Abstract
We study traveling wave solutions to a general fourth-order differential equation that is a singular perturbation of the Fisher–Kolmogorov equation. We apply the geometric method for singularly perturbed systems to show that for every positive wavespeed there exists a traveling wave. Also, we find that there exists a critical wavespeed c* which divides these solutions into monotonic (c⩾c*) and oscillatory (c<c*) solutions. We show that the monotonic fronts are locally stable under perturbations in appropriate weighted Sobolev spaces by using various energy functionals.
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