Backscattering and Nonparaxiality Arrest Collapse of Damped Nonlinear Waves

The critical nonlinear SchrSdinger equation (NLS) models the propagation of intense laser light ill Kerr media. This equation is derived from tile more conq)reheasive nonlinear Hehnholtz e(tuation (NLH) t)y employing the paraxial at)t)roximation and negle(:ting th(' backscattered waves. It is known that if the input power of tile laser beam (i.e., L2 norm of the ilfitial soh,ti(m) is sufficiently high, then the NLS inodcl predicts that the beanl will self-fo(:us to a l)oint (i.e., collal):m) at a finite propagation distance. Matheulati(:ally, this behavior corresponds to the formation of a singularity in the solution of the NLS. A key question which has been open for many years is whelher the solution to t im NLH, i.e., tile "parent" equation. may nonetheless exist and remain regular everywhere, in particular for those initial conditions (input powers) that lead to blowup in the NLS. In the current study we a(hh'ess this (tue_tion by introducing linear daml)ing into t)oth models and sul)sequently (:omparing the mnnerical solutions of the damped NLH (t)oundary-value problem) with tile corresponding solutions of the (lan_l)ed NLS (initii,l-value t)roblem). Linear daml)ing is introduced in much the same way as done when analyzing the clas:dcal constant-coefficient Hehnholtz equation using tile limiting absorption principle. Numerically, we have f(,und that it provides a very effi('i(,nl tool for controlling tile solutions of both the NLH an(t NLS. In t)artic flar, we have been able to identify initial conditions for which the NLS solution does beeonle singular, whereas the NLH solution still remains regular everywhere. _\> believe that our finding of a larger domain of ( xistence for the NLH than that for tile NLS is accounted for by precisely those mechanisms that have bee_L neglected when deriving the NLS from the NLH, i.e., nonparaxiality and backscattering.