Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations
Discrete Lagrangian multiform theory is a variational perspective on lattice equations that are integrable in the sense of multidimensional consistency. The Lagrangian multiforms for the equations of the ABS classification formed the start of this theory, but the Lagrangian multiforms that are usual...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2025 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2025
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/213518 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations. Jacob J. Richardson and Mats Vermeeren. SIGMA 21 (2025), 058, 30 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862716406981722112 |
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| author | Richardson, Jacob J. Vermeeren, Mats |
| author_facet | Richardson, Jacob J. Vermeeren, Mats |
| citation_txt | Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations. Jacob J. Richardson and Mats Vermeeren. SIGMA 21 (2025), 058, 30 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Discrete Lagrangian multiform theory is a variational perspective on lattice equations that are integrable in the sense of multidimensional consistency. The Lagrangian multiforms for the equations of the ABS classification formed the start of this theory, but the Lagrangian multiforms that are usually considered in this context produce equations that are slightly weaker than the ABS equations. In this work, we present alternative Lagrangian multiforms that have Euler-Lagrange equations equivalent to the ABS equations. In addition, the treatment of the ABS Lagrangian multiforms in the existing literature fails to acknowledge that the complex functions in their definitions have branch cuts. The choice of branch affects both the existence of an additive three-leg form for the ABS equations and the closure property of the Lagrangian multiforms. We give counterexamples for both these properties, but we recover them by including integer-valued fields, related to the branch choices, in the action sums.
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| first_indexed | 2026-03-20T13:21:50Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-213518 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T13:21:50Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Richardson, Jacob J. Vermeeren, Mats 2026-02-18T11:23:17Z 2025 Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations. Jacob J. Richardson and Mats Vermeeren. SIGMA 21 (2025), 058, 30 pages 1815-0659 2020 Mathematics Subject Classification: 39A36; 37J70; 37J06 arXiv:2501.13012 https://nasplib.isofts.kiev.ua/handle/123456789/213518 https://doi.org/10.3842/SIGMA.2025.058 Discrete Lagrangian multiform theory is a variational perspective on lattice equations that are integrable in the sense of multidimensional consistency. The Lagrangian multiforms for the equations of the ABS classification formed the start of this theory, but the Lagrangian multiforms that are usually considered in this context produce equations that are slightly weaker than the ABS equations. In this work, we present alternative Lagrangian multiforms that have Euler-Lagrange equations equivalent to the ABS equations. In addition, the treatment of the ABS Lagrangian multiforms in the existing literature fails to acknowledge that the complex functions in their definitions have branch cuts. The choice of branch affects both the existence of an additive three-leg form for the ABS equations and the closure property of the Lagrangian multiforms. We give counterexamples for both these properties, but we recover them by including integer-valued fields, related to the branch choices, in the action sums. The impetus for this work was provided by the critical questions asked by an anonymous referee of the paper that has now become Part II of the present work [17]. We are grateful for their detailed feedback and constructive criticism. We would like to thank Prof Frank Nijhoff and Dr. Vincent Caudrelier for helpful discussions on the topic of this work and many related subjects. JR acknowledges funding from the Engineering and Physical Sciences Research Council DTP, Crowther Endowment, and School of Mathematics at the University of Leeds. MV is supported by the Engineering and Physical Sciences Research Council [EP/Y006712/1]. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations Article published earlier |
| spellingShingle | Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations Richardson, Jacob J. Vermeeren, Mats |
| title | Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations |
| title_full | Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations |
| title_fullStr | Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations |
| title_full_unstemmed | Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations |
| title_short | Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations |
| title_sort | discrete lagrangian multiforms for abs equations i: quad equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213518 |
| work_keys_str_mv | AT richardsonjacobj discretelagrangianmultiformsforabsequationsiquadequations AT vermeerenmats discretelagrangianmultiformsforabsequationsiquadequations |