Rotating Wave Solutions to Lattice Dynamical Systems I: The Anti-Continuum Limit

TL;DR

This paper proves the existence of rotating wave solutions in a discrete lattice Ginzburg-Landau system without relying on continuous symmetries, focusing on the anti-continuum limit and linking finite lattice phase systems to infinite lattice solutions.

Contribution

It introduces a novel approach to establish rotating wave solutions in lattice dynamical systems lacking Euclidean symmetry, extending phase system analysis to infinite lattices.

Findings

Existence of rotating wave solutions in the anti-continuum limit.

Extension of finite lattice phase solutions to infinite lattices.

Foundation for studying dynamics of rotating waves without symmetry assumptions.

Abstract

Rotating waves are a fascinating feature of a wide array of complex systems, particularly those arising in the study of many chemical and biological processes. With many rigorous mathematical investigations of rotating waves relying on the model exhibiting a continuous Euclidean symmetry, this work is aimed at understanding these nonlinear waves in the absence of such symmetries. Here we will consider a spatially discrete lattice dynamical system of Ginzburg-Landau type and prove the existence of rotating waves in the anti-continuum limit. This result is achieved by providing a link between the work on phase systems stemming from the study of identically coupled oscillators on finite lattices to carefully track the solutions as the size of the lattice grows. It is shown that in the infinite square lattice limit of these phase systems that a rotating wave solution exists, which can be extended to the Ginzburg-Landau system of study here. The results of this work provide a necessary first step in the investigation of rotating waves as solutions to lattice dynamical systems in an effort to understand the dynamics of such solutions outside of the idealized situation where the underlying symmetry of a differential equation can be exploited.