Integrable nonlocal finite-dimensional Hamiltonian systems related to the Ablowitz-Kaup-Newell-Segur system
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916104154120192 |
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| author | Xia, Baoqiang Zhou, Ruguang |
| author_facet | Xia, Baoqiang Zhou, Ruguang |
| contents | The method of nonlinearization of the Lax pair is developed for the Ablowitz-Kaup-Newell-Segur (AKNS) equation in the presence of space-inverse reductions. As a result, we obtain a new type of finite-dimensional Hamiltonian systems: they are nonlocal in the sense that the inverse of the space variable is involved. For such nonlocal Hamiltonian systems, we show that they preserve the Liouville integrability and they can be linearized on the Jacobi variety. We also show how to construct the algebro-geometric solutions to the AKNS equation with space-inverse reductions by virtue of our nonlocal finite-dimensional Hamiltonian systems. As an application, algebro-geometric solutions to the AKNS equation with the Dirichlet and with the Neumann boundary conditions, and algebro-geometric solutions to the nonlocal nonlinear Schrödinger (NLS) equation are obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11259 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integrable nonlocal finite-dimensional Hamiltonian systems related to the Ablowitz-Kaup-Newell-Segur system Xia, Baoqiang Zhou, Ruguang Exactly Solvable and Integrable Systems The method of nonlinearization of the Lax pair is developed for the Ablowitz-Kaup-Newell-Segur (AKNS) equation in the presence of space-inverse reductions. As a result, we obtain a new type of finite-dimensional Hamiltonian systems: they are nonlocal in the sense that the inverse of the space variable is involved. For such nonlocal Hamiltonian systems, we show that they preserve the Liouville integrability and they can be linearized on the Jacobi variety. We also show how to construct the algebro-geometric solutions to the AKNS equation with space-inverse reductions by virtue of our nonlocal finite-dimensional Hamiltonian systems. As an application, algebro-geometric solutions to the AKNS equation with the Dirichlet and with the Neumann boundary conditions, and algebro-geometric solutions to the nonlocal nonlinear Schrödinger (NLS) equation are obtained. |
| title | Integrable nonlocal finite-dimensional Hamiltonian systems related to the Ablowitz-Kaup-Newell-Segur system |
| topic | Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2401.11259 |