Non-existence of elliptic travelling wave solutions of the complex Ginzburg–Landau equation

Abstract

We give a simple proof that, for generic parameter values, the cubic complex one-dimensional Ginzburg–Landau equation has no elliptic travelling wave solutions. This is contrary to the expectations of Musette and Conte, in Physica D 181 (2003) 70–79, that elliptic solutions of zero codimension should exist. The method of proof, based on the residue theorem, is very general, and can be applied to determine necessary conditions for the existence of elliptic travelling waves for any autonomous partial differential equation. As another application, we prove that Kudryashov’s codimension-one elliptic solution of the generalized Kuramoto–Sivashinsky equation is the only one possible.