Numerical Method for Finding 3D Solitons of the NonlinearSchrodinger Equation in the Axially Symmetric Case

摘要

A system of two nonlinear Schrodinger equations is considered that governs the frequencydoubling of femtosecond pulses propagating in an axially symmetric medium with quadratic and cubicnonlinearity. A numerical method is proposed to find soliton solutions of the problem, which is previ-ously reformulated as an eigenvalue problem. The practically important special case of a singleSchrodinger equation is discussed. Since three-dimensional solitons in the case of cubic nonlinearityare unstable with respect to small perturbations in their shape, a stabilization method is proposedbased on weak modulations of the cubic nonlinearity coefficient and variations in the length of thefocalizing layers. It should be emphasized that, according to the literature, stabilization was previouslyachieved by alternating layers with oppositely signed nonlinearities or by using nonlinear layers withstrongly varying nonlinearities (of the same sign). In the case under study, it is shown that weak mod-ulation leads to an increase in the length of the medium by more than 4 times without light wave col-lapse. To find the eigenfunctions and eigenvalues of the nonlinear problem, an efficient iterative pro-cess is constructed that produces three-dimensional solitons on large grids.