\input zb-basic
\input zb-ioport
\iteman{io-port 03935604}
\itemau{Terman, David}
\itemti{Directed graphs and traveling waves.}
\itemso{Trans. Am. Math. Soc. 289, 809-847 (1985).}
\itemab
The differential equation studied in this interesting paper is the following: $u\sb t=u\sb{xx}+f(u).$ It might arise in mathematical biology as population genetics, ecology and nerve conduction. The existence of traveling wave solutions is considered. With a few conditions on f the results demonstrate that there may exist zero, exactly one, a finite number or an infinite number of traveling wave solutions which connect two stable rest points. The technique proving these is the following: The phase planes are identified with an array of integers. The phase planes may be complicated, but the arrays of integers become relatively simple so directed graphs can be constructed. Using the directed graphs one determines how many waves connect two stable rest points.
\itemrv{J.Schoenenberger-Deuel}
\itemcc{}
\itemut{traveling wave solutions; stable rest points; phase planes; directed graphs}
\itemli{doi:10.2307/2000264}
\end