Exact Solutions for Domain Walls in Coupled Complex Ginzburg - Landau Equations

TL;DR

This paper derives exact analytical solutions for domain walls in coupled complex Ginzburg-Landau equations, revealing diverse configurations and stability properties of front pairs in nonlinear dissipative media.

Contribution

It introduces a novel factorization method and employs computer-assisted algebra to find exact coupled-front solutions in the CGLE system, advancing understanding of domain wall dynamics.

Findings

Exact solutions for coupled domain walls are obtained.

The solutions include multiple free parameters for diverse configurations.

Numerical simulations confirm stability properties of the solutions.

Abstract

The complex Ginzburg Landau equation (CGLE) is a ubiquitous model for the evolution of slowly varying wave packets in nonlinear dissipative media. A front (shock) is a transient layer between a plane-wave state and a zero background. We report exact solutions for domain walls, i.e., pairs of fronts with opposite polarities, in a system of two coupled CGLEs, which describe transient layers between semi-infinite domains occupied by each component in the absence of the other one. For this purpose, a modified Hirota bilinear operator, first proposed by Bekki and Nozaki, is employed. A novel factorization procedure is applied to reduce the intermediate calculations considerably. The ensuing system of equations for the amplitudes and frequencies is solved by means of computer-assisted algebra. Exact solutions for mutually-locked front pairs of opposite polarities, with one or several free parameters, are thus generated. The signs of the cubic gain/loss, linear amplification/attenuation, and velocity of the coupled-front complex can be adjusted in a variety of configurations. Numerical simulations are performed to study the stability properties of such fronts.