Features of Discrete Integrability
35
26. A.V. Mikhailov, J.P. Wang, P. Xenitidis, Recursion operators, conservation laws, and integrability conditions for difference equations. Teore. Mat. Fiz. 1, 23–49 (2011). Transl. Theore.
Math. Phys. 167, 421–443 (2011). arXiv:1004.5346
27. A.V. Mikhailov, J.P. Wang, P. Xenitidis, Cosymmetries and Nijenhuis recursion operators for
difference equation. Nonlinearity 24, 2079–2097 (2011). arXiv:1009.2403
28. R.I. Yamilov, Symmetries as integrability criteria for differential difference equations. J. Phys.
A Math. Gen. 39, R541–R623 (2006)
29. P. Hydon. Difference Equations by Differential Equation Methods. Number 27 in Cambridge
Monographs on Applied and Computational Mathematics (Cambridge University Press, 2014)
30. O. Rasin, P. Hydon, Symmetries of integrable difference equations on the quad graph. Stud.
Appl. Math. 119, 253–269 (2007)
31. P.D. Xenitidis, V.G. Papageorgiou, Symmetries and integrability of discrete equations defined
on a black–white lattice. J. Phys. A Math. Theor. 42, 454025 (2009)
32. D.K. Demskoy, C.-M. Viallet, Algebraic entropy for semi-discrete equations. J. Phys. A Math.
Theor. 45, 352001 (2012). arXiv:1206.1214
33. I.R. Shafarevich, Basic Algebraic Geometry. Number 231 in Grundlehren der mathematischen
Wissenschaften (Springer, Berlin, 1977)
34. G.R.W. Quispel, J.A.G. Roberts, C.J. Thompson, Integrable mappings and soliton equations.
Phys. Lett. A 126, 419 (1988)
35. G.R.W. Quispel, J.A.G. Roberts, C.J. Thompson, Integrable mappings and soliton equations II.
Physica D34, 183–192 (1989)
36. H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé
equations. Commun. Math. Phys. 220(1), 165–229 (2001)
37. J.J. Duistermaat, Discrete Integrable Systems: QRT Maps and Elliptic Surfaces. Springer
Monographs in Mathematics (Springer, New York, 2010)
38. M.P. Bellon, C.-M. Viallet, Algebraic entropy. Commun. Math. Phys. 204, 425–437 (1999).
arXiv:chao-dyn/9805006
39. B. Grammaticos, A. Ramani, V. Papageorgiou, Do integrable mappings have the Painlevé
property? Phys. Rev. Lett. 67, 1825–1827 (1991)
40. J. Hietarinta, C.-M. Viallet, Singularity confinement and chaos in discrete systems. Phys. Rev.
Lett. 81(2), 325–328 (1998). arXiv:solv-int/9711014
41. C.M. Viallet, On the algebraic structure of rational discrete dynamical systems. J. Phys. A
Math. Theor. 48, 16FT01 (2015)
42. C.-M. Viallet, Algebraic Entropy for Lattice Equations. arXiv:math-ph/0609043
43. S. Tremblay, B. Grammaticos, A. Ramani, Integrable lattice equations and their growth
properties. Phys. Lett. A 278, 319–324 (2001)
44. D. Levi, R. Yamilov, On a nonlinear integrable difference equation on the square 3Dinconsistent. Ufa Math. J. 1, 03 (2009). arXiv:0902.2126
45. S. Lobb, F. Nijhoff, Lagrangian multiforms and multidimensional consistency. J. Phys. A
Math. Theor. 42(45), 454013 (2009)
46. Y. Suris, Variational symmetries and pluri-Lagrangian systems, in Dynamical Systems, Number
Theory and Applications. A Festschrift in Honor of Armin Leutbecher’s 80th Birthday, ed. by
Th. Hagen, F. Rupp, J. Scheurle. (World Scientific, 2016), pp. 255–266 arXiv:1307.2639
47. J. Hietarinta, C.M. Viallet, Weak Lax pairs for lattice equations. Nonlinearity 25, 1955–1966
(2012). arXiv:1105.3329
48. S. Butler, M. Hay. Simple Identification of Fake Lax Pairs (2013). arXiv:1311.2406
35
26. A.V. Mikhailov, J.P. Wang, P. Xenitidis, Recursion operators, conservation laws, and integrability conditions for difference equations. Teore. Mat. Fiz. 1, 23–49 (2011). Transl. Theore.
