All meromorphic traveling waves of cubic and quintic complex Ginzburg-Landau equations

TL;DR

This paper classifies all meromorphic traveling wave solutions of the one-dimensional cubic and quintic complex Ginzburg-Landau equations, providing explicit formulas and completing previous solution lists.

Contribution

It proves the finiteness of such solutions using Eremenko's theorem and explicitly derives all solutions, including five obtained by a new method.

Findings

Finiteness of meromorphic traveling waves proven

All solutions explicitly derived and listed

Completes previous solution classifications

Abstract

For both cubic and quintic nonlinearities of the one-dimensional complex Ginzburg-Landau evolution equation, we prove by a theorem of Eremenko the finiteness of the number of traveling waves whose squared modulus has only poles in the complex plane, and we provide all their closed form expressions. Among these eleven solutions, five are provided by the method used. This allows us to complete the list of solutions previously obtained by other authors.