inflection points. It is shown that traveling wave solutions with the same wave speed can converge to the three different limiting values behind the wave. The Evans function technique is used to conduct linear stability analysis. We demonstrate that traveling wave solutions can be linearly stable for each of the three possible cases. Thus, we found linearly stable solutions in the form of a traveling wave, which correspond to admissible discontinuities in hyperbolic model. These discontinuities have the same speed but different states behind them.