On trilinear and quadrilinear equations associated with the lattice Gel'fand-Dikii hierarchy
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914905148358656 |
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| author | van der Kamp, P. H. Nijhoff, F. W. McLaren, D. I. Quispel, G. R. W |
| author_facet | van der Kamp, P. H. Nijhoff, F. W. McLaren, D. I. Quispel, G. R. W |
| contents | Introduced in 2012, by Zhang, Zhao, and Nijhoff, the trilinear Boussinesq equation is the natural form of the equation for the $τ$-function of the lattice Boussinesq system. In this paper we study various aspects of this equation: its highly nontrivial derivation from the bilinear lattice AKP equation under dimensional reduction, a quadrilinear dual lattice equation, conservation laws, and periodic reductions leading to higher-dimensional integrable maps and their Laurent property. Furthermore, we consider a higher Gel'fand-Dikii lattice system, its periodic reductions and Laurent property. As a special application, from both a trilinear Boussinesq recurrence as well as a higher Gel'fand-Dikii system of three bilinear recurrences, we establish Somos-like integer sequences. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_11175 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On trilinear and quadrilinear equations associated with the lattice Gel'fand-Dikii hierarchy van der Kamp, P. H. Nijhoff, F. W. McLaren, D. I. Quispel, G. R. W Exactly Solvable and Integrable Systems Mathematical Physics Combinatorics Introduced in 2012, by Zhang, Zhao, and Nijhoff, the trilinear Boussinesq equation is the natural form of the equation for the $τ$-function of the lattice Boussinesq system. In this paper we study various aspects of this equation: its highly nontrivial derivation from the bilinear lattice AKP equation under dimensional reduction, a quadrilinear dual lattice equation, conservation laws, and periodic reductions leading to higher-dimensional integrable maps and their Laurent property. Furthermore, we consider a higher Gel'fand-Dikii lattice system, its periodic reductions and Laurent property. As a special application, from both a trilinear Boussinesq recurrence as well as a higher Gel'fand-Dikii system of three bilinear recurrences, we establish Somos-like integer sequences. |
| title | On trilinear and quadrilinear equations associated with the lattice Gel'fand-Dikii hierarchy |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Combinatorics |
| url | https://arxiv.org/abs/2407.11175 |