On the existence of heteroclinic connections

Abstract

We consider a potential W : R m → R with two different global minima a − , a + and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian systemhas a family of T -periodic solutions u T which, along a sequence T j → +∞, converges locally to a heteroclinic solution that connects a − to a + . We then focus on the elliptic systemthat we interpret as an infinite dimensional analogous of (0.1), where x plays the role of time and W is replaced by the action functional J R (u) = R ( 1 2 |u y | 2 + W (u))dy. We assume that J R has two different global minimizersū − ,ū + : R → R m in the set of maps that connect a − to a + . We work in a symmetric context and prove, via a minimization procedure, that (0.2) has a family of solutions u L : R 2 → R m , which is L-periodic in x, converges to a ± as y → ±∞ and, along a sequence L j → +∞, converges locally to a heteroclinic solution that connectsū − toū + .

Metric methods for heteroclinic connections

“…Appendix A. About the condition (H3a) Condition (H3a) was crucial in [18] and in finite-dimensional heteroclinic connection problems (see also [15] where this condition is cited). It was also introduced in [11] for applications to weighted distances in Wasserstein spaces.…”

Nondegeneracy of heteroclinic orbits for a class of potentials on the plane

“…The first existence proofs of a heteroclinic connection in the vector case were given by Rabinowitz [15] and by Sternberg [18,17]. For more recent developments on the heteroclinic connection problem we refer to [6,3,5,14,19,8,9,4]. The aforementioned works provide sufficient conditions for the existence of heteroclinic orbits, in various settings.…”

Double layered solutions to the extended Fisher–Kolmogorov P.D.E.

“…The same viewpoint is also adopted in the present paper, and we refer to [6,16] (resp. [2,7,21]) for further applications to other types of equations (resp. for the construction of periodic in t solutions satisfying the boundary conditions (9b)).…”