  For the first time exact analytical solutions to the eikonal equations in
(1+1) dimensions with a refractive index being a saturated function of
intensity are constructed. It is demonstrated that the solutions exhibit
collapse; an explicit analytical expression for the self-focusing position,
where the intensity tends to infinity, is found. Based on an approximated Lie
symmetry group, solutions to the eikonal equations with arbitrary nonlinear
refractive index are constructed. Comparison between exact and approximate
solutions is presented. Approximate solutions to the nonlinear Schrodinger
equation in (1+2) dimensions with arbitrary refractive index and initial
intensity distribution are obtained. A particular case of refractive index
consisting of Kerr refraction and multiphoton ionization is considered. It is
demonstrated that the beam collapse can take place not only at the beam axis
but also in an off-axis ring region around it. An analytical condition
distinguishing these two cases is obtained and explicit formula for the
self-focusing position is presented.
