Some integrable maps and their Hirota bilinear forms

TL;DR

This paper introduces a new family of integrable birational maps, explores their Hirota bilinear forms, and establishes their connection to discrete integrable equations, Poisson structures, and Liouville integrability.

Contribution

It extends known integrable maps by introducing a two-parameter family, derives their Hirota bilinear forms, and constructs associated Lax pairs and Poisson brackets.

Findings

The tau function satisfies a homogeneous Laurent recurrence.

The maps exhibit quadratic degree growth confirmed by tropical analysis.

The bilinear equations are reductions of the discrete KP equation.

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.