Meromorphic traveling wave solutions of the complex cubic-quintic Ginzburg-Landau equation

TL;DR

This paper classifies meromorphic solutions of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation, showing they are elliptic or degenerate elliptic, and provides explicit decompositions of these solutions.

Contribution

It proves all meromorphic solutions are elliptic or degenerate elliptic and offers explicit decompositions of the new elliptic solutions using the subequation method.

Findings

All meromorphic solutions are elliptic or degenerate elliptic.

Explicit decompositions of the new elliptic solutions are provided.

The solutions are characterized using Clunie's lemma.

Abstract

We look for singlevalued solutions of the squared modulus M of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using Clunie's lemma, we first prove that any meromorphic solution M is necessarily elliptic or degenerate elliptic. We then give the two canonical decompositions of the new elliptic solution recently obtained by the subequation method.