Abarenkova, N., Anglès d'Auriac, J.-Ch., Boukraa, S., Hassani, S. and Maillard, J.-M. 1999. Topological entropy and Arnold complexity for two-dimensional mappings. Phys. Lett. A, 262(1), 44–49.Google Scholar

Ablowitz, M. J. and Clarkson, P. A. 1991. Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series, vol. 149. Cambridge, UK: Cambridge University Press.

Ablowitz, M. J. and Fokas, A. S. 2003. Complex Variables: Introduction and Applications. Cambridge, UK: Cambridge University Press.

Ablowitz, M. J. and Herbst, B. M. 1990. On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation. SIAM J. Appl. Math., 50(2), 339–351.Google Scholar

Ablowitz, M. J. and Ladik, J. F. 1976. A nonlinear difference scheme and inverse scattering. Stud. Appl. Math., 55(3), 213–229.Google Scholar

Ablowitz, M. J. and Ladik, J. F. 1976/77. On the solution of a class of nonlinear partial difference equations. Stud. Appl. Math., 57(1), 1–12.Google Scholar

Ablowitz, M. J. and Segur, H. 1977. Exact linearization of a Painlevé transcendent. Phys. Rev. Lett., 38(20), 1103.Google Scholar

Ablowitz, M. J. and Segur, H. 1981. Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, vol. 4. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).

Ablowitz, M. J., Ramani, A. and Segur, H. 1978. Nonlinear evolution equations and ordinary differential equations of Painlevé type. Lett. Nuovo Cimento, 23(9), 333–338.Google Scholar

Ablowitz, M. J., Ramani, A. and Segur, H. 1980a. A connection between nonlinear evolution equations and ordinary differential equations of P-type. I. J. Math. Phys., 21(4), 715–721.Google Scholar

Ablowitz, M. J., Ramani, A. and Segur, H. 1980b. A connection between nonlinear evolution equations and ordinary differential equations of P-type. II. J. Math. Phys., 21(5), 1006–1015.Google Scholar

Ablowitz, M. J., Halburd, R. and Herbst, B. 2000. On the extension of the Painlevé property to difference equations. Nonlinearity, 13(3), 889–905.Google Scholar

Abramowitz, M. and Stegun, I. A. 1970. Handbook of Mathematical Functions. NewYork: Dover.

Adler, M. and van Moerbeke, P. 1999. Vertex operator solutions to the discrete KPhierarchy. Commun. Math. Phys., 203(1), 185–210.Google Scholar

Adler, V. E. 1998. Bäcklund transformation for the Krichever-Novikov equation. Int. Math. Res. Not., 1998(1), 1–4.Google Scholar

Adler, V. E. 2001. Discrete equations on planar graphs. J. Phys. A: Math. Gen., 34(48), 10453–10460.Google Scholar

Adler, V. E. 2006. Some incidence theorems and integrable discrete equations. Discrete Comput. Geom., 36(3), 489–498.Google Scholar

Adler, V. E. and Suris, Yu. B. 2004. Q4: integrable master equation related to an elliptic curve. Int. Math. Res. Not., 2004(47), 2523–2553.Google Scholar

Adler, V. E. and Veselov, A. P. 2004. Cauchy problem for integrable discrete equations on quad-graphs. Acta Appl. Math., 84(2), 237–262.Google Scholar

Adler, V. E., Marikhin, V. G. and Shabat, A. B. 2001. Lagrangian chains and canonical Bäcklund transformations. Theor. Math. Phys., 129(2), 1448–1465.Google Scholar

Adler, V. E., Bobenko, A. I. and Suris, Yu. B. 2003. Classification of integrable equations on quad-graphs. The consistency approach. Commun. Math. Phys., 233(3), 513–543.Google Scholar

Adler, V. E., Bobenko, A. I. and Suris, Yu. B. 2004. Geometry of Yang–Baxter maps: pencils of conics and quadrirational mappings. Commun. Anal. Geom., 12(5), 967–1007.Google Scholar

Adler, V. E., Bobenko, A. I. and Suris, Yu. B. 2009. Discrete nonlinear hyperbolic equations: classification of integrable cases. Funktsional. Anal. i Prilozhen., 43(1), 3–21.Google Scholar

Adler, V. E., Bobenko, A. I. and Suris, Yu. B. 2012. Classification of integrable discrete equations of octahedron type. Int. Math. Res. Not. 2012(8), 1822–1889.Google Scholar

Agafonov, S. I. 2003. Imbedded circle patterns with the combinatorics of the square grid and discrete Painlevé equations. Discrete Comput. Geom., 29(2), 305–319.Google Scholar

