Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Directed graphs and traveling waves


Author: David Terman
Journal: Trans. Amer. Math. Soc. 289 (1985), 809-847
MSC: Primary 35K55; Secondary 05C20, 92A09
DOI: https://doi.org/10.1090/S0002-9947-1985-0784015-6
MathSciNet review: 784015
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 _{\vert u\vert \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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35K55, 05C20, 92A09

Retrieve articles in all journals with MSC: 35K55, 05C20, 92A09


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0784015-6
Keywords: Reaction-diffusion equation, traveling wave solution, directed graphs
Article copyright: © Copyright 1985 American Mathematical Society