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Existence of traveling wavefront solutions for the discrete Nagumo equation. (English) Zbl 0752.34007
It is known that the Nagumo equation $\partial u/\partial t+D{\partial }^{2}u/\partial {x}^{2}+f\left(u\right)=0$ has a so-called travelling wave front. This means that there exists a function $U$ such that $U\left(-\infty \right)=0$, $U\left(\infty \right)=1$ and that $u\left(x,t\right)=U\left(x/\sqrt{D}+ct\right)$, $c>0$ 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 ${\stackrel{˙}{u}}_{n}=d\left({u}_{n-1}-2{u}_{n}+{u}_{n+1}\right)+f\left({u}_{n}\right)$, $n\in ℤ$, $d>0$. 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 $U\left(-\infty \right)=0$, $U\left(\infty \right)=1$, $U\left(x\right)>0,$ $\forall x\in ℝ$, and such that ${u}_{n}\left(t\right)=U\left(n+ct\right)$, $c>0$, 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.
##### MSC:
 34A35 ODE of infinite order 34A45 Theoretical approximation of solutions of ODE 35K57 Reaction-diffusion equations 65M06 Finite difference methods (IVP of PDE)