Agafonov, S. I. 2005. Asymptotic behavior of discrete holomorphic maps zc and log(z) . J. Nonlinear Math. Phys., 12, Supp 2, 1–14.Google Scholar

Agafonov, S. I., and Bobenko, A. I. 2000. Discret. zγ and Painlevé equations. Int. Math. Res. Not., 2000(4), 165–193.Google Scholar

Agafonov, S. I. and Bobenko, A. I. 2003. Hexagonal circle patterns with constant intersection angles and discrete Painlevé and Riccati equations. J. Math. Phys., 44(8), 3455–3469.Google Scholar

Ahlfors, L. V. 1953. Complex Analysis, an Introduction to the Theory of Analytic Functions of One Complex Variable. New York: McGraw-Hill Book Company.

Airault, H., McKean, H. P. and Moser, J. 1977. Rational and elliptic solutions of the Korteweg–de Vries equation and a related many-body problem. Commun. Pure Appl. Math., 30(1), 95–148.Google Scholar

Akhiezer, N. I. 1990. Elements of the Theory of Elliptic Functions. Translations of Mathematical Monographs, vol. 79. Providence, RI: American Mathematical Society. Translated from the second Russian edition by H. H., McFaden.

Akritas, A. G., Akritas, E. K. and Malaschonok, G. I. 1996. Various proofs of Sylvester's (determinant) identity. Math. Comput. Simul., 42(4), 585–593.Google Scholar

Ando, H., Hay, M., Kajiwara, K. and Masuda, T. 2014. An explicit formula for the discrete power function associated with circle patterns of Schramm type. Funkcial. Ekvac., 57(1), 1–41.Google Scholar

Andrews, G. E., Askey, R. and Roy, R. 1999. Special Functions. Encyclopedia of Mathematics and Its Applications. Cambridge, UK: Cambridge University Press.

Anglès d'Auriac, J.-Ch., Maillard, J.-M., and Viallet, C. M. 2006. On the complexity of some birational transformations. J. Phys. A: Math. Gen., 39(14), 3641.Google Scholar

Arinkin, D. and Borodin, A. 2006. Moduli spaces o. d-connections and difference Painlevé equations. Duke Math. J., 134(3), 515–556.Google Scholar

Arnold, V. I. 1990. Dynamics of complexity of intersections. Bol. Soc. Brasil. Mat. (N.S.), 21(1), 1–10.Google Scholar

Arnold, V. I. 1997. Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60. New York: Springer.

Atkinson, J. 2008a. Bäcklund transformations for integrable lattice equations. J. Phys. A: Math. Theor., 41(13), 135202, 8pp.

Atkinson, J. 2008b. Integrable Lattice Equations: Connection to the Möbius Group, Bäcklund Transformations and Solutions. Ph.D. thesis, University of Leeds.

Atkinson, J. 2012. A multidimensionally consistent version of Hirota's discrete KdV equation. J. Phys. A: Math. Theor., 45(22), 222001.Google Scholar

Atkinson, J. and Joshi, N. 2013. Singular-boundary reductions of type-Q ABS equations. Int. Math. Res. Not., 2013(7), 1451–1481.Google Scholar

Atkinson, J. and Nieszporski, M. 2014. Multi-quadratic quad equations: integrable cases from a factorized-discriminant hypothesis. Int. Math. Res. Not., 2014(15), 4215–4240.Google Scholar

Atkinson, J. and Nijhoff, F. W. 2010. A constructive approach to the soliton solutions of integrable quadrilateral lattice equations. Commun Math. Phys., 299(2), 283–304.Google Scholar

Atkinson, J., Hietarinta, J. and Nijhoff, F. W. 2007. Seed and soliton solutions for Adler's lattice equation. J. Phys. A: Math. Theor., 40(1), F1–F8.Google Scholar

Atkinson, J., Hietarinta, J. and Nijhoff, F. W. 2008. Soliton solutions for Q3. J. Phys. A: Math. Theor., 41(14), 142001, 11pp.Google Scholar

Atkinson, J., Lobb, S. B. and Nijhoff, F. W. 2012. An integrable multicomponent quadequation and its Lagrangian formulation. Theor. Math. Phys., 173(3), 1644–1653.Google Scholar

Babelon, O., Bernard, D. and Talon, M. 2003. Introduction to Classical Integrable Systems. Cambridge, UK: Cambridge University Press.

