The nonlinear Schroedinger limit of the complex Ginzburg-Landau equation.

Abstract

This work consists of a study of the complex Ginzburg-Landau equation (CGL) as a perturbation of the nonlinear Schrodinger equation (NLS) in one dimension under periodic boundary conditions. Using an averaging technique which is similar to a Melnikov method for pde's, necessary conditions are derived for the persistence of NLS solutions under the CGL perturbation. For the traveling wave solutions, these conditions are derived for a general nonlinearity and written explicitly as two equations for the two continuous parameters which determine the NLS traveling wave. It is shown using a Melnikov argument that in this case these two conditions are sufficient provided they satisfy a transversality condition. As a concrete example, the equations for the parameters are solved numerically in the important case of the CGL equation with a cubic nonlinearity. For the case of the CGL equation with a general power nonlinearity, it is proved that the NLS homoclinic orbits to rotating waves are destroyed by the CGL perturbation. Special attention is dedicated to the cubic case. For this nonlinearity, the NLS equation is a completely integrable Hamiltonian system and a much larger family of its solutions can be written explicitly. The necessary conditions for the persistence of the NLS isospectral manifold are written explicitly as a system of equations for the simple periodic eigenvalues. As an example, the conditions for an even genus two solution are written down as a system of three equations with three unknowns.