# Variable coefficient complex Ginzburg-Landau equation

## TL;DR

This paper introduces an analytical method to derive exact solutions for the variable coefficient complex Ginzburg-Landau equation by transforming it into the nonlinear Schrödinger equation along complex characteristic curves.

## Contribution

It presents a novel analytical approach that transforms the variable coefficient CGLE into the NLSE, enabling exact solutions for complex, spatially extended systems.

## Key findings

- Transformations yield an imaginary time advection equation.
- Variable coefficient CGLE is reducible to NLSE on complex curves.
- Exact solutions of NLSE generate solutions for the original CGLE.

## Abstract

The complex Ginzburg-Landau equation (CGLE) is a general model of spatially extended nonequilibrium systems. In this paper, an analytical method for a variable coefficient CGLE is presented to obtain exact solutions. Variable transformations for space and time variables with coefficient functions yield an imaginary time advection equation related to a complex valued characteristic curve. The variable coefficient CGLE is transformed into the nonlinear Schr{"\o}dinger equation (NLSE) on the complex valued characteristic curve. This result indicates that the analytical solutions of the NLSE generate that of the variable coefficient CGLE.