Self-similar parabolic beam generation and propagation

The (1+1)-dimensional and (2+1)-dimensional amplified nonlinear Schrodinger equations incorporating diffraction, Kerr nonlinearity, and gain are solved analytically and numerically. An asymptotic solution is found corresponding to self-similar propagation of a beam with parabolic amplitude and phase profiles. While the (1+1)-dimensional solution is directly analogous to parabolic pulse propagation in nonlinear dispersive media, the existence of self-similar propagation in (2+1) dimensions is a nontrivial question, given that spatial solitons are unstable in bulk media with nonsaturating nonlinearities. We show that self-similar parabolic beams are possible in such media with gain and a negative nonlinear index.