Motion of spiral waves in the Complex Ginzburg-Landau equation

TL;DR

This paper analyzes the dynamics of multiple spiral waves in the complex Ginzburg-Landau equation, deriving laws of motion for their centers depending on their separation, with implications for understanding wave interactions.

Contribution

It provides new asymptotic laws describing the motion of spiral wave centers in the complex Ginzburg-Landau equation near the vortex limit, considering different separation scales.

Findings

Interaction changes from along the line of centers to perpendicular as separation increases

Interaction strength is algebraic at small separations and exponential at large separations

Asymptotic wavenumber and frequency depend on spiral center positions and evolve slowly

Abstract

Solutions of the general cubic complex Ginzburg-Landau equation comprising multiple spiral waves are considered. For parameters close to the vortex limit, and for a system of spiral waves with well-separated centres, laws of motion of the centres are found which vary depending on the order of magnitude of the separation of the centres. In particular, the direction of the interaction changes from along the line of centres to perpendicular to the line of centres as the separation increases, with the strength of the interaction algebraic at small separations and exponentially small at large separations. The corresponding asymptotic wavenumber and frequency are determined. These depend on the positions of the centres of the spirals, and so evolve slowly as the spirals move.