Alternate Evans Functions and Viscous Shock Waves

The Evans function is known as a helpful tool for locating the spectrum of some variational diierential operators. This is of special interest regarding the stability analysis of traveling waves, such as reaction-diiusion waves, solitary waves, viscous shock waves, etc., and has been used in numerous contexts. The rst aim of this paper is to present an overview of the various ways to deene an Evans function for an abstract diierential operator. Not all of these alternatives are new, but we show consistent connections between them. Subsequently, we focus on viscous shock waves, extending the work of Gardner and Zumbrun in several directions. In particular, we: (i) show some advantages of alternate Evans functions in practical computations; (ii) carry out a reened analysis in case of neutral stability; (iii) show how to treat systems of size n > 2, thus resolving a problem left open by Gardner and Zumbrun.