Perturbative nonlinear J-matrix method of scattering in two dimensions

TL;DR

This paper develops a perturbative nonlinear extension of the J-matrix scattering method in two dimensions, enabling analysis of nonlinear Schrödinger equations with bifurcation phenomena.

Contribution

It introduces a novel perturbative formulation for nonlinear scattering in 2D using the J-matrix method, including numerical implementation and bifurcation analysis.

Findings

Derived results for cubic and quintic nonlinearities.

Observed bifurcation with two stable solutions at certain energies.

Demonstrated the method's applicability to nonlinear Schrödinger equations.

Abstract

We introduce a perturbative formulation for a nonlinear extension of the J-matrix method of scattering in two dimensions. That is, we obtain the scattering matrix for the time-independent nonlinear Schr\"odinger equation in two dimensions with circular symmetry. The formulation relies on the linearization of products of orthogonal polynomials and on the utilization of the tools of the J-matrix method. Gauss quadrature integral approximation is instrumental in the numerical implementation of the approach. We present the theory for a general \psi ^{2n + 1} nonlinearity, where n is a natural number, and obtain results for the cubic and quintic nonlinearities, \psi ^3 and \psi ^5. At certain value(s) of the energy, we observe the occurrence of bifurcation with two stable solutions. This curious and interesting phenomenon is a clear signature and manifestation of the underlying nonlinearity.