OSCILLATORY INSTABILITY OF TRAVELING WAVES FOR A KDV-BURGERS EQUATION

The stability of traveling wave solutions of a generalization of the K dV-Burgers equation: partial derivative(t)u + u(p) partial derivative( x)u + partial derivative(x)3u = alpha partial derivative(x)2u, is stud ied as the parameters p and alpha are varied. The eigenvalue problem f or the linearized evolution of perturbations is analyzed by numericall y computing Evans' function, D(lambda), an analytic function whose zer os correspond to discrete eigenvalues. In particular, the number of un stable eigenvalues in the complex plane is evaluated by computing the winding number of D(lambda). Analytical and numerical evidence suggest s that a Hopf bifurcation occurs for oscillatory traveling wave profil es in certain parameter ranges. Dynamic simulations suggest that the b ifurcation is subcritical - no stable time periodic solution is found.