Directed graphs and traveling waves

Abstract:

The existence of traveling wave solutions for equations of the form ${u_t} = {u_{xx}} + F\prime (u)$ is considered. All that is assumed about $F$ is that it is sufficiently smooth, ${\lim _{|u| \to \infty }}F(u) = - \infty$, $F$ has only a finite number of critical points, each of which is nondegenerate, and if $A$ and $B$ are distinct critical points of $F$, then $F(A) \ne F(B)$. The results demonstrate that, for a given function $F$, there may exist zero, exactly one, a finite number, or an infinite number of waves which connect two fixed, stable rest points. The main technique is to identify the phase planes, which arise naturally from the problem, with an array of integers. While the phase planes may be very complicated, the arrays of integers are always quite simple to analyze. Using the arrays of integers one is able to construct a directed graph; each path in the directed graph indicates a possible ordering, starting with the fastest, of which waves must exist. For a large class of functions $F$ one is then able to use the directed graphs in order to determine how many waves connect two stable rest points.