Degenerate elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation

TL;DR

This paper derives and analyzes a subset of traveling wave solutions for the quintic complex Ginzburg-Landau equation, providing compact forms and conditions for boundedness and singularity through phase diagram analysis.

Contribution

It introduces a novel method to reduce the QCGLE to ODEs and explicitly solves for specific cases, offering new insights into solution structures.

Findings

Derived compact traveling wave solutions for QCGLE

Identified conditions for bounded and singular solutions

Numerical examples illustrating the solutions

Abstract

A subset of traveling wave solutions of the quintic complex Ginzburg-Landau equation (QCGLE) is presented in compact form. The approach consists of the following parts. - Reduction of the QCGLE to a system of two ordinary differential equations (ODEs) by a traveling wave ansatz. - Solution of the system for two (ad hoc) cases relating phase and amplitude. - Presentation of the solution for both cases in compact form. - Presentation of constraints for bounded and for singular positive solutions by analyzing the analytical properties of the solution by means of a phase diagram approach. The results are exemplified numerically