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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 $$\partial u/\partial t+D\partial^ 2 u/\partial x^ 2+f(u)=0$$ 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 $$\dot u_ n=d(u_{n-1}- 2u_ n+u_{n+1})+f(u_ n)$$, $$n\in\mathbb{Z}$$, $$d>0$$. 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 $$U(-\infty)=0$$, $$U(\infty)=1$$, $$U(x)>0,$$ $$\forall x\in\mathbb{R}$$, and such that $$u_ n(t)=U(n+ct)$$, $$c>0$$, 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.

##### MSC:
 34A35 Ordinary differential equations of infinite order 34A45 Theoretical approximation of solutions to ordinary differential equations 35K57 Reaction-diffusion equations 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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