Forced translational symmetry-breaking for abstract evolution equations: the organizing center for blocking of travelling waves

TL;DR

This paper investigates how breaking translational symmetry in differential equations affects traveling waves, revealing a cone-shaped region in parameter space where wave propagation fails, using center manifold theory.

Contribution

It applies the center manifold theorem to analyze symmetry-breaking effects on traveling waves in abstract evolution equations, identifying conditions leading to wave blocking.

Findings

Propagation failure occurs in a cone in the (c,ε) parameter space.

Symmetry-breaking causes wave blocking near the symmetric case.

Method can be adapted to parameter-independent equations like Fisher-KPP.

Abstract

We consider two parameter families of differential equations on a Banach space X, where the parameters c and $\epsilon$ are such that: (1) when $\epsilon=0$, the differential equations are symmetric under the action of the group of one-dimensional translations SE(1) acting on X, whereas when $\epsilon\neq 0$, this translation symmetry is broken, (2) when $\epsilon=0$, the symmetric differential equations admit a smooth family of relative equilibria (travelling waves) parametrized by the drift speed c, with $c=0$ corresponding to steady-states. Under certain hypotheses on the differential equations and on the Banach space X, we use the center manifold theorem of Sandstede, Scheel and Wulff to study the effects of the symmetry-breaking perturbation on the above family of relative equilibria. In particular, we show that the phenomenon commonly referred to as propagation failure, or wave blocking occurs in a cone in the $(c,\epsilon)$ parameter space which emanates from the point $(c,\epsilon)=(0,0)$. We also discuss how our methods can be adapted to perturbations of parameter-independent differential equations (such as the Fisher-KPP) which admit families of relative equilibria parametrized by drift speed.