Babelon, O., Talon, M. and Peyranére, M. C. 2010. Kowalevski's analysis of the swinging Atwood's machine. J. Phys. A: Math. Theor., 43(8), 085207.Google Scholar

Bäcklund, A. V. 1883. Om ytor med konstant negative krökning. Lund Univ. Årsskrift, 19, 1–48.Google Scholar

Baker, G. A. 1975. Essentials of Padé Approximants. New York: Academic Press.

Barnett, D. C., Halburd, R. G., Morgan, W. and Korhonen, R. J. 2007. Nevanlinna theory for th. q-difference operator and meromorphic solutions of q-difference equations. Proc. R. Soc. Edinburgh Sect. A, 137(3), 457–474.Google Scholar

Bauer, F. L. 1959. Sequential reduction to tridiagonal form. J. Soc. Indust. Appl. Math., 7, 107–113.Google Scholar

Baxter, R. J. 1982. Exactly Solved Models in Statistical Mechanics. London: Academic Press Inc.

Bazhanov, V. V. and Sergeev, S. M. 2012a. Elliptic gamma-function and multi-spin solutions of the Yang–Baxter equation. Nucl. Phys. B, 856(2), 475–496.

Bazhanov, V. V. and Sergeev, S. M. 2012b. A master solution of the quantum Yang–Baxter equation and classical discrete integrable equations. Adv. Theor. Math. Phys., 16(1), 65–95.Google Scholar

Bellon, M. P. and Viallet, C.-M. 1999. Algebraic entropy. Commun. Math. Phys., 204(2), 425–437.Google Scholar

Bender, C.M. and Orszag, S. A. 1978. Advanced Mathematical Methods for Scientists and Engineers. New York: McGraw-Hill.

Bessis, D., Orszag, S. A. and Zuber, J.-B. 1980. Quantum field theory techniques in graphical enumeration. Adv. Appl. Math., 1(2), 109–157.Google Scholar

Bianchi, L. 1899. Vorlesungen über Differentialgeometrie. Leipzig: B.G. Teubner.

Birkhoff, G. D. 1913. The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations. Proc. Am. Acad. Arts Sci., 49(9), 521–568.Google Scholar

Bluman, G. W. and Cole, J. D. 1974. Similarity Methods for Differential Equations, vol. 13. Berlin: Springer.

Bobenko, A. I. 1999. Discrete conformal maps and surfaces. In P., Clarkson and F. W., Nijhoff (eds), Symmetries and Integrability of Difference Equations. London Mathematical Society Lecture Note Series, vol. 255, Cambridge, UK: Cambridge University Press, 97–108.

Bobenko, A. I., and Pinkall, U. 1996a. Discrete isothermic surfaces. J. Reine Angew. Math., 475, 187–208.Google Scholar

Bobenko, A. I., and Pinkall, U. 1996b. Discrete surfaces with constant negative Gaussian curvature and the Hirota equation. J. Differ. Geom., 43(3), 527–611.Google Scholar

Bobenko, A. I. and Suris, Yu. B. 2002. Integrable systems on quad-graphs. Int. Math. Res. Not., 2002(11), 573–611.Google Scholar

Bobenko, A. I. and Suris, Yu. B. 2008. Discrete Differential Geometry. Graduate Studies in Mathematics, vol. 98. Providence, RI: American Mathematical Society.

Bobenko, A. I. and Suris, Yu. B. 2010. On the Lagrangian structure of integrable quadequations. Lett. Math. Phys., 92, 17–31.Google Scholar

Bobenko, A. I. and Suris, Yu. B. 2015. Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems. Commun. Math. Phys., 336(1), 199–215.Google Scholar

Bobenko, A. I.,Mercat, C., and Suris, Yu. B. 2005. Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green's function. J. Reine Angew. Math., 2005(583), 117–161.Google Scholar

Boelen, L. and van Assche, W. 2010. Discrete Painlevé equations for recurrence coefficients of semiclassical Laguerre polynomials. Proc. Am. Math. Soc., 138(4), 1317–1331.Google Scholar

Boelen, L., Smet, C. and van Assche, W. 2010. q-discrete Painlevé equations for recurrence coefficients of modified q-Freud orthogonal polynomials. J. Differ. Equ. Appl., 16(1), 37–53.Google Scholar

Bogdanov, L. V. and Konopelchenko, B. G. 1998a. Analytic-bilinear approach to integrable hierarchies. I. Generalized KP hierarchy. J. Math. Phys., 39(9), 4683–4700.Google Scholar

Bogdanov, L. V. and Konopelchenko, B. G. 1998b. Analytic-bilinear approach to integrable hierarchies. II. Multicomponent KP and 2D Toda lattice hierarchies. J. Math. Phys., 39(9), 4701–4728.Google Scholar

Boiti, M., Pempinelli, F., Prinari, B. and Spire, A. 2001. An integrable discretization of KdV at large times. Inverse Prob., 17(3), 515–526.Google Scholar

Boll, R. 2011. Classification of 3D consistent quad-equations. J. Nonlinear Math. Phys., 18(3), 337–365.Google Scholar

Boll, R. 2012. On multidimensional consistent systems of asymmetric quad-equations. arXiv e-prints arXiv:1201.1203.

