The existence of travelling waves of the discrete Nagumo equation

Abstract

This thesis considers
solutions to the discrete Nagumo equation u˙ n = d(un−1 − 2un + un+1) + f(un),
n ∈ Z. For sufficiently large d, the
solutions are of the form un(t) = U(n + ct) with c > 0. This thesis contains
the proof of existence of traveling wave solutions of the discrete Nagumo
equations and originates from Bertram Zinner’s article ”Existence of Traveling
Wavefront Solutions for the Discrete Nagumo equation” [Zin90]. In the first
chapter, all the prerequisite knowledge needed to understand the proof, such as
Brouwer’s fixed point theorem, is presented. The proof starts by considering
f(un) as a linear function and thus simplifying the problem. The simplified
problem is converted into a fixed point problem by considering a Poincar´e map which
can be solved using Brouwer’s fixed point theorem. Finally, the proof ends by confirming
that the solutions of the approximated, simplified problem have a limit point
which corresponds to the traveling wave solutions of the discrete Nagumo
equation