Computation and Stability of Traveling Waves in Second Order Evolution Equations

TL;DR

This paper extends the freezing method to second order damped wave equations to analyze traveling wave stability and computation, supported by rigorous theory and numerical examples.

Contribution

It introduces a novel extension of the freezing method to second order equations, enabling stable computation of traveling waves in damped wave systems.

Findings

The method converges to wave speed under spectral conditions.

Numerical examples confirm theoretical predictions.

The approach applies to Nagumo and FitzHugh-Nagumo models.

Abstract

The topic of this paper are nonlinear traveling waves occuring in a system of damped waves equations in one space variable. We extend the freezing method from first to second order equations in time. When applied to a Cauchy problem, this method generates a comoving frame in which the solution becomes stationary. In addition it generates an algebraic variable which converges to the speed of the wave, provided the original wave satisfies certain spectral conditions and initial perturbations are sufficiently small. We develop a rigorous theory for this effect by recourse to some recent nonlinear stability results for waves in first order hyperbolic systems. Numerical computations illustrate the theory for examples of Nagumo and FitzHugh-Nagumo type.