Complex Ginzburg–Landau equations as perturbations of nonlinear Schrödinger equations: A Melnikov approach

We study the persistence of quasiperiodic and homoclinic solutions of generalized nonlinear Schrödinger equations under Ginzburg–Landau perturbations. In this paper, the first of a series, Melnikov criteria for the persistence of quasiperiodic and homoclinic solutions are derived directly from the governing partial differential equations via an averaging technique. For families of tori of quasiperiodic solutions, such as rotating waves and traveling waves, that arise within critical sets of linear combinations of conserved functionals, we find that usually only isolated tori will satisfy these selection criteria. Moreover, in some simple cases these criteria are sufficient to conclude that a torus persists. We also demonstrate the nonpersistence of solutions that are homoclinic to rotating waves under a broad class of Ginzburg–Landau perturbations which satisfy a convexity condition.