New solutions in the theory of self-focusing with saturating nonlinearity
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New Solutions in the Theory of SelfFocusing with Saturating Nonlinearity V. F. Kovaleva,*, K. I. Popovb, and V. Yu. Bychenkovc a
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russia b Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario K1N 6N5, Canada cLebedev Physical Institute, Russian Academy of Sciences, Leninskii pr. 53, Moscow, 119991 Russia *email: [email protected] Received April 20, 2011
Abstract—On the basis of an approximate analytic solution of a Cauchy problem for a nonlinear Schrödinger (NLS) equation describing steadystate light beams in a medium with saturating nonlinearity by the method of renormgroup (RG) symmetries, a classification of selffocusing solutions is given depending on two con trol parameters: the relative contributions of diffraction and nonlinearity and the saturation strength of the nonlinearity. The existence of tubetype selffocusing solutions is proved for an entering beam with Gaussian radial distribution of intensity. Numerical simulation is carried out that allows one to verify the theory devel oped and to determine its applicability limits. DOI: 10.1134/S1063776112010128
1. INTRODUCTION
described by numerical simulation [12–15] because the relevant equations belong to the class of noninte grable systems, which allow, if any, only qualitative analysis. The main results of the numerical simulation confirm that selffocusing in the saturatingnonlin earity regime exhibits qualitatively different properties compared with those in a medium with cubic nonlin earity; in particular, these are the limited growth of the beam intensity on its axis and the flattening of the phase front near the beam axis (see, for example, [5, p. 109]).
This year marks the 50th anniversary of the discov ery, by Askar’yan, of the phenomenon of selffocusing of light in a nonlinear medium [1]. Since then, starting with Talanov’s paper [2], the problems of mathemati cally rigorous or mathematically validated approxi mate description of the phenomenon of selffocusing of a wave beam have been solved for various types of nonlinearity of a medium and various models of light propagation [3–8]. The most comprehensive study of selffocusing of wave beams has been carried out for nonlinear media with cubic nonlinearity. The results obtained were represented, on the one hand, by exact and approximate analytical theories and, on the other hand, by numerous numerical calculations. The description of these results can be found in various sur veys and monographs [5–8]. From the mathematical point of view, selffocusing for cubic nonlinearity, as well as for other types of algebraic nonlinearity, mani fests itself in the emergence of a singularity in the solu tion of the equations that are used to describe the spa tial distribution of the electric field of a light beam. There are a number of papers devoted to the condition under which a singularity arises and to the detailed analysis of the field
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