Alternate Evans functions and viscous shock waves (Q2706219)

The spectrum of a finite matrix consists of the zeros of its characteristic polynomial. For differential operators studied in the context of asymptotic stability analysis of traveling waves, the role of this polynomial is taken over by the so-called Evans function D. The paper starts in fact by a concise review of its possible definition(s), with emphasis on an application in the study of the viscous shock waves.NEWLINENEWLINENEWLINEIn the context of various applications of Evans functions, the paper is a more or less immediate continuation of the work by \textit{R. A. Gardner} and \textit{K. Zumbrun} [Commun. Pure Appl. Math. 51, No. 7, 797-855 (1998; Zbl 0933.35136)], comparing the merits and shortcomings related to different definitions of the Evans functions in practical computations, and preferring the use of the homotopy to the original rescaling approach. The authors emphasize the useful role, the so-called ``dual'' and ``mixed'' type, of the definition of D. There are two directions of the new development of its applications, viz., the improvement of the stability analysis (especially for the so-called Lax shock) and an extension of the formalism to the general system of size \(n > 2\) (giving, in fact, a proof of the missing lemma in the general theory).