Existence of traveling wavefront solutions for the discrete Nagumo equation

Abstract

In this paper we show that the discrete Nagumo equation n = d( u n −1 +2 u n + n + 1 )+ f( u n ), n ε Z has a traveling wave solution for sufficiently strong coupling d. The problem is at first simplified into a fixed point problem which can be solved by Brouwer's fixed point theorem. The solutions of the simplified problem are then continued via index-theory to solutions of approximate problems. In the final step it is proven that the solutions of the approximate problems have a limit point which corresponds to a solution of the original problem.