A Modulation Method for Self-focusing in the Perturbed Critical Nonlinear Schr Odinger Equation

In this Letter we introduce a systematic perturbation method for analyzing the e ect of small perturbations on critical self focusing by reducing the perturbed critical nonlinear Schrodinger equation (PNLS) to a simpler system of modulation equations that do not depend on the transverse variables. The modulation equations can be further simpli ed, depending on whether PNLS is power conserving or not. An importantand somewhat surprising result is that various small defocusingperturbations lead to a universal form for the modulation equations, whose solutions have slowly decaying focusing-defocusing oscillations. PACS 42.65.J keywords: self-focusing, nonlinear Schrodinger equation, modulation theory