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F. Achleitner, P. Szmolyan:
"Saddle-node bifurcation of viscous profiles";
in: "ASC Report 21/2010", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2010, ISBN: 978-3-902627-03-2.

Traveling wave solutions of viscous conservation laws, that are associated to Lax
shocks of the inviscid equation, have generically a transversal viscous pro le. In
the case of a non-transversal viscous pro le we show by using Melnikov theory
that a parametrized perturbation of the pro le 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.

viscous conservation law, traveling wave, bifurcation, spectral stability, Evans

http://www.asc.tuwien.ac.at/preprint/2010/asc21x2010.pdf