Abstract.The formation of singularities of self-focusing solutions of the nonlinear Schr odinger equation NLS in critical dimension is characterized by a delicate balance between the focusing nonlinearity and di raction Laplacian, and is thus very sensitive to small perturbations. In this paper we i n troduce a systematic perturbation theory for analyzing the e ect of additional small terms on self focusing, in which the perturbed critical NLS is reduced to a simpler system of modulation equations that do not depend on the spatial variables transverse to the beam axis. The modulation equations can be further simpli ed, depending on whether the perturbed NLS is power conserving or not. We review previous applications of modulation theory and present several new ones that include: Dispersive saturating nonlinearities, self-focusing with Debye relaxation, the Davey Stewartson equations, self-focusing in optical ber arrays and the e ect of randomness. An important and somewhat surprising result is that various small defocusing perturbations lead to a generic form of the modulation equations, whose solutions have slowly decaying focusing-defocusing oscillations. In the special case of the unperturbed critical NLS, modulation theory leads to a new adiabatic law for the rate of blowup which is accurate from the early stages of self-focusing and remains valid up to the singularity point. This adiabatic law preserves the lens transformation property of critical NLS and it leads to an analytic formula for the location of the singularity as a function of the initial pulse power, radial distribution and focusing angle. The asymptotic limit of this law agrees with the known loglog blowup behavior. However, the loglog behavior is reached only after huge ampli cations of the initial amplitude, at which point the physical basis of NLS is in doubt. We also include 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 location of Gaussian pulses.

We determine the threshold power for self-focusing collapse both in a bulk medium and in a hollow-core waveguide for various spatial profiles. We find that the threshold power for collapse in the waveguide is always equal to the lower-bound prediction for a bulk medium.

No abstract

Models in marketing with asymmetric reference effects lead to nonsmooth optimization problems and differential games which cannot be solved using standard methods. In this study, we introduce a new method for calculating explicitly optimal strategies, open-loop equilibria, and closed-loop equilibria of such nonsmooth problems. Application of this method to the case of asymmetric reference-price effects with loss-aversive consumers leads to the following conclusions: (1) When the planning horizon is infinite, after an introductory stage the optimal price stabilizes at a steady-state price, which is slightly below the optimal price in the absence of reference-price effects. (2) The optimal strategy is the same as in the symmetric case, but with the loss parameter determined by the initial reference-price. (3) Competition does not change the qualitative behavior of the optimal strategy. (4) Adopting an appropriate constant-price strategy results in a minute decline in profits.

Analyses of many different types of nonlinear wave equations indicate that a collapsing wave will transform into a universal blowup profile regardless of its initial shape; that is, the amplitude of the wave increases as the spatial extent decreases in a self-similar fashion. We show experimentally that the spatial profile of a collapsing optical wave evolves to a specific circularly symmetric shape, known as the Townes profile, for elliptically shaped or randomly distorted input beams. These results represent the first experimental confirmation of this universal collapsing behavior and provide deeper insight into the high-power filamentation of femtosecond laser pulses in air.