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 nonlinear Schrodinger equation. Using modulation theory [Fibich and Papanicolaou, Phys. Lett. A 239 (1998) 167], we derive a reduced system of ordinary differential equations that describes self-focusing in CGL. Analysis of the reduced system shows that in the physical regime of the parameters there is no blowup in CGL. Rather, the solution focuses once and then defocuses. The validity of the analysis is verified by comparison of numerical solutions of CGL with those of the reduced system. (C) 1998 Elsevier Science B.V.