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Zbl 0752.34007
Zinner, Bertram
Existence of traveling wavefront solutions for the discrete Nagumo equation. (English)
[J] J. Differ. Equations 96, No.1, 1-27 (1992). ISSN 0022-0396

It is known that the Nagumo equation $\partial u/\partial t+D\partial\sp 2 u/\partial x\sp 2+f(u)=0$ has a so-called travelling wave front. This means that there exists a function $U$ such that $U(-\infty)=0$, $U(\infty)=1$ and that $u(x,t)=U(x/\sqrt D+ct)$, $c>0$ is a solution. In this paper so-called ``discrete Nagumo equation'' is considered. In fact it is an infinite system of ODE's of the form $\dot u\sb n=d(u\sb{n-1}- 2u\sb n+u\sb{n+1})+f(u\sb n)$, $n\in\bbfZ$, $d>0$. The author proves (in a rigorous way) that under certain conditions on $f$ a similar travelling wave front exists also for the discrete case. Namely it is proved that there exists a function $U$, satisfying the conditions $U(-\infty)=0$, $U(\infty)=1$, $U(x)>0,$ $\forall x\in\bbfR$, and such that $u\sb n(t)=U(n+ct)$, $c>0$, is a solution of the discrete Nagumo equation, provided that $d$ is large enough. It is to be stressed that the proof given here has a clear approximational aspect.
[K.Moszyński (Warszawa)]

MSC 2000:: *34A35 ODE of infinite order
34A45 Theoretical approximation of solutions of ODE
35K57 Reaction-diffusion equations
65M06 Finite difference methods (IVP of PDE)

Keywords: travelling wave front; discrete Nagumo equation; infinite system of ODE's

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