Instability of oscillatory shock profile solutions of the generalized Burgers-KdV equation

Oscillatory shock profile solutions of the Burgers-KdV equation are studied in the limit as the viscosity epsilon --> 0. A rigorous proof of their instability is obtained by showing that the linearized operator about such a solution has many unstable eigenvalues for sufficiently small epsilon. The result is obtained by applying a topological method introduced by Alexander, Gardner and Jones to extract spectral information about the perturbed wave with slightly positive viscosity from particular ''pieces'' of the underlying wave which are approximated by a solitary wave solution of the reduced equation with no viscous dissipation.