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

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 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.

Keywords: Hirota–Miwa equation; Liouville integrable maps; Somos sequences; cluster algebras.

 Funding Agency Grant Number Engineering and Physical Sciences Research Council EP/P50421X/1EP/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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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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Citation: Andrew N. W. Hone, Theodoros E. Kouloukas, Chloe Ward, “On Reductions of the Hirota–Miwa Equation”, SIGMA, 13 (2017), 057, 17 pp.

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This publication is cited in the following articles:
1. A. N. W. Hone, T. E. Kouloukas, G. R. W. Quispel, “Some integrable maps and their Hirota bilinear forms”, J. Phys. A-Math. Theor., 51:4 (2018), 044004
2. Runliang Lin, Yukun Du, “Generalized Darboux transformation for the discrete Kadomtsev–Petviashvili equation with self-consistent sources”, Theoret. and Math. Phys., 196:3 (2018), 1320–1332
3. C. A. Evripidou, G. R. W. Quispel, J. A. G. Roberts, “Poisson structures for difference equations”, J. Phys. A-Math. Theor., 51:47 (2018), 475201
4. Pogrebkov A., “Hirota Difference Equation and Darboux System: Mutual Symmetry”, Symmetry-Basel, 11:3 (2019), 436
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