Numerical simulation of traveling waves in the FitzHugh-Nagumo system via equilibria of nonlocal equations

The FitzHugh-Nagumo system have a special kind of solution named traveling wave, which has a form u( x,t)= phi (x+ct) and w (x,t) = psi (x+ct), and furthermore, it is a stable solution. We aim to obtain a numerical characterization of its profile (phi,psi) and propagation speed c. Changing variables, we transform the problem of finding those solutions of the problem of finding an equilibrium in a nonlocal system of equations. This procedure allows us to compute simultaneously the traveling wave profiles and its propagation speed avoiding moving meshes, as we illustrate with several numerical examples. We show that the solutions of this nonlocal equation exponentially converge to a traveling wave of the original problem and the nonlocal term exponentially converges to the speed of propagation. With numerical examples and by using the open software FreeFem++, we will illustrate that the solutions of the system of partial differential equations in nonlocal coordinates converge to a traveling wave of the original problem. The nonlocal coordinate system also allows for exact calculation of the propagation speed.