Approximating travelling waves by equilibria of non local equations
For researchers studying travelling waves in parabolic PDEs, this provides a method to compute wave profiles and speeds using stationary solutions, avoiding moving mesh techniques.
The paper transforms a parabolic evolution equation with a travelling wave solution into a non-local equation with a stationary wave of the same profile, and shows that on bounded intervals the unique stationary solution converges to the travelling wave as the interval length increases, enabling simultaneous computation of wave profile and speed without moving meshes.
We consider an evolution equation of parabolic type in R having a travelling wave solution. We perform an appropriate change of variables which transforms the equation into a non local evolution one having a travelling wave solution with zero speed of propagation with exactly the same profile as the original one. We analyze the relation of the new equation with the original one in the entire real line. We also analyze the behavior of the non local problem in a bounded interval with appropriate boundary conditions and show that it has a unique stationary solution which is asymptotically stable for large enough intervals and that converges to the travelling wave as the interval approaches the entire real line. This procedure allows to compute simultaneously the travelling wave profile and its propagation speed avoiding moving meshes, as we illustrate with several numerical examples.