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

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.