Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension

The formation of singularities of self-focusing solutions of the nonlinear Schrodinger equation (NLS) in critical dimension is characterized by a deli cate balance between the focusing nonlinearity and diffraction (Laplacian), and is thus very sensitive to small perturbations. In this paper we introd uce a systematic perturbation theory for analyzing the effect of additional small terms on self-focusing, in which the perturbed critical NLS is reduc ed to a simpler system of modulation equations that do not depend on the sp atial variables transverse to the beam axis. The modulation equations can b e further simplified, depending on whether the perturbed NLS is power conse rving or not. We review previous applications of modulation theory and pres ent several new ones that include dispersive saturating nonlinearities, sel f-focusing with Debye relaxation, the Davey-Stewartson equations, self-focu sing in optical fiber arrays, and the effect of randomness. An important an d somewhat surprising result is that various small defocusing perturbations lead to a generic form of the modulation equations, whose solutions have s lowly decaying focusing-defocusing oscillations. In the special case of the unperturbed critical NLS, modulation theory leads to a new adiabatic law f or the rate of blowup which is accurate from the early stages of self-focus ing and remains valid up to the singularity point. This adiabatic law prese rves the lens transformation property of critical NLS and it leads to an an alytic formula for the location of the singularity as a function of the ini tial pulse power, radial distribution, and focusing angle. The asymptotic l imit of this law agrees with the known loglog blowup behavior. However, the loglog behavior is reached only after huge amplifications of the initial a mplitude, at which point the physical basis of NLS is in doubt. We also inc lude in this paper a new condition for blowup of solutions in critical NLS and an improved version of the Dawes-Marburger formula for the blowup locat ion of Gaussian pulses.