Heteroclinic Travelling Waves of Gradient Diffusion Systems

TL;DR

This paper proves the existence of heteroclinic travelling wave solutions for gradient diffusion systems using a variational approach, characterizing the wave and speed through an energy functional with an artificial constraint.

Contribution

It introduces a variational method with an artificial constraint to establish existence and characterize heteroclinic travelling waves in gradient systems.

Findings

Existence of travelling wave solutions connecting minima of W.

Variational characterization of wave speed and profile.

Method applicable to systems with multiple minima.

Abstract

We establish existence of travelling waves to the gradient system $u_t = u_{zz} - \nabla W(u)$ connecting two minima of $W$ when $u : \R \times (0,\infty) \larrow \R^N$, that is, we establish existence of a pair $(U,c) \in [C^2(\R)]^N \by (0,\infty)$, satisfying \[ \{{array}{l} U_{xx} - \nabla W (U) = - c U_x U(\pm \infty) = a^{\pm}, {array}. \] where $a^{\pm}$ are local minima of the potential $W \in C_{\textrm{loc}}^2(\R^N)$ with $W(a^-)< W(a^+)=0$ and $N \geq 1$. Our method is variational and based on the minimization of the functional $E_c (U) = \int_{\R}\Big\{{1/2}|U_x|^2 + W(U) \Big\}e^{cx} dx$ in the appropriate space setup. Following Alikakos-Fusco \cite{A-F}, we introduce an artificial constraint to restore compactness and force the desired asymptotic behavior, which we later remove. We provide variational characterizations of the travelling wave and the speed. In particular, we show that $E_c(U)=0$.