Computing the refined stability condition

The classical (inviscid) stability analysis of shock waves is based on the Lopatinskiĭ determinant, $\Delta$—a function of frequencies whose zeros determine the stability of the underlying shock. A careful analysis of $\Delta$ shows that in some cases the stable and unstable regions of parameter space are separated by an open set of parameters. Zumbrun and Serre [Indiana Univ. Math. J. 48 (1999), 937–992] have shown that by taking account of viscous effects not present in the definition of $\Delta$, it is possible to determine the precise location in the open, neutral set of parameter space at which stability is lost. In particular, they show that the transition to instability under suitably localized perturbations is determined by an “effective viscosity” coefficient given in terms of the second derivative of the associated Evans function, the viscous analogue of $\Delta$. Here, in the simplest possible setting, we propose and implement two approaches toward the practical computation of this coefficient. Moreover, in a special case, we derive an exact solution of the relevant differential equations.