Classification of Traveling Wave Solutions of Reaction-Diffusion Systems.
Konstantin Mischaikow · 1985
A classification scheme is presented for traveling wave solutions of reaction diffusion systems of the form x sub t = x sub alpha alpha + Del V(x) where t, are elements of R x is an element of R superscript n and V: R superscript n approaches R. The important assumptions on V are that the limit as the absolute value of x approaches infinity of V(x) is minus infinity, that the set (xbarV(x) - Q) is convex for Q sufficiently large that V has a finite number of critical points, and that if M sub 1 and M sub 2 are critical points of V then V(M sub 1) not equal V(M sub 2). The primary tools used are the Conley index and connection matrix. The classifications are given via paths in graphs whose vertices and edges are connection matrices. These results are then used to prove the existence of an infinite number of traveling wave solutions for a specific example.