Math. Phys. 167, 421–443 (2011). arXiv:1004.5346
27. A.V. Mikhailov, J.P. Wang, P. Xenitidis, Cosymmetries and Nijenhuis recursion operators for
difference equation. Nonlinearity 24, 2079–2097 (2011). arXiv:1009.2403
28. R.I. Yamilov, Symmetries as integrability criteria for differential difference equations. J. Phys.
A Math. Gen. 39, R541–R623 (2006)
29. P. Hydon. Difference Equations by Differential Equation Methods. Number 27 in Cambridge
Monographs on Applied and Computational Mathematics (Cambridge University Press, 2014)
30. O. Rasin, P. Hydon, Symmetries of integrable difference equations on the quad graph. Stud.
Appl. Math. 119, 253–269 (2007)
31. P.D. Xenitidis, V.G. Papageorgiou, Symmetries and integrability of discrete equations defined
on a black–white lattice. J. Phys. A Math. Theor. 42, 454025 (2009)
32. D.K. Demskoy, C.-M. Viallet, Algebraic entropy for semi-discrete equations. J. Phys. A Math.
Theor. 45, 352001 (2012). arXiv:1206.1214
33. I.R. Shafarevich, Basic Algebraic Geometry. Number 231 in Grundlehren der mathematischen
Wissenschaften (Springer, Berlin, 1977)
34. G.R.W. Quispel, J.A.G. Roberts, C.J. Thompson, Integrable mappings and soliton equations.
Phys. Lett. A 126, 419 (1988)
35. G.R.W. Quispel, J.A.G. Roberts, C.J. Thompson, Integrable mappings and soliton equations II.
Physica D34, 183–192 (1989)
36. H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé
equations. Commun. Math. Phys. 220(1), 165–229 (2001)
37. J.J. Duistermaat, Discrete Integrable Systems: QRT Maps and Elliptic Surfaces. Springer
Monographs in Mathematics (Springer, New York, 2010)
38. M.P. Bellon, C.-M. Viallet, Algebraic entropy. Commun. Math. Phys. 204, 425–437 (1999).
arXiv:chao-dyn/9805006
39. B. Grammaticos, A. Ramani, V. Papageorgiou, Do integrable mappings have the Painlevé
property? Phys. Rev. Lett. 67, 1825–1827 (1991)
40. J. Hietarinta, C.-M. Viallet, Singularity confinement and chaos in discrete systems. Phys. Rev.
Lett. 81(2), 325–328 (1998). arXiv:solv-int/9711014
41. C.M. Viallet, On the algebraic structure of rational discrete dynamical systems. J. Phys. A
Math. Theor. 48, 16FT01 (2015)
42. C.-M. Viallet, Algebraic Entropy for Lattice Equations. arXiv:math-ph/0609043
43. S. Tremblay, B. Grammaticos, A. Ramani, Integrable lattice equations and their growth
properties. Phys. Lett. A 278, 319–324 (2001)
44. D. Levi, R. Yamilov, On a nonlinear integrable difference equation on the square 3Dinconsistent. Ufa Math. J. 1, 03 (2009). arXiv:0902.2126
45. S. Lobb, F. Nijhoff, Lagrangian multiforms and multidimensional consistency. J. Phys. A
Math. Theor. 42(45), 454013 (2009)
46. Y. Suris, Variational symmetries and pluri-Lagrangian systems, in Dynamical Systems, Number
Theory and Applications. A Festschrift in Honor of Armin Leutbecher’s 80th Birthday, ed. by
Th. Hagen, F. Rupp, J. Scheurle. (World Scientific, 2016), pp. 255–266 arXiv:1307.2639
47. J. Hietarinta, C.M. Viallet, Weak Lax pairs for lattice equations. Nonlinearity 25, 1955–1966
(2012). arXiv:1105.3329
48. S. Butler, M. Hay. Simple Identification of Fake Lax Pairs (2013). arXiv:1311.2406