Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 004, 21 pages      arXiv:1705.00298      https://doi.org/10.3842/SIGMA.2018.004
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

Reconstructing a Lattice Equation: a Non-Autonomous Approach to the Hietarinta Equation

Giorgio Gubbiotti ab and Christian Scimiterna b
a) School of Mathematics and Statistics, F07, The University of Sydney, New South Wales 2006, Australia
b) Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre and Sezione INFN di Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy

Received April 30, 2017, in final form December 15, 2017; Published online January 09, 2018

Abstract
In this paper we construct a non-autonomous version of the Hietarinta equation [Hietarinta J., J. Phys. A: Math. Gen. 37 (2004), L67-L73] and study its integrability properties. We show that this equation possess linear growth of the degrees of iterates, generalized symmetries depending on arbitrary functions, and that it is Darboux integrable. We use the first integrals to provide a general solution of this equation. In particular we show that this equation is a sub-case of the non-autonomous $Q_{\rm V}$ equation, and we provide a non-autonomous Möbius transformation to another equation found in [Hietarinta J., J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 223-230] and appearing also in Boll's classification [Boll R., Ph.D. Thesis, Technische Universität Berlin, 2012].

Key words: quad-equations; Darboux integrability; algebraic entropy; generalized symmetries; exact solutions.

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