On the heteroclinic connection problem for multi-well gradient systems

TL;DR

This paper introduces a geometric variational method to establish the existence of heteroclinic connections in multi-well gradient systems, simplifying previous approaches and relying on geodesics in a degenerate metric.

Contribution

It presents a new, geometric variational approach to find heteroclinic connections in systems with multiple potential wells, improving upon prior methods by Sternberg and others.

Findings

Geodesics minimize length in a degenerate metric related to the potential W.

Heteroclinic connections correspond to geodesics avoiding intermediate wells.

The approach simplifies the existence proof of heteroclinic orbits in multi-well systems.

Abstract

We revisit the existence problem of heteroclinic connections in $\mathbb{R}^N$ associated with Hamiltonian systems involving potentials $W:\mathbb{R}^N\to \mathbb{R}$ having several global minima. Under very mild assumptions on $W$ we present a simple variational approach to first find geodesics minimizing length of curves joining any two of the potential wells, where length is computed with respect to a degenerate metric having conformal factor $\sqrt{W}.$ Then we show that when such a minimizing geodesic avoids passing through other wells of the potential at intermediate times, it gives rise to a heteroclinic connection between the two wells. This work improves upon the approach of P.Sternberg in $\texttt{Vector-valued local minimizers of nonconvex}$ $\texttt{variational problems}$, and represents a more geometric alternative to the approaches for finding such connections described, for example, by N.D. Alikakos and G.Fusco in $\texttt{On the connection problem for potentials with}$ $\texttt{several global minima}$, by S.V. Bolotin in $\texttt{Libration motions of natural dynamical systems}$, by J. Byeon, P. Montecchiari, and P. Rabinowitz in $\texttt{A double well potential}$ $\texttt{system}$, and by P. Rabinowitz in $\texttt{Homoclinic and heteroclinic orbits for a class of Hamiltonian}$ $\texttt{systems}$.