Some integrable maps and their Hirota bilinear forms

A N W Hone,T E Kouloukas,G R W Quispel

doi:10.1088/1751-8121/aa9b52

A N W Hone, T E Kouloukas + Show 1 more

Open Access

https://doi.org/10.1088/1751-8121/aa9b52

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Abstract

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.

Highlights

In recent work [4], Demskoi, Tran, van der Kamp and Quispel (DTKQ) introduced a oneparameter family of birational maps, given by the Nth-order difference equation un + un+1 + . . . + un+N un+1un+2 · · · un+N−1 = α, (1)for each integer N2
We have shown that the key to understanding the integrability of the family of maps considered in [4] is to introduce an additional parameter β, as in (2), and lift to one dimension higher, eliminating this parameter to obtain (19)
For the case of even N, we have found that the Liouville integrability of (19) follows from the corresponding results for reductions of Hirota’s lattice KdV equation, considered in previous work

Summary

Introduction

N+1 2 independent irst integrals, explicitly derived in terms of multi-sums of products, and from a conjectured formula for the degrees dn of the iterates (quadratic in the index n) it was inferred that limn→∞ n−1 log dn = 0 for each N, so that the corresponding map should have vanishing algebraic entropy in the sense of [15]. These results suggested that (1) should correspond to a inite-dimensional system that is integrable in the Liouville sense [27, 41]. We make a further comment on the connection with Svinin’s work in our conclusions

Singularity coninement and Laurentiication

Proof and consequences of the main theorem

The case of even N

Conclusions