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We study the catastrophic stationary self-focusing (collapse) of laser beamin nonlinear Kerr media. The width of a self-similar solutions near collapsedistance $z=z_c$ obeys $(z_c-z)^{1/2}$ scaling law with the well-known leadingorder modification of loglog type $\propto (\ln|\ln(z_c-z)|)^{-1/2}$. We showthat the validity of the loglog modification requires double-exponentiallylarge amplitudes of the solution $\sim {10^{10}}^{100}$, which is unrealisticto achieve in either physical experiments or numerical simulations. We derive anew equation for the adiabatically slow parameter which determines the systemself-focusing across a large range of solution amplitudes. Based on thisequation we develop a perturbation theory for scaling modifications beyond theleading loglog. We show that for the initial pulse with the optical powermoderately above ($\lesssim 1.2$) the critical power of self-focusing, the newscaling agrees with numerical simulations beginning with amplitudes around onlythree times above of the initial pulse.

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