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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2511.14519 |
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| _version_ | 1866913142807724032 |
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| author | Taiwo, T. J. Alhaidari, A. D. Khawaja, U. Al |
| author_facet | Taiwo, T. J. Alhaidari, A. D. Khawaja, U. Al |
| contents | 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 ψ^{2n + 1} nonlinearity, where n is a natural number, and obtain results for the cubic and quintic nonlinearities, ψ^3 and ψ^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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14519 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perturbative nonlinear J-matrix method of scattering in two dimensions Taiwo, T. J. Alhaidari, A. D. Khawaja, U. Al Quantum Physics Exactly Solvable and Integrable Systems 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 ψ^{2n + 1} nonlinearity, where n is a natural number, and obtain results for the cubic and quintic nonlinearities, ψ^3 and ψ^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. |
| title | Perturbative nonlinear J-matrix method of scattering in two dimensions |
| topic | Quantum Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2511.14519 |