The Complex Ginzburg-Landau Equation as a Pertubation of the Nonlinear Schrödinger Equation: A Melnikov Approach
Thursday, October 2, 1997 - 11:00am - 12:00pm
The generalized complex Ginzburg-Landau (CGL) equation has a long history in physics as a generic amplitude equation near the onset of instabilities that lead to chaotic dynamics in numerous physical systems. We study it as a damped-driven perturbation of the nonlinear Schrödinger (NLS) equation, which has a Hamiltonian structure. We give conditions that quasi-periodic and homoclinic structures in the NLS phase space must satisfy if they are to persist under the CGL perturbation. We show that for NLS rotating and traveling waves, these conditions genericly determine which of them persist. When the CGL has a global Lyapunov function, a complete bifurcation analysis of persisting solutions can be carried out and their stability analyzed. When the focusing NLS equation is perturbed, we show that none of the NLS orbits homoclinic to rotating waves persist, but rather distinct homoclinic structures are produced by the CGL perturbation.