On the c-k constrained KP and BKP hierarchies: the Fermionic pictures, solutions and additional symmetries
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929257970663424 |
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| author | Tian, Kelei Li, Song Yi, Ge Xu, Ying Cheng, Jipeng |
| author_facet | Tian, Kelei Li, Song Yi, Ge Xu, Ying Cheng, Jipeng |
| contents | In this paper, we study two generalized constrained integrable hierarchies, which are called the $c$-$k$ constrained KP and BKP hierarchies. The Fermionic picture of the $c$-$k$ constrained KP hierarchy is given. We give some solutions for the $c$-$k$ constrained KP hierarchy by using the free Fermion operators and define its additional symmetries. Its additional flows form a subalgebra of the Virasoro algebra. Furthermore, the additional flows acting on eigenfunctions $q_{i}(t)$ and adjoint eigenfunctions $r_{i}(t)$ of the $c$-$k$ constrained KP hierarchy are presented. Next, we define the $c$-$k$ constrained BKP hierarchy and obtain its bilinear identity and solutions. The algebra formed by the additional symmetric flow of the $c$-$k$ constrained BKP hierarchy that we defined is still a subalgebra of the Virasoro algebra and it is a subalgebra of the algebra formed by the additional flows of the $c$-$k$ constrained KP hierarchy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17503 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the c-k constrained KP and BKP hierarchies: the Fermionic pictures, solutions and additional symmetries Tian, Kelei Li, Song Yi, Ge Xu, Ying Cheng, Jipeng Exactly Solvable and Integrable Systems Mathematical Physics In this paper, we study two generalized constrained integrable hierarchies, which are called the $c$-$k$ constrained KP and BKP hierarchies. The Fermionic picture of the $c$-$k$ constrained KP hierarchy is given. We give some solutions for the $c$-$k$ constrained KP hierarchy by using the free Fermion operators and define its additional symmetries. Its additional flows form a subalgebra of the Virasoro algebra. Furthermore, the additional flows acting on eigenfunctions $q_{i}(t)$ and adjoint eigenfunctions $r_{i}(t)$ of the $c$-$k$ constrained KP hierarchy are presented. Next, we define the $c$-$k$ constrained BKP hierarchy and obtain its bilinear identity and solutions. The algebra formed by the additional symmetric flow of the $c$-$k$ constrained BKP hierarchy that we defined is still a subalgebra of the Virasoro algebra and it is a subalgebra of the algebra formed by the additional flows of the $c$-$k$ constrained KP hierarchy. |
| title | On the c-k constrained KP and BKP hierarchies: the Fermionic pictures, solutions and additional symmetries |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics |
| url | https://arxiv.org/abs/2402.17503 |