Some integrable maps and their Hirota bilinear forms

Hone, Andrew N.W. and Kouloukas, Theodoros E. and Quispel, G.R.W. (2017) Some integrable maps and their Hirota bilinear forms. Journal of Physics A: Mathematical and Theoretical, 51 (4). ISSN 1751-8113. E-ISSN 1751-8121. (doi: (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided)

PDF - Publisher pdf

Creative Commons Licence
This work is licensed under a Creative Commons Attribution 4.0 International License.
Download (666kB) Preview
PDF (This is the final version submitted to the publisher, with minor corrections made.) - Author's Accepted Manuscript
Restricted to Repository staff only
Contact us about this Publication Download (352kB)
Official URL


We introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the Laurent property, and the tropical (or ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. We prove that the tau function also satisfies two different bilinear equations, each of which is a reduction of the Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence). Furthermore, these bilinear equations are related to reductions of particular two-dimensional integrable lattice equations, of discrete KdV or discrete Toda type. These connections, as well as the cluster algebra structure of the bilinear equations, allow a direct construction of Poisson brackets, Lax pairs and first integrals for the birational maps. As a consequence of the latter results, we show how each member of the family can be lifted to a system that is integrable in the Liouville sense, clarifying observations made previously in the original DTKQ case.

Item Type: Article
Projects: [UNSPECIFIED] Cluster algebras with periodicity and discrete dynamics over finite fields
Uncontrolled keywords: Hirota bilinear form, Liouville integrability, discrete KdV, discrete Toda lattice, tropical
Subjects: Q Science > QA Mathematics (inc Computing science)
Q Science > QA Mathematics (inc Computing science) > QA801 Analytic mechanics
Q Science > QC Physics > QC20 Mathematical Physics
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Andrew N W Hone
Date Deposited: 17 Nov 2017 05:33 UTC
Last Modified: 19 Jan 2018 10:20 UTC
Resource URI: (The current URI for this page, for reference purposes)
Hone, Andrew N.W.:
  • Depositors only (login required):


Downloads per month over past year