Slowly varying fully nonlinear wavetrains in the Ginzburg-Landau equation

Following the ideas of Howard and Kopell 9] a perturbation theory is developed for slowly varying fully nonlinear wavetrains (i.e. solutions which appear locally as travelling waves, but with frequencies and wavelengths which may vary widely on long length and time scales). This perturbation theory is applied to the Ginzburg-Landau equation. The motion and stability of slowly varying wavetrains is shown to be governed by a simple wave equation which can develop shocks corresponding to rapid changes in wavenumber. Numerical results supporting this theory are presented. A shock structure is proposed and numerically verified. These results together with a winding invariant valid in the limit of slow variation suggest that over a large range of parameters many initial conditions relax to uniform wavetrains. The evolution of a marginally diffusively stable wavetrain is also examined; it is argued that the evolution is governed by a perturbed Korteweg-de Vries equation.