A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations
A pluri-Lagrangian structure is an attribute of integrability for lattice equations and for hierarchies of differential equations. It combines the notion of multi-dimensional consistency (in the discrete case) or commutativity of the flows (in the continuous case) with a variational principle. Recen...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2019 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2019
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/210178 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations / M. Vermeeren // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 31 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862568423955890176 |
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| author | Vermeeren, M. |
| author_facet | Vermeeren, M. |
| citation_txt | A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations / M. Vermeeren // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 31 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A pluri-Lagrangian structure is an attribute of integrability for lattice equations and for hierarchies of differential equations. It combines the notion of multi-dimensional consistency (in the discrete case) or commutativity of the flows (in the continuous case) with a variational principle. Recently, we developed a continuum limit procedure for pluri-Lagrangian systems, which we now apply to most of the ABS list and some members of the lattice Gelfand-Dickey hierarchy. We obtain pluri-Lagrangian structures for many hierarchies of integrable PDEs for which such structures were previously unknown. This includes the Krichever-Novikov hierarchy, the double hierarchy of sine-Gordon and modified KdV equations, and a first example of a continuous multi-component pluri-Lagrangian system.
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| first_indexed | 2025-12-07T21:24:39Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210178 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:39Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Vermeeren, M. 2025-12-03T14:26:18Z 2019 A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations / M. Vermeeren // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 31 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K10; 39A14 arXiv: 1811.01855 https://nasplib.isofts.kiev.ua/handle/123456789/210178 https://doi.org/10.3842/SIGMA.2019.044 A pluri-Lagrangian structure is an attribute of integrability for lattice equations and for hierarchies of differential equations. It combines the notion of multi-dimensional consistency (in the discrete case) or commutativity of the flows (in the continuous case) with a variational principle. Recently, we developed a continuum limit procedure for pluri-Lagrangian systems, which we now apply to most of the ABS list and some members of the lattice Gelfand-Dickey hierarchy. We obtain pluri-Lagrangian structures for many hierarchies of integrable PDEs for which such structures were previously unknown. This includes the Krichever-Novikov hierarchy, the double hierarchy of sine-Gordon and modified KdV equations, and a first example of a continuous multi-component pluri-Lagrangian system. This research was supported by the DFG through the SFB/TRR 109, ‘Discretization in Geometry and Dynamics’. The author is grateful to the anonymous referees for their insightful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations Article published earlier |
| spellingShingle | A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations Vermeeren, M. |
| title | A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations |
| title_full | A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations |
| title_fullStr | A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations |
| title_full_unstemmed | A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations |
| title_short | A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations |
| title_sort | variational perspective on continuum limits of abs and lattice gd equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210178 |
| work_keys_str_mv | AT vermeerenm avariationalperspectiveoncontinuumlimitsofabsandlatticegdequations AT vermeerenm variationalperspectiveoncontinuumlimitsofabsandlatticegdequations |