Stationary Flows Revisited
In this paper, we revisit the subject of stationary flows of Lax hierarchies of a coupled KdV class. We explain the main ideas in the standard KdV case and then consider the dispersive water waves (DWW) case, with respectively 2 and 3 Hamiltonian representations. Each Hamiltonian representation give...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2023 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2023
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211869 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Stationary Flows Revisited. Allan P. Fordy and Qing Huang. SIGMA 19 (2023), 015, 34 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862668219165179904 |
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| author | Fordy, Allan P. Huang, Qing |
| author_facet | Fordy, Allan P. Huang, Qing |
| citation_txt | Stationary Flows Revisited. Allan P. Fordy and Qing Huang. SIGMA 19 (2023), 015, 34 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper, we revisit the subject of stationary flows of Lax hierarchies of a coupled KdV class. We explain the main ideas in the standard KdV case and then consider the dispersive water waves (DWW) case, with respectively 2 and 3 Hamiltonian representations. Each Hamiltonian representation gives us a different form of stationary flow. Comparing these, we construct Poisson maps, which, being non-canonical, give rise to bi-Hamiltonian representations of the stationary flows. An alternative approach is to use the Miura maps, which we do in the case of the DWW hierarchy, which has two ''modifications''. This structure gives us 3 sequences of Poisson-related stationary flows. We use the Poisson maps to build a tri-Hamiltonian representation of each of the three stationary hierarchies. One of the Hamiltonian representations allows a multi-component squared eigenfunction expansion, which gives degrees of freedom Hamiltonians, with first integrals. A Lax representation for each of the stationary flows is derived from the coupled KdV matrices. In the case of 3 degrees of freedom, we give a generalisation of our Lax matrices and Hamiltonian functions, which allows a connection with the rational Calogero-Moser (CM) system. This gives a coupling of the CM system with other potentials, along with a Lax representation. We present the particular case of coupling one of the integrable Hénon-Heiles systems to CM.
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| first_indexed | 2026-03-16T11:49:32Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211869 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T11:49:32Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Fordy, Allan P. Huang, Qing 2026-01-14T09:53:40Z 2023 Stationary Flows Revisited. Allan P. Fordy and Qing Huang. SIGMA 19 (2023), 015, 34 pages 1815-0659 2020 Mathematics Subject Classification: 35Q53; 37K10; 70H06 arXiv:2210.12771 https://nasplib.isofts.kiev.ua/handle/123456789/211869 https://doi.org/10.3842/SIGMA.2023.015 In this paper, we revisit the subject of stationary flows of Lax hierarchies of a coupled KdV class. We explain the main ideas in the standard KdV case and then consider the dispersive water waves (DWW) case, with respectively 2 and 3 Hamiltonian representations. Each Hamiltonian representation gives us a different form of stationary flow. Comparing these, we construct Poisson maps, which, being non-canonical, give rise to bi-Hamiltonian representations of the stationary flows. An alternative approach is to use the Miura maps, which we do in the case of the DWW hierarchy, which has two ''modifications''. This structure gives us 3 sequences of Poisson-related stationary flows. We use the Poisson maps to build a tri-Hamiltonian representation of each of the three stationary hierarchies. One of the Hamiltonian representations allows a multi-component squared eigenfunction expansion, which gives degrees of freedom Hamiltonians, with first integrals. A Lax representation for each of the stationary flows is derived from the coupled KdV matrices. In the case of 3 degrees of freedom, we give a generalisation of our Lax matrices and Hamiltonian functions, which allows a connection with the rational Calogero-Moser (CM) system. This gives a coupling of the CM system with other potentials, along with a Lax representation. We present the particular case of coupling one of the integrable Hénon-Heiles systems to CM. This work was supported by the National Natural Science Foundation of China (grant no. 11871396). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Stationary Flows Revisited Article published earlier |
| spellingShingle | Stationary Flows Revisited Fordy, Allan P. Huang, Qing |
| title | Stationary Flows Revisited |
| title_full | Stationary Flows Revisited |
| title_fullStr | Stationary Flows Revisited |
| title_full_unstemmed | Stationary Flows Revisited |
| title_short | Stationary Flows Revisited |
| title_sort | stationary flows revisited |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211869 |
| work_keys_str_mv | AT fordyallanp stationaryflowsrevisited AT huangqing stationaryflowsrevisited |