On the integrable six-wave interaction system and its space-time shifted reduction
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911204247601152 |
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| author | Ablowitz, Mark J. Gupta, Ramesh Musslimani, Ziad H. Ossi, Nicholas J. |
| author_facet | Ablowitz, Mark J. Gupta, Ramesh Musslimani, Ziad H. Ossi, Nicholas J. |
| contents | The multi-dimensional six-wave interaction system is derived in the context of nonlinear optics. Starting from Maxwell's equations, a reduced system of equations governing the dynamics of the electric and polarization fields are obtained. Using a space-time multi-scale asymptotic expansion, a hierarchy of coupled equations describing the spatio-temporal evolution of the perturbed electric and polarization fields are derived. The leading order equation admits a six-wave ansatz satisfying a triad resonance condition. By removing secular terms at next order, a first order in space and time quadratically nonlinear coupled six-wave interaction system is obtained. This resulting system is tied to its integrable counterpart which was postulated by Ablowitz and Haberman in the 1970s. A reduction to a space-time shifted nonlocal three-wave system is presented. The resulting system is solved using the inverse scattering transform, which employs nonlocal symmetries between the associated eigenfunctions and scattering data; soliton solutions are then obtained. Finally, an infinite set of conservation laws for the six-wave system is derived; one is shown to be connected to its Hamiltonian structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10312 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the integrable six-wave interaction system and its space-time shifted reduction Ablowitz, Mark J. Gupta, Ramesh Musslimani, Ziad H. Ossi, Nicholas J. Exactly Solvable and Integrable Systems The multi-dimensional six-wave interaction system is derived in the context of nonlinear optics. Starting from Maxwell's equations, a reduced system of equations governing the dynamics of the electric and polarization fields are obtained. Using a space-time multi-scale asymptotic expansion, a hierarchy of coupled equations describing the spatio-temporal evolution of the perturbed electric and polarization fields are derived. The leading order equation admits a six-wave ansatz satisfying a triad resonance condition. By removing secular terms at next order, a first order in space and time quadratically nonlinear coupled six-wave interaction system is obtained. This resulting system is tied to its integrable counterpart which was postulated by Ablowitz and Haberman in the 1970s. A reduction to a space-time shifted nonlocal three-wave system is presented. The resulting system is solved using the inverse scattering transform, which employs nonlocal symmetries between the associated eigenfunctions and scattering data; soliton solutions are then obtained. Finally, an infinite set of conservation laws for the six-wave system is derived; one is shown to be connected to its Hamiltonian structure. |
| title | On the integrable six-wave interaction system and its space-time shifted reduction |
| topic | Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2510.10312 |