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On reductions of the Hirota-Miwa equation

Hone, Andrew N.W., Kouloukas, Theodoros E., Ward, Chloe (2017) On reductions of the Hirota-Miwa equation. Symmetries, Integrability and Geometry: Methods and Applications, 13 (57). pp. 1-17. ISSN 1815-0659. (doi:10.3842/SIGMA.2017.057) (KAR id:61659)


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.

Item Type: Article
DOI/Identification number: 10.3842/SIGMA.2017.057
Projects: Cluster algebras with periodicity and discrete dynamics over finite fields
Uncontrolled keywords: Hirota–Miwa equation; Liouville integrable maps; Somos sequences; cluster algebras
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Funders: Engineering and Physical Sciences Research Council (
Depositing User: Andrew Hone
Date Deposited: 09 May 2017 11:40 UTC
Last Modified: 04 Mar 2024 16:38 UTC
Resource URI: (The current URI for this page, for reference purposes)

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Hone, Andrew N.W..

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Kouloukas, Theodoros E..

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