Discrete Lagrangian Multiforms for ABS Equations II: Tetrahedron and Octahedron Equations
We present four types of discrete Lagrangian 2-form associated with the integrable quad equations of the ABS list. These include the triangle Lagrangian that has traditionally been used in the Lagrangian multiform description of ABS equations, the trident Lagrangian that was central to Part I of thi...
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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/123456789/213517 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Discrete Lagrangian Multiforms for ABS Equations II: Tetrahedron and Octahedron Equations. Jacob J. Richardson and Mats Vermeeren. SIGMA 21 (2025), 059, 27 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862580443805646848 |
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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 II: Tetrahedron and Octahedron Equations. Jacob J. Richardson and Mats Vermeeren. SIGMA 21 (2025), 059, 27 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present four types of discrete Lagrangian 2-form associated with the integrable quad equations of the ABS list. These include the triangle Lagrangian that has traditionally been used in the Lagrangian multiform description of ABS equations, the trident Lagrangian that was central to Part I of this paper, and two Lagrangians that have not been studied in the multiform setting. Two of the Lagrangian 2-forms have the quad equations, or a system equivalent to the quad equations, as their Euler-Lagrange equations, and one produces the tetrahedron equations. This is in contrast to the triangle Lagrangian 2-form, which produces equations that are weaker than the quad equations (they are equivalent to two octahedron equations). We use relations between the Lagrangian 2-forms to prove that the system of quad equations is equivalent to the combined system of tetrahedron and octahedron equations. Furthermore, for each of the Lagrangian 2-forms, we study the double zero property of the exterior derivative. In particular, this gives a possible variational interpretation to the octahedron equations.
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| first_indexed | 2026-03-21T12:06:05Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-213517 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T12:06:05Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
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| spelling | Richardson, Jacob J. Vermeeren, Mats 2026-02-18T11:23:09Z 2025 Discrete Lagrangian Multiforms for ABS Equations II: Tetrahedron and Octahedron Equations. Jacob J. Richardson and Mats Vermeeren. SIGMA 21 (2025), 059, 27 pages 1815-0659 2020 Mathematics Subject Classification: 39A36; 37J70; 37J06 arXiv:2403.16845 https://nasplib.isofts.kiev.ua/handle/123456789/213517 https://doi.org/10.3842/SIGMA.2025.059 We present four types of discrete Lagrangian 2-form associated with the integrable quad equations of the ABS list. These include the triangle Lagrangian that has traditionally been used in the Lagrangian multiform description of ABS equations, the trident Lagrangian that was central to Part I of this paper, and two Lagrangians that have not been studied in the multiform setting. Two of the Lagrangian 2-forms have the quad equations, or a system equivalent to the quad equations, as their Euler-Lagrange equations, and one produces the tetrahedron equations. This is in contrast to the triangle Lagrangian 2-form, which produces equations that are weaker than the quad equations (they are equivalent to two octahedron equations). We use relations between the Lagrangian 2-forms to prove that the system of quad equations is equivalent to the combined system of tetrahedron and octahedron equations. Furthermore, for each of the Lagrangian 2-forms, we study the double zero property of the exterior derivative. In particular, this gives a possible variational interpretation to the octahedron equations. The authors are grateful to Professor Frank Nijhoff and Dr. Vincent Caudrelier for their continued support over many years, and to the organisers and supporters of the BIRS-IASM workshop on Lagrangian Multiform Theory and Pluri-Lagrangian Systems in October 2023, where this work was started. We thank the anonymous referees for their constructive criticism on the initial version of this paper, which inspired much of the work presented in Part I. 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 II: Tetrahedron and Octahedron Equations Article published earlier |
| spellingShingle | Discrete Lagrangian Multiforms for ABS Equations II: Tetrahedron and Octahedron Equations Richardson, Jacob J. Vermeeren, Mats |
| title | Discrete Lagrangian Multiforms for ABS Equations II: Tetrahedron and Octahedron Equations |
| title_full | Discrete Lagrangian Multiforms for ABS Equations II: Tetrahedron and Octahedron Equations |
| title_fullStr | Discrete Lagrangian Multiforms for ABS Equations II: Tetrahedron and Octahedron Equations |
| title_full_unstemmed | Discrete Lagrangian Multiforms for ABS Equations II: Tetrahedron and Octahedron Equations |
| title_short | Discrete Lagrangian Multiforms for ABS Equations II: Tetrahedron and Octahedron Equations |
| title_sort | discrete lagrangian multiforms for abs equations ii: tetrahedron and octahedron equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213517 |
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