### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 057, 17 pages      arXiv:1705.01094      https://doi.org/10.3842/SIGMA.2017.057
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

### On Reductions of the Hirota-Miwa Equation

Andrew N.W. Hone, Theodoros E. Kouloukas and Chloe Ward
School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, UK

Received May 02, 2017, in final form July 17, 2017; Published online July 23, 2017

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.

Key words: Hirota-Miwa equation; Liouville integrable maps; Somos sequences; cluster algebras.

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