### Thursday, May 22

1:30 PM-2:30 PM

*Chair: Vered Rom-Kedar, Weizmann Institute of Science, Israel*
*Ballroom I, II, III - Level B*

## IP11

Complex Ginzburg-Landau Equations as Perturbations of Nonlinear Schrödinger
Equations

General complex Ginzburg-Landau (CGL) equations are considered as perturbations of general nonlinear Schrödinger (NLS) equations over a one dimensional periodic domain. We identify some structures in the phase space of the NLS equations that persist under the CGL perturbation, where "persists" means to be approximated in the C infty topology by a CGL structure as the perturbation tends to zero. Such structures necessarily play an important role in the dynamics of the CGL global attractor. We find criteria for NLS rotating waves and traveling waves to persist and, in some cases, determine their dynamical stability. In the Lyapunov case this characterizes all omega-limit sets of the flow and a complete bifurcation analysis can be carried out. 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.

**C. David Levermore**

*Department of Mathematics*

*University of Arizona*

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*TMP, 4/4/97*