### 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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References

1. Adler V.E., Bobenko A.I., Suris Yu.B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys. 233 (2003), 513-543, nlin.SI/0202024.
2. Adler V.E., Bobenko A.I., Suris Yu.B., Discrete nonlinear hyperbolic equations: classification of integrable cases, Funct. Anal. Appl. 43 (2009), 3-17, arXiv:0705.1663.
3. Adler V.E., Startsev S.Ya., On discrete analogues of the Liouville equation, Theoret. and Math. Phys. 121 (1999), 1484-1495, solv-int/9902016.
4. Atkinson J., Bäcklund transformations for integrable lattice equations, J. Phys. A: Math. Theor. 41 (2008), 135202, 8 pages, arXiv:0801.1998.
5. Atkinson J., Integrable lattice equations: Connection to the Möbius group, Bäcklund transformations and solutions, Ph.D. Thesis, The University of Leeds, 2008.
6. Bellon M.P., Viallet C.M., Algebraic entropy, Comm. Math. Phys. 204 (1999), 425-437, chao-dyn/9805006.
7. Bobenko A.I., Suris Yu.B., Integrable systems on quad-graphs, Int. Math. Res. Not. 2002 (2002), 573-611, nlin.SI/0110004.
8. Boll R., Classification of 3D consistent quad-equations, J. Nonlinear Math. Phys. 18 (2011), 337-365, arXiv:1009.4007.
9. Boll R., Corrigendum: Classification of 3D consistent quad-equations, J. Nonlinear Math. Phys. 19 (2012), 1292001, 3 pages.
10. Boll R., Classification and Lagrangian structure of 3D consistent quad-equations, Ph.D. Thesis, Technische Universität Berlin, 2012.
11. Bridgman T., Hereman W., Quispel G.R.W., van der Kamp P.H., Symbolic computation of Lax pairs of partial difference equations using consistency around the cube, Found. Comput. Math. 13 (2013), 517-544, arXiv:1308.5473.
12. Butler S., Hay M., Simple identification of fake Lax pairs, arXiv:1311.2406.
13. Butler S., Hay M., Two definitions of fake Lax pairs, AIP Conf. Proc. 1648 (2015), 180006, 5 pages.
14. Calogero F., Degasperis A., Spectral transform and solitons. Vol. I. Tools to solve and investigate nonlinear evolution equations, Studies in Mathematics and its Applications, Vol. 13, North-Holland Publishing Co., Amsterdam - New York, 1982.
15. Calogero F., Nucci M.C., Lax pairs galore, J. Math. Phys. 32 (1991), 72-74.
16. Diller J., Dynamics of birational maps of ${\bf P}^2$, Indiana Univ. Math. J. 45 (1996), 721-772.
17. Diller J., Favre C., Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), 1135-1169.
18. Doliwa A., Santini P.M., Multidimensional quadrilateral lattices are integrable, Phys. Lett. A 233 (1997), 365-372, solv-int/9612007.
19. Falqui G., Viallet C.M., Singularity, complexity, and quasi-integrability of rational mappings, Comm. Math. Phys. 154 (1993), 111-125, hep-th/9212105.
20. Garifullin R.N., Gudkova E.V., Habibullin I.T., Method for searching higher symmetries for quad-graph equations, J. Phys. A: Math. Theor. 44 (2011), 325202, 16 pages, arXiv:1104.0493.
21. Garifullin R.N., Yamilov R.I., Generalized symmetry classification of discrete equations of a class depending on twelve parameters, J. Phys. A: Math. Theor. 45 (2012), 345205, 23 pages, arXiv:1203.4369.
22. Garifullin R.N., Yamilov R.I., Integrable discrete nonautonomous quad-equations as Bäcklund auto-transformations for known Volterra and Toda type semidiscrete equations, J. Phys. Conf. Ser. 621 (2015), 012005, 18 pages, arXiv:1405.1835.
23. Grammaticos B., Halburd R.G., Ramani A., Viallet C.M., How to detect the integrability of discrete systems, J. Phys. A: Math. Theor. 42 (2009), 454002, 30 pages.
24. Grammaticos B., Ramani A., Viallet C.M., Solvable chaos, Phys. Lett. A 336 (2005), 152-158, math-ph/0409081.
25. Gubbiotti G., Integrability of difference equations through algebraic entropy and generalized symmetries, in Symmetries and Integrability of Difference Equations, Editors D. Levi, R. Verge-Rebelo, P. Winternitz, CRM Ser. Math. Phys., Springer, Cham, 2017, 75-151.
26. Gubbiotti G., Integrability properties of quad equations consistent on the cube, Ph.D. Thesis, Università degli Studi Roma Tre, 2017.
27. Gubbiotti G., Hay M., A SymPy module to calculate algebraic entropy for difference equations and quadrilateral partial difference equations, in preparation.
28. Gubbiotti G., Scimiterna C., Levi D., Algebraic entropy, symmetries and linearization of quad equations consistent on the cube, J. Nonlinear Math. Phys. 23 (2016), 507-543, arXiv:1603.07930.
29. Gubbiotti G., Scimiterna C., Levi D., Linearizability and a fake Lax pair for a nonlinear nonautonomous quad-graph equation consistent around the cube, Theoret. and Math. Phys. 189 (2016), 1459-1471.
30. Gubbiotti G., Scimiterna C., Levi D., On partial differential and difference equations with symmetries depending on arbitrary functions, Acta Polytechnica 56 (2016), 193-201, arXiv:1512.01967.
31. Gubbiotti G., Scimiterna C., Levi D., The non-autonomous YdKN equation and generalized symmetries of Boll equations, J. Math. Phys. 58 (2017), 053507, 18 pages, arXiv:1510.07175.
