Detection and construction of an elliptic solution to the complex cubic-quintic Ginzburg-Landau equation

TL;DR

This paper explores methods for detecting and constructing elliptic solutions to the complex cubic-quintic Ginzburg-Landau equation, focusing on overcoming specific mathematical challenges in the process.

Contribution

It demonstrates how to apply residue criteria and subequation methods to find elliptic solutions for the complex Ginzburg-Landau equation.

Findings

Successfully applied residue sum criterion to elliptic solutions.

Constructed a first order differential subequation for the equation.

Provided a framework for solving complex amplitude evolution equations.

Abstract

In evolution equations for a complex amplitude, the phase obeys a much more intricate equation than the amplitude. Nevertheless, general methods should be applicable to both variables. On the example of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation (CGL5), we explain how to overcome the difficulties arising in two such methods: (i) the criterium that the sum of residues of an elliptic solution should be zero, (ii) the construction of a first order differential equation admitting the given equation as a differential consequence (subequation method).