A Modulation Method for Self-focusing in the Perturbed

In this Letter we introduce a systematic perturbation method for analyzing the eeect of small perturbations on critical self focusing by reducing the perturbed critical nonlinear Schrr odinger equation (PNLS) to a simpler system of modulation equations that do not depend on the transverse variables. The modulation equations can be further simpliied, depending on whether PNLS is power conserving or not. An importantand somewhat surprising result is that various small defocusingper-turbations lead to a universal form for the modulation equations, whose solutions have slowly decaying focusing-defocusing oscillations. @y 2 ; 0 < 1 ; arises in various physical models in nonlinear optics 1 , plasma physics and uid dynamics (e.g. table 1). When = 0 eq. (1) reduces to the critical nonlinear Schrr odinger equation (CNLS) ii z + ? + jj 2 = 0 : (2) We recall that for the nonlinear Schrr odinger equation with a general nonlinearity and transverse dimension D ii z + @ 2 @x 2 1 + + @ 2 @x 2 D + jj 2 = 0 ; we distinguish between three diierent cases: 1) When D < 2, the subcritical case, diiraction always dominates and focusing singularities do not form. 2) In the supercritical case D > 2, there is a large class of smooth initial amplitudes for which a focusing singularity forms in nite distance z. Since in supercritical self-focusing the nonlinearity dominates over diiraction, addition of small perturbations to the equation has a small eeect. 3) In the critical case D = 2 (as in the case of eq. 2), solutions can also become singular in a nite z. However, in this borderline case between subcritical and supercritical self-focusing, singularity formation is characterized by a near-balance between the focusing nonlinearity and diiraction. As a result, critical self-focusing is extremely sensitive to small perturbations, which can have a large eeect and can even lead to the arrest of collapse. Self-focusing is a genuinely nonlinear phenomenon and standard linearization methods cannot be used to analyze singularity formation in eqs. (1) and (2). In addition, methods such as the inverse scattering transform (IST), which is so successful in the 1D cubic subcritical case, cannot be applied to eq. (2), because (2) is not integrable. Self-focusing in (1) or (2) is, moreover, a local phenomenon which cannot be accurately captured by global estimates. For these reasons, despite considerable progress the present theory of critical …