The Exact Ground State of the Frenkel-Kontorova Model with Repeated Parabolic Potential: I. Basic Results

TL;DR

This paper derives exact energies and configurations for a one-dimensional Frenkel-Kontorova model with a periodic parabolic potential, reducing the problem to convex function minimization on a finite simplex.

Contribution

It provides a novel analytical approach to find exact solutions for the Frenkel-Kontorova model with a specific periodic potential structure.

Findings

Exact energies and configurations are obtained for the model.

The problem reduces to convex function minimization on a finite simplex.

Basic results establish a foundation for further analysis of the model.

Abstract

The problem of finding the exact energies and configurations for the Frenkel-Kontorova model consisting of particles in one dimension connected to their nearest-neighbors by springs and placed in a periodic potential consisting of segments from parabolas of identical (positive) curvature but arbitrary height and spacing, is reduced to that of minimizing a certain convex function defined on a finite simplex.