It is a natural idea to study the stability of shock waves by analyzing the spectrum of the linearized evolution operator. The Evans function approach to such problems provides a general geometric framework to study and exploit spectral properties of the linearized problem. Briefly speaking, the Evans function is an analytic function of the spectral parameter whose zeros to the right of the essential spectrum correspond to eigenvalues. A shock wave is spectrally stable, if the spectrum of the related linear operator consists of eigenvalues with negative real part and the eigenvalue zero. Zumbrun and collaborators have shown that spectral stability of viscous shock wave implies its nonlinear stability.
We study the generic case of a saddle-node bifurcation of viscous shock waves, where the family of viscous shock waves can be described via the Melnikov function. By relating the derivatives of the Melnikov function with derivatives of the Evans function, we prove a change of stability within the family of viscous shock waves. We apply our results to an example in magnetohydrodynamics.