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: ∂ 1 u+ u p ∂ x u+ ∂ 3 x u= α∂ 2 x u, is studied as the parameters p and α are varied. The eigenvalue problem for the linearized evolution of perturbations is analyzed by numerically computing Evans' function, D(λ), 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(λ). 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.