32. Gubbiotti G., Scimiterna C., Levi D., A two-periodic generalization of the $Q_{\rm V}$ equation, J. Integrable Syst. 2 (2017), xyx004, 13 pages.
33. Gubbiotti G., Scimiterna C., Yamilov R.I., Darboux integrability of trapezoidal $H^4$ and $H^4$ families of lattice equations II: General solutions, arXiv:1704.05805.
34. Gubbiotti G., Yamilov R.I., Darboux integrability of trapezoidal $H^4$ and $H^4$ families of lattice equations I: First integrals, J. Phys. A: Math. Theor. 50 (2017), 345205, 26 pages, arXiv:1608.03506.
35. Hay M., A completeness study on discrete, $2\times2$ Lax pairs, J. Math. Phys. 50 (2009), 103516, 29 pages, arXiv:0806.3940.
36. Hay M., A completeness study on certain $2\times 2$ Lax pairs including zero terms, SIGMA 7 (2011), 089, 12 pages, arXiv:1104.0084.
37. Hietarinta J., A new two-dimensional lattice model that is `consistent around a cube', J. Phys. A: Math. Gen. 37 (2004), L67-L73, nlin.SI/0311034.
38. Hietarinta J., Searching for CAC-maps, J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 223-230.
39. Hietarinta J., Viallet C.M., Searching for integrable lattice maps using factorization, J. Phys. A: Math. Theor. 40 (2007), 12629-12643, arXiv:0705.1903.
40. Hietarinta J., Viallet C.M., Weak Lax pairs for lattice equations, Nonlinearity 25 (2012), 1955-1966, arXiv:1105.3329.
41. Joshi N., Kitaev A.V., Treharne P.A., On the linearization of the first and second Painlevé equations, J. Phys. A: Math. Theor. 42 (2009), 055208, 18 pages, arXiv:0806.0271.
42. Lando S.K., Lectures on generating functions, Student Mathematical Library, Vol. 23, Amer. Math. Soc., Providence, RI, 2003.
43. Levi D., Yamilov R.I., The generalized symmetry method for discrete equations, J. Phys. A: Math. Theor. 42 (2009), 454012, 18 pages, arXiv:0902.4421.
44. Levi D., Yamilov R.I., Generalized symmetry integrability test for discrete equations on the square lattice, J. Phys. A: Math. Theor. 44 (2011), 145207, 22 pages, arXiv:1011.0070.
45. Nijhoff F.W., Lax pair for the Adler (lattice Krichever-Novikov) system, Phys. Lett. A 297 (2002), 49-58, nlin.SI/0110027.
46. Nijhoff F.W., Walker A.J., The discrete and continuous Painlevé VI hierarchy and the Garnier systems, Glasg. Math. J. 43A (2001), 109-123, nlin.SI/0001054.
47. Ramani A., Joshi N., Grammaticos B., Tamizhmani T., Deconstructing an integrable lattice equation, J. Phys. A: Math. Gen. 39 (2006), L145-L149.
48. Rasin O.G., Hydon P.E., Symmetries of integrable difference equations on the quad-graph, Stud. Appl. Math. 119 (2007), 253-269.
49. Roberts J.A.G., Tran D.T., Algebraic entropy of (integrable) lattice equations and their reductions, arXiv:1703.01069.
50. Russakovskii A., Shiffman B., Value distribution for sequences of rational mappings and complex dynamics, Indiana Univ. Math. J. 46 (1997), 897-932, math.CV/9604204.
51. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
52. Startsev S.Ya., Darboux integrable discrete equations possessing an autonomous first-order integral, J. Phys. A: Math. Theor. 47 (2014), 105204, 16 pages, arXiv:1310.2282.
53. Startsev S.Ya., On relationships between symmetries depending on arbitrary functions and integrals of discrete equations, J. Phys. A: Math. Theor. 50 (2017), 50LT01, 12 pages, arXiv:1611.02235.
54. SymPy Development Team, SymPy: Python library for symbolic mathematics, 2016, http://www.sympy.org.
55. Takenawa T., Discrete dynamical systems associated with root systems of indefinite type, Comm. Math. Phys. 224 (2001), 657-681, nlin.SI/0103016.
56. Tremblay S., Grammaticos B., Ramani A., Integrable lattice equations and their growth properties, Phys. Lett. A 278 (2001), 319-324, arXiv:0709.3095.
57. Veselov A.P., Growth and integrability in the dynamics of mappings, Comm. Math. Phys. 145 (1992), 181-193.
58. Viallet C.M., Algebraic entropy for lattice equations, math-ph/0609043.
59. Viallet C.M., Integrable lattice maps: $Q_{\rm V}$, a rational version of $Q_4$, Glasg. Math. J. 51 (2009), 157-163, arXiv:0802.0294.
60. Viallet C.M., On the algebraic structure of rational discrete dynamical systems, J. Phys. A: Math. Theor. 48 (2015), 16FT01, 21 pages, arXiv:1501.06384.
61. Xenitidis P.D., Papageorgiou V.G., Symmetries and integrability of discrete equations defined on a black-white lattice, J. Phys. A: Math. Theor. 42 (2009), 454025, 13 pages, arXiv:0903.3152.
62. Yamilov R., Symmetries as integrability criteria for differential difference equations, J. Phys. A: Math. Gen. 39 (2006), R541-R623.
63. Zhiber A.V., Sokolov V.V., Exactly integrable hyperbolic equations of Liouville type, Russian Math. Surveys 56 (2001), no. 1, 61-101.