Description of the self-focusing and collapse effects by a modified nonlinear Schrödinger equation

Abstract

The nonlinear Schrödinger equation is completed with the term which takes into account the changes of the propagation constant during propagation. A numerical scheme for the new equation, which is a modification of the Crank-Nicholson scheme, is developed. This numerical scheme maintains the constancy of the on-axis phase of the envelope function and the invariant of the equation corresponding to the energy flow. Some examples are given of numerical simulations for gaussian initial conditions which show the absence of the collapse and multifocusing structure of the beam in a Kerr-law nonlinear medium. Show more