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Some integrable maps and their Hirota bilinear forms. (English) Zbl 1384.37071
The authors introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) if one of the parameters is set to zero [D. K. Demskoi et al., J. Phys. A, Math. Theor. 45, No. 13, Article ID 135202, 10 p. (2012; Zbl 1311.39001)]. 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 ultra-discrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. They 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 discrete KP equation, or octahedron recurrence). Furthermore, these bilinear equations are related to reductions of particular two-dimensional integrable lattice equations. 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. It is also shown 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.
The main result of the paper is Theorem 1.1 that reads as follows: suppose that $u_{n}=\frac{\tau_{n+3}\tau_{n}}{\tau_{n+2}\tau_{n+1}}\tag{1}$ is a solution of $\left(\sum_{j=0}^{N}u_{n+j}+\beta\right)\prod_{k=1}^{N-1}u_{n+k}=\alpha. \tag{2}$ Then $$\tau_{n}$$ satisfies the bilinear equation $\tau_{n+N+2}\tau_{n} = \gamma_{n} \tau_{n+N+1}\tau_{n+1} + \alpha \tau_{n+N}\tau_{n+2} , \tag{3}$ where the quantity $$\gamma_{n}$$ is $$2$$-periodic; and conversely, the equation (3) for $$\tau_{n}$$, with the $$2$$-periodic coefficient $$\gamma_{n}$$, has a first integral $$\beta$$ such that $$u_{n}$$ given by (1) satisfies (2). Moreover, if $$u_{n}$$ is given by (1), then when $$N$$ is even (2) has a first integral $$K$$ such that $$\tau_{n}$$ satisfies $\tau_{n+2N+1}\tau_{n} = -\alpha \tau_{n+2N} \tau_{n+1} + K \tau_{n+N+1} \tau_{n+N},$ while for $$N$$ odd (2) has a first integral $$K$$ such that $\tau_{n+2N+2}\tau_{n} = \alpha^{2} \tau_{n+2N} \tau_{n+2} + K \tau^{2}_{n+N+1}.$

##### MSC:
 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 47A07 Forms (bilinear, sesquilinear, multilinear) 39A10 Additive difference equations 39A14 Partial difference equations
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