Oscillatory instability of traveling waves for a KdV-Burgers equation

Abstract

The stability of traveling wave solutions of a generalization of the KdV-Burgers equation: [part]1u+up[part]xu+[part]3xu=[alpha][part]2xu, is studied as the parameters p and [alpha] are varied. The eigenvalue problem for the linearized evolution of perturbations is analyzed by numerically computing Evans' function, D([lambda]), an analytic function whose zeros correspond to discrete eigenvalues. In particular, the number of unstable eigenvalues in the complex plane is evaluated by computing the winding number of D([lambda]). Analytical and numerical evidence suggests that a Hopf bifurcation occurs for oscillatory traveling wave profiles in certain parameter ranges. Dynamic simulations suggest that the bifurcation is subcritical periodic solution is found.