Open level lines of a superposition of periodic potentials on a plane

TL;DR

This paper studies the behavior of open level lines in superimposed periodic potentials on a plane, linking it to the Novikov problem and classifying lines as regular or chaotic, with implications for physical systems.

Contribution

It provides a general description of open level lines in superimposed periodic potentials, connecting to the Novikov problem and introducing a topological classification.

Findings

Classification of level lines into regular and chaotic types

Connection to the Novikov problem for quasi-periodic potentials

Relevance to physical systems with superimposed periodic potentials

Abstract

We consider here open level lines of potentials resulting from the superposition of two different periodic potentials on the plane. This problem can be considered as a particular case of the Novikov problem on the behavior of open level lines of quasi-periodic potentials on the plane with four quasi-periods. At the same time, the formulation of this problem may have many additional features that arise in important physical systems related to it. Here we will try to give a general description of the emerging picture both in the most general case and in the presence of additional restrictions. The main approach to describing the possible behavior of the open level lines will be based on their division into topologically regular and chaotic level lines.