Fordy, AP orcid.org/0000-0002-2523-0262 and Huang, Q (2023) Stationary Flows Revisited. Symmetry, Integrability and Geometry: Methods and Applications, 19. 015. ISSN 1815-0659
Abstract
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 N 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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © The Authors 2023. This is an open access article under the terms of the Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0). |
Keywords: | KdV hierarchy; stationary flows; bi-Hamiltonian; complete integrability; Hénon-Heiles; Calogero-Moser |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 17 Apr 2023 11:06 |
Last Modified: | 25 Jun 2023 23:19 |
Published Version: | http://dx.doi.org/10.3842/SIGMA.2023.015 |
Status: | Published |
Publisher: | National Academy of Science of Ukraine |
Identification Number: | 10.3842/SIGMA.2023.015 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:198217 |