Stationary reduction method based on nonisospectral deformation of orthogonal polynomials, and discrete Painlevé-type equations
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866909791450824704 |
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| author | Yue, Xiao-Lu Chang, Xiang-Ke Hu, Xing-Biao |
| author_facet | Yue, Xiao-Lu Chang, Xiang-Ke Hu, Xing-Biao |
| contents | In this work, we propose a new approach called ``stationary reduction method based on nonisospectral deformation of orthogonal polynomials" for deriving discrete Painlevé-type (d-P-type) equations. We apply this approach to (bi)orthogonal polynomials satisfying ordinary orthogonality, $(1,m)$-biorthogonality, generalized Laurent biorthogonality, Cauchy biorthogonality and partial-skew orthogonality. As a result, several seemingly novel classes of high order d-P-type equations or integrable difference systems with potential relations with new d-P-type equations, along with their particular solutions and respective Lax pairs, are derived. Notably, the derived integrable difference system related to the Cauchy biorthogonality is a stationary reduction of a nonisospectral generalization involving the first two flows of the Toda hierarchy of CKP type. Additionally, the integrable difference system related to the partial-skew orthogonality is associated with the nonisospectral Toda hierarchy of BKP type, and it is found to admit a solution expressed in terms of Pfaffians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09875 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stationary reduction method based on nonisospectral deformation of orthogonal polynomials, and discrete Painlevé-type equations Yue, Xiao-Lu Chang, Xiang-Ke Hu, Xing-Biao Exactly Solvable and Integrable Systems Mathematical Physics 33C47, 33E17, 34M55, 39A36, 37K10 In this work, we propose a new approach called ``stationary reduction method based on nonisospectral deformation of orthogonal polynomials" for deriving discrete Painlevé-type (d-P-type) equations. We apply this approach to (bi)orthogonal polynomials satisfying ordinary orthogonality, $(1,m)$-biorthogonality, generalized Laurent biorthogonality, Cauchy biorthogonality and partial-skew orthogonality. As a result, several seemingly novel classes of high order d-P-type equations or integrable difference systems with potential relations with new d-P-type equations, along with their particular solutions and respective Lax pairs, are derived. Notably, the derived integrable difference system related to the Cauchy biorthogonality is a stationary reduction of a nonisospectral generalization involving the first two flows of the Toda hierarchy of CKP type. Additionally, the integrable difference system related to the partial-skew orthogonality is associated with the nonisospectral Toda hierarchy of BKP type, and it is found to admit a solution expressed in terms of Pfaffians. |
| title | Stationary reduction method based on nonisospectral deformation of orthogonal polynomials, and discrete Painlevé-type equations |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics 33C47, 33E17, 34M55, 39A36, 37K10 |
| url | https://arxiv.org/abs/2407.09875 |