Frenkel-Kontorova Models, Pinned Particle Configurations and Burgers Shocks

TL;DR

This paper explores the connection between particle configurations in Frenkel-Kontorova models and Burgers equation shocks, providing analytical solutions for specific potentials and insights into energy states and discommensurations.

Contribution

It establishes a detailed analytical link between lowest energy particle configurations and Burgers shocks, including exact solutions for parabolic potentials.

Findings

Analytical solutions for lowest energy configurations

Characterization of excited states with discommensurations

Insight into the role of shocks in energy separation

Abstract

We analyze the relationship between the lowest energy configurations of an infinite harmonic chain of particles in a periodic potential and the evolution of characteristics in a periodically-forced inviscid Burgers equation. The shock discontinuities in the the Burgers evolution arise from thermodynamical considerations and play an important role as they separate out flows related to lowest energy configurations from those associated with higher energies. We study in detail the exactly solvable case of an external potential consisting of parabolic segments, and calculate analytically the lowest energy configurations, as well as excited states containing discommensurations.