
SIGMA, 2017, Volume 13, 057, 17 pp.
(Mi sigma1257)




This article is cited in 4 scientific papers (total in 4 papers)
On Reductions of the Hirota–Miwa Equation
Andrew N. W. Hone^{}, Theodoros E. Kouloukas^{}, Chloe Ward^{} ^{} School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, UK
Abstract:
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 halfintegers, 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.
Keywords:
Hirota–Miwa equation; Liouville integrable maps; Somos sequences; cluster algebras.
Funding Agency 
Grant Number 
Engineering and Physical Sciences Research Council 
EP/P50421X/1 EP/M004333/1 
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. 
DOI:
https://doi.org/10.3842/SIGMA.2017.057
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Bibliographic databases:
MSC: 70H06; 37K10; 39A20; 39A14; 13F60 Received: May 2, 2017; in final form July 17, 2017; Published online July 23, 2017
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Andrew N. W. Hone, Theodoros E. Kouloukas, Chloe Ward, “On Reductions of the Hirota–Miwa Equation”, SIGMA, 13 (2017), 057, 17 pp.
Citation in format AMSBIB
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This publication is cited in the following articles:

A. N. W. Hone, T. E. Kouloukas, G. R. W. Quispel, “Some integrable maps and their Hirota bilinear forms”, J. Phys. AMath. Theor., 51:4 (2018), 044004

Runliang Lin, Yukun Du, “Generalized Darboux transformation for the discrete Kadomtsev–Petviashvili equation with selfconsistent sources”, Theoret. and Math. Phys., 196:3 (2018), 1320–1332

C. A. Evripidou, G. R. W. Quispel, J. A. G. Roberts, “Poisson structures for difference equations”, J. Phys. AMath. Theor., 51:47 (2018), 475201

Pogrebkov A., “Hirota Difference Equation and Darboux System: Mutual Symmetry”, SymmetryBasel, 11:3 (2019), 436

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