Homoclinic solutions for nonlocal equations and applications to the theory of atom dislocation

TL;DR

This paper proves the existence of homoclinic solutions for nonlocal fractional Laplacian equations with applications to atomic dislocation models in crystals, revealing new stable configurations and equilibria.

Contribution

It introduces new existence results for homoclinic solutions in nonlocal equations with gradient forcing, extending the Peierls-Nabarro model to fractional operators.

Findings

Existence of crystal configurations with atoms at extrema in unstable positions.

Solutions reach equilibrium at infinity with small perturbations.

Results apply to both homogeneous and inhomogeneous potentials.

Abstract

We establish the existence of homoclinic solutions for suitable systems of nonlocal equations whose forcing term is of gradient type. The elliptic operator under consideration is the fractional Laplacian and the potentials that we take into account are of two types: the first one is a spatially homogeneous function with a strict local maximum at the origin, the second one is a spatially inhomogeneous potential satisfying the Ambrosetti-Rabinowitz condition coupled to a quadratic term with spatially dependent growth at infinity. The existence of these special solutions has interesting consequences for the theory of atomic edge dislocations in crystals according to the Peierls-Nabarro model and its generalization to fractional equations. Specifically, for the first type of potentials, the results obtained give the existence of a crystal configuration with atoms located at both extrema in an unstable rest position, up to an arbitrarily small modification of the structural potential and a "pinch" of a particle at any given position. For the second type of potentials, the results obtained also entail the existence of a crystal configuration reaching an equilibrium at infinity, up to an arbitrarily small superquadratic perturbation of the classical Peierls-Nabarro potential.