title: Saddle-node bifurcation of viscous profiles.
abstract: Traveling wave solutions of viscous conservation laws, that are associated to Lax shocks of the inviscid equation, have generically a transversal viscous profile. In the case of a non-transversal viscous profile we show by using Melnikov theory that a parametrized perturbation of the profile equation leads generically to a saddle-node bifurcation of these solutions. An example of this bifurcation in the context of magnetohydrodynamics is given. The spectral stability of the traveling waves generated in the saddle-node bifurcation is studied via an Evans function approach. It is shown that generically one real eigenvalue of the linearization of the viscous conservation law around the parametrized family of traveling waves changes its sign at the bifurcation point. Hence this bifurcation describes the basic mechanism of a stable traveling wave which becomes unstable in a saddle-node bifurcation.
authors.affiliation: Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria.
authors.forename: Franz
authors.initials: F
authors.lastname: Achleitner
authors.forename: Peter
authors.initials: P
authors.lastname: Szmolyan
keywords: Bifurcation; Evans function; Spectral stability; Traveling wave; Viscous conservation law