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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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