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

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 $Q_{\mathrm{V}}$ 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].

In order that the manuscript be reasonably self contained, we have used some introductory material from our paper arXiv:1704.05805 to provide background for the presentation made here

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• Backlund transformations for integrable lattice equations
• Darboux Integrability of Trapezoidal $H^4$ and $H^6$ Families of Lattice Equations II: General Solutions
• Darboux integrability of trapezoidal $H^4$ and $H^6$ families of lattice equations I: First integrals
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Cited by in corpus (2)

• Darboux Integrability of Trapezoidal $H^4$ and $H^6$ Families of Lattice Equations II: General Solutions
• On Darboux non-integrability of the Hietarinta equation