Space Curves and Solitons of the KP Hierarchy. I. The -th Generalized KdV Hierarchy
It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta functions associated with hyperelliptic curves, and that soliton solutions can be obtained by rational (singular) limits of the corresponding curves. In this paper, we discuss a class of KP...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2021 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2021
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211164 |
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| Zitieren: | Space Curves and Solitons of the KP Hierarchy. I. The -th Generalized KdV Hierarchy. Yuji Kodama and Yuancheng Xie. SIGMA 17 (2021), 024, 43 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862603587861872640 |
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| author | Kodama, Yuji Xie, Yuancheng |
| author_facet | Kodama, Yuji Xie, Yuancheng |
| citation_txt | Space Curves and Solitons of the KP Hierarchy. I. The -th Generalized KdV Hierarchy. Yuji Kodama and Yuancheng Xie. SIGMA 17 (2021), 024, 43 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta functions associated with hyperelliptic curves, and that soliton solutions can be obtained by rational (singular) limits of the corresponding curves. In this paper, we discuss a class of KP solitons in connection with space curves, which are labeled by certain types of numerical semigroups. In particular, we show that some class of the (singular and complex) KP solitons of the -th generalized KdV hierarchy with ≥ 2 is related to the rational space curves associated with the numerical semigroup ⟨, +1, …, +⟩, where m≥1 and 1 ≤ ≤ −1. We also calculate the Schur polynomial expansions of the τ-functions for those KP solitons. Moreover, we construct smooth curves by deforming the singular curves associated with the soliton solutions. For these KP solitons, we also construct the space curve from a commutative ring of differential operators in the sense of the well-known Burchnall-Chaundy theory.
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| first_indexed | 2026-03-14T03:46:07Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211164 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T03:46:07Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
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| spelling | Kodama, Yuji Xie, Yuancheng 2025-12-25T13:19:26Z 2021 Space Curves and Solitons of the KP Hierarchy. I. The -th Generalized KdV Hierarchy. Yuji Kodama and Yuancheng Xie. SIGMA 17 (2021), 024, 43 pages 1815-0659 2020 Mathematics Subject Classification: 37K40; 37K10; 14H70; 14H50 arXiv:1912.06768 https://nasplib.isofts.kiev.ua/handle/123456789/211164 https://doi.org/10.3842/SIGMA.2021.024 It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta functions associated with hyperelliptic curves, and that soliton solutions can be obtained by rational (singular) limits of the corresponding curves. In this paper, we discuss a class of KP solitons in connection with space curves, which are labeled by certain types of numerical semigroups. In particular, we show that some class of the (singular and complex) KP solitons of the -th generalized KdV hierarchy with ≥ 2 is related to the rational space curves associated with the numerical semigroup ⟨, +1, …, +⟩, where m≥1 and 1 ≤ ≤ −1. We also calculate the Schur polynomial expansions of the τ-functions for those KP solitons. Moreover, we construct smooth curves by deforming the singular curves associated with the soliton solutions. For these KP solitons, we also construct the space curve from a commutative ring of differential operators in the sense of the well-known Burchnall-Chaundy theory. One of the authors (YK) would like to thank Jing-PingWang and Sasha Mikhailov for useful discussions at the beginning stage of the present work. He also appreciated their financial support by the EPSRC grants EP/P012698/1 and EP/P1012655/1 during his stay at the University of Kent and the University of Leeds. He also thanks Atsushi Nakayashiki for many helpful and useful comments on the Sato Grassmannians and his interest in the present work, and Shigeki Matsutani for valuable comments on space curves and numerical semigroups. We would like to thank Herb Clemens for the useful discussion on the singularity of space curves. We would also like to thank the referees for their valuable comments and suggestions. The present work is partially supported by NSF grant DMS-1714770. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Space Curves and Solitons of the KP Hierarchy. I. The -th Generalized KdV Hierarchy Article published earlier |
| spellingShingle | Space Curves and Solitons of the KP Hierarchy. I. The -th Generalized KdV Hierarchy Kodama, Yuji Xie, Yuancheng |
| title | Space Curves and Solitons of the KP Hierarchy. I. The -th Generalized KdV Hierarchy |
| title_full | Space Curves and Solitons of the KP Hierarchy. I. The -th Generalized KdV Hierarchy |
| title_fullStr | Space Curves and Solitons of the KP Hierarchy. I. The -th Generalized KdV Hierarchy |
| title_full_unstemmed | Space Curves and Solitons of the KP Hierarchy. I. The -th Generalized KdV Hierarchy |
| title_short | Space Curves and Solitons of the KP Hierarchy. I. The -th Generalized KdV Hierarchy |
| title_sort | space curves and solitons of the kp hierarchy. i. the -th generalized kdv hierarchy |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211164 |
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