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Forced Translational Symmetry-Breaking for Abstract Evolution Equations

Volume 4, Number 2 (2013), 16 - 43

Forced Translational Symmetry-Breaking for Abstract Evolution Equations

Communicated By: 
Alain Haraux
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Abstract

In many mathematical models (e.g. reaction-diffusion systems, integro-differential equations, delayed partial differential equations), travelling waves are observed as solutions. Typically, when the underlying medium of propagation is homogeneous, the propagation speed of the travelling wave is constant throughout the medium. However, heterogeneities can modulate the wave speed. For “large” heterogeneities (and/or “small” propagation speeds), the effect of the modulation can be so large that propagation through the site of heterogeneity is impossible. This phenomenon is commonly referred to as propagation failure, or wave blocking. In this paper, we will show that for a large class of models, this phenomenon originates from a codimension two organizing center in a general class of abstract evolution equations in which the translation symmetry (which is characteristic of a homogeneous medium) is broken by a small perturbation. We consider two parameter families of differential equations on a Banach space X, where the parameters c and ε are such that: • when ε = 0, the differential equations are symmetric under an action of the group of onedimensional translations SE(1), whereas when ε , 0, this translation symmetry is broken, • when ε = 0, the symmetric differential equations admit a smooth family of relative equilibria (travelling waves) with trivial isotropy 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 center manifold reduction to study the effects of the symmetry-breaking perturbation on the above family of relative equilibria. In particular, we show propagation failure occurs in a wedge in the (c,ε) parameter space which emanates from the point (c,ε) = (0, 0).