On Reductions of the Hirota-Miwa Equation
The Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence) is a bilinear partial difference equation in three independent variables. It is integrable in the sense that it arises as the compatibility condition of a linear system (Lax pair). The Hirota-Miwa equatio...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2017 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2017
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/148768 |
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| Cite this: | On Reductions of the Hirota-Miwa Equation / A.N.W. Hone, T.E. Kouloukas, C. Ward // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 29 назв. — англ. |
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Hone, A.N.W. Kouloukas, T.E. Ward, C. 2019-02-18T18:48:50Z 2019-02-18T18:48:50Z 2017 On Reductions of the Hirota-Miwa Equation / A.N.W. Hone, T.E. Kouloukas, C. Ward // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 70H06; 37K10; 39A20; 39A14; 13F60 DOI:10.3842/SIGMA.2017.057 https://nasplib.isofts.kiev.ua/handle/123456789/148768 The Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence) is a bilinear partial difference equation in three independent variables. It is integrable in the sense that it arises as the compatibility condition of a linear system (Lax pair). The Hirota-Miwa equation has infinitely many reductions of plane wave type (including a quadratic exponential gauge transformation), defined by a triple of integers or half-integers, which produce bilinear ordinary difference equations of Somos/Gale-Robinson type. Here it is explained how to obtain Lax pairs and presymplectic structures for these reductions, in order to demonstrate Liouville integrability of some associated maps, certain of which are related to reductions of discrete Toda and discrete KdV equations. This paper is a contribution to the Special Issue on Symmetries and Integrability of Dif ference Equations. The full collection is available at http://www.emis.de/journals/SIGMA/SIDE12.html. Some of these results first appeared in the Ph.D. Thesis [28], which was supported by EPSRC studentship EP/P50421X/1. ANWH is supported by EPSRC fellowship EP/M004333/1. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Reductions of the Hirota-Miwa Equation Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
On Reductions of the Hirota-Miwa Equation |
| spellingShingle |
On Reductions of the Hirota-Miwa Equation Hone, A.N.W. Kouloukas, T.E. Ward, C. |
| title_short |
On Reductions of the Hirota-Miwa Equation |
| title_full |
On Reductions of the Hirota-Miwa Equation |
| title_fullStr |
On Reductions of the Hirota-Miwa Equation |
| title_full_unstemmed |
On Reductions of the Hirota-Miwa Equation |
| title_sort |
on reductions of the hirota-miwa equation |
| author |
Hone, A.N.W. Kouloukas, T.E. Ward, C. |
| author_facet |
Hone, A.N.W. Kouloukas, T.E. Ward, C. |
| publishDate |
2017 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
The Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence) is a bilinear partial difference equation in three independent variables. It is integrable in the sense that it arises as the compatibility condition of a linear system (Lax pair). The Hirota-Miwa equation has infinitely many reductions of plane wave type (including a quadratic exponential gauge transformation), defined by a triple of integers or half-integers, which produce bilinear ordinary difference equations of Somos/Gale-Robinson type. Here it is explained how to obtain Lax pairs and presymplectic structures for these reductions, in order to demonstrate Liouville integrability of some associated maps, certain of which are related to reductions of discrete Toda and discrete KdV equations.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/148768 |
| citation_txt |
On Reductions of the Hirota-Miwa Equation / A.N.W. Hone, T.E. Kouloukas, C. Ward // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 29 назв. — англ. |
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2025-12-07T18:55:09Z |
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2025-12-07T18:55:09Z |
| _version_ |
1850876844266487808 |