Perturbative nonlinear J-matrix method of scattering in two dimensions

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 Schrödinger 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 nonlinearity, where n is a natural number, and obtain results for the cubic and quintic nonlinearities, and 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