Global Bifurcation of Spiral Wave Solutions to the Complex Ginzburg-Landau Equation

TL;DR

This paper proves the existence of unbounded branches of rotating spiral wave solutions with any number of arms in the complex Ginzburg-Landau equation using equivariant degree theory, providing rigorous support for these patterns in various systems.

Contribution

It introduces a novel application of the $ ext{T}^2$-equivariant degree to establish spiral wave solutions, overcoming limitations of previous methods.

Findings

Existence of unbounded branches of spiral solutions proven.

Applicable to any number of arms in the solutions.

Provides rigorous mathematical justification for pattern formation.

Abstract

We use the $\mathbb T^2$-equivariant degree to establish the existence of unbounded branches of rotating spiral wave solutions with any number of arms for the complex Ginzburg Landau equation GLe on the planar unit disc, leveraging the spatial symmetries inherent to the problem and avoiding limiting constraints encountered in previous studies (Dai 2021) that utilized the classical Leray-Schauder degree. Our results provide rigorous mathematical justification for the formation and persistence of these fundamental patterns, which are ubiquitous in physical, chemical, and biological systems but have, until now, eluded formal proof under general conditions.