Structural Stability of Travelling Waves in an Integrodifferential Equation, Dyn

In this article we study the structural stability of travelling waves of an integrodifferential equation, which can be viewed as the nonlocal analogue to the usual reaction-diffusion system u ̇ = uxx + f(u). More precisely, we are interested in the question whether a travelling wave solution persists under small perturbations of the equation. Since the travelling wave equation is a functional differential equation of mixed type, a deeper understanding of the intersection of stable and unstable manifold of the steady state in mixed type equations turns out to be crucial. As one of the main results we prove the existence of stable and unstable manifolds for general functional differential equations. We apply our results to the one-dimensional equation of elasticity with nonlocal energy. In particular we prove that a travelling wave is structural stable if and only if the underlying shock wave is compressive.