Monotone gradient dynamics and location of stationary (p,q)-configurations

TL;DR

This paper uses the monotone property of gradient dynamics in the Frenkel-Kontorova model to identify stationary states and forbidden regions in the configuration space, revealing new types of configurations in generalized models.

Contribution

It introduces a method to locate stationary states and forbidden regions in (p,q)-configuration space, and shows the existence of non-minimizing ordered configurations in generalized models.

Findings

Identified ordered and unordered stationary states in (p,q)-configurations.

Mapped forbidden regions for stationary states.

Discovered generalized configurations that are neither action minimizing nor minimaximizing.

Abstract

Exploiting the monotone property of the gradient dynamics of the Frenkel-Kontorova model, we locate in the space of (p,q)-configurations the ordered and unordered stationary states, as well as forbidden regions for such states. Moreover we show that some generalized Frenkel--Kontorova models (associated to multiharmonic standard maps) can have ordered (p,q)--configurations that are neither action minimizing nor minimaximizing, and give their location with respect to the set of (p,q)--minimizers and minimaximizers.