Heteroclinic solutions for a generalized Frenkel-Kontorova model by minimization methods of Rabinowitz and Stredulinsky

TL;DR

This paper develops a variational approach to find heteroclinic solutions in a generalized Frenkel-Kontorova model, revealing complex solution structures using methods by Rabinowitz and Stredulinsky.

Contribution

It introduces a variational construction for heteroclinic and periodic solutions in the model, extending previous methods to generate more complex configurations.

Findings

Existence of heteroclinic solutions with rational rotation vectors.

Construction of solutions heteroclinic in multiple directions.

Method to generate increasingly complex solutions.

Abstract

We study heteroclinic solutions of a generalized Frenkel-Kontorova model. Using the methods of Rabinowitz and Stredulinsky, we prove that if the rotation vector of the configuration is rational and if there is an adjacent pair of periodic configurations, then there is a solution that is heteroclinic in one fixed direction and periodic in other directions. Furthermore, if the above heteroclinic solutions have an adjacent pair, then there is a solution that is heteroclinic in two directions and periodic in other directions. The procedure can be repeated to produce more complex solutions. Thus we obtain a variational construction for these minimal and Birkhoff solutions.