Approximating travelling waves by equilibria of non local equations

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.

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