Preprint of the project: SFB 701: Spectral Structures and Topological Methods in Mathematics - Project B3

Numerical Analysis of equivariant evolution equations

16-022 Wolf-Jürgen Beyn, Denny Otten, Jens Rottmann-Matthes.
Computation and Stability of Traveling Waves in Second Order Evolution Equations


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.