Self-focusing in the complex Ginzburg-Landau limit of the critical nonlinear Schrodinger equation

We analyze serf-focusing and singularity formation in the complex Ginzburg- Landau equation (CGL) in the regime where it;is close to the critical nonli near Schrodinger equation. Using modulation theory [Fibich and Papanicolaou , Phys. Lett. A 239 (1998) 167], we derive a reduced system of ordinary dif ferential equations that describes self-focusing in CGL. Analysis of the re duced system shows that in the physical regime of the parameters there is n o blowup in CGL. Rather, the solution focuses once and then defocuses. The validity of the analysis is verified by comparison of numerical solutions o f CGL with those of the reduced system. (C) 1998 Elsevier Science B.V.