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Reconstructing a lattice equation: a non-autonomous approach to the Hietarinta equation. (English) Zbl 1387.37062

The main aim of this paper is to construct a nonautonomous version of the Hietarinta equation [J. Hietarinta, J. Phys. A, Math. Gen. 37, No. 6, L67–L73 (2004; Zbl 1044.81101)] and study its integrability properties. The authors show that this equation possesses linear growth of the degrees of iterates, generalized symmetries depending on arbitrary functions, and that it is Darboux integrable. They use the first integrals to provide a general solution of this equation. In particular it is shown that this equation is a sub-case of the nonautonomous \(Q_V\) equation, and they also provide a nonautonomous Möbius transformation to another equation found in [J. Hietarinta, J. Nonlinear Math. Phys. 12, 223–230 (2005; Zbl 1091.37020)] and appearing also in R. Boll’s classification [Classification and Lagrangian structure of 3D consistent quad-equations. Berlin: Technische Universität (Ph.D. Thesis) (2012)].

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
39A14 Partial difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
37K60 Lattice dynamics; integrable lattice equations

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References:

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