Small-Scale Light Structures in a Kerr Medium

A system of equations has been proposed for a monochromatic weakly nonlinear light wave in a Kerr medium. This system is equivalent up to the third order in electric field to the known equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kern 1pt} {\text{curl}}\,{\text{curl}}{\mathbf{E}} = k_{0}<^>{2}[{\mathbf{E}} + \alpha {\text{|}}{\mathbf{E}}{{{\text{|}}}<^>{2}}{\mathbf{E}} + \beta ({\mathbf{E}} \cdot {\mathbf{E}}){\mathbf{E}}{\text{*}}]$$\end{document}, but the new equations are much more convenient for numerical computation. Optical fields with small structures of two or three wavelengths have been simulated using this system. It has been found that a stable self-focused light beam (a two-dimensional vector soliton) in some parametric domain is possible even without modification of nonlinearity. "Inelastic" collisions between two such narrow beams with opposite circular polarizations have been calculated. Furthermore, examples of interacting optical vortices, spatial separation of the circular polarizations, and the Kelvin-Helmholtz instability have been given for defocusing nonlinearity.