Beyond leading-order logarithmic scaling in the catastrophic self-focusing of a laser beam in Kerr media

We study the catastrophic stationary self-focusing (collapse) of a laser beam in nonlinear Kerr media. The width of self-similar solutions near the collapse distance $z={z}_{c}$ obeys the ${({z}_{c}\ensuremath{-}z)}^{1/2}$ scaling law with the well-known leading-order modification of loglog type $\ensuremath{\propto}(\mathrm{ln}|\mathrm{ln}({z}_{c}\ensuremath{-}z){|)}^{\ensuremath{-}1/2}$. We show that the validity of the loglog modification requires double-exponentially large amplitudes of the solution $\ensuremath{\sim}$${{10}^{10}}^{100}$, which is unrealistic to achieve in either physical experiments or numerical simulations. We derive an equation for the adiabatically slow parameter which determines the system self-focusing across a large range of solution amplitudes. Based on this equation we develop a perturbation theory for scaling modifications beyond the leading loglog. We show that, for the initial pulse with the optical power moderately above ($\ensuremath{\lesssim}$$1.2$) the critical power of self-focusing, the scaling agrees with numerical simulations beginning with amplitudes around only three times above the initial pulse.