Boll, R., Petrera, M. and Suris, Yu. B. 2013a. What is integrability of discrete variational systems? Proc. R. Soc. London, Ser. A, 470, 30550.Google Scholar

Boll, R., Petrera, M. and Suris, Yu. B. 2013b. Multi-time Lagrangian 1-forms for families of Bäcklund transformations: Toda-type systems. J. Phys. A: Math. Theor., 46(27), 275204.Google Scholar

Boll, R., Petrera, M., and Suris, Yu. B. 2015. Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems. J. Phys. A: Math. Theor., 48(8), 085203.Google Scholar

Boll, R., Petrera, M. and Suris, Yu. B. 2016. On integrability of discrete variational systems. Octahedron relations. Int. Math. Res. Not., 2016(3), 645–668.Google Scholar

Boole, G. 1860. A Treatise on the Calculus of Finite Differences. London: McMillan & Co. Borodin, A. 2004. Isomonodromy transformations of linear systems of difference equations. Ann. Math., 1141–1182.

Boukraa, S., and Maillard, J.-M. 1995. Factorization properties of birational mappings. Physica A, 220(3), 403–470.Google Scholar

Boukraa, S., Maillard, J.-M. and Rollet, G. 1994. Integrable mappings and polynomial growth. Physica A, 209(1), 162–222.Google Scholar

Boussinesq, J. 1877. Essai sur la théorie des eaux courantes. Mémoires présentés par divers savants à l'Académie des sciences de l'Institut national de France, vol. XXIII. Imprimerie Nationale.

Brezin, E. and Kazakov, V. A. 1990. Exactly solvable field theories of closed strings. Phys. Lett., 236(2), 144–150.Google Scholar

Brezinski, C. 1977. Accélération de la Convergence en Analyse Numérique. Lecture Notes in Mathematics, vol. 584. Berlin Heidelberg: Springer-Verlag.

Brezinski, C. 2013. Padé-Type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics. Basel: Birkhäuser.

Brezinski, C., He, Y., Hu, X. B., Redivo-Zaglia, M. and Sun, J. Q. 2012. Multiste. ε-algorithm, Shanks' transformation, and the Lotka-Volterra system by Hirota's method. Math. Comput., 81(279), 1527–1549.Google Scholar

Brualdi, R. A. and Schneider, H. 1983. Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley. Linear Algebra Appl., 52–53(0), 769–791.Google Scholar

Bruschi, M., Ragnisco, O., Santini, P. M. and Tu, G. Z. 1991. Integrable symplectic maps. Physica D, 49(3), 273–294.Google Scholar

Bustamante, M. D. and Hojman, S. A. 2003. Multi-Lagrangians, hereditary operators and Lax pairs for the Korteweg–de Vries positive and negative hierarchies. J. Math. Phys., 44(10), 4652–4671.Google Scholar

Butler, S. 2012a. A discrete inverse scattering transform for Q3δ. arXiv preprint arXiv:1210.1869.

Butler, S. 2012b. Multidimensional inverse scattering of integrable lattice equations. Nonlinearity, 25(6), 1613.Google Scholar

Butler, S. and Joshi, N. 2010. An inverse scattering transform for the lattice potential KdV equation. Inverse Prob., 26(11), 115012.Google Scholar

Cadzow, J. A. 1970. Discrete calculus of variations. J. Control, 11(3), 393–407.Google Scholar

Calogero, F. 1969. Solution of a three-body problem in one dimension. J. Math. Phys., 10(12), 2191–2196.Google Scholar

Calogero, F. and Degasperis, A. 1982. Spectral Transform and Solitons, vol. 1. Amsterdam: North-Holland Publ.

Cao, C., and Xu, X. 2012. A finite genus solution of the H1 model. J. Phys. A: Math. Theor., 45(5), 055213.Google Scholar

Cao, C. and Zhang, G. 2012. A finite genus solution of the Hirota equation via integrable symplectic maps. J. Phys. A: Math. Theor., 45(9), 095203.Google Scholar

Capel, H. W. and Sahadevan, R. 2001. A new family of four-dimensional symplectic and integrable mappings. Physica A, 289(1–2), 86–106.Google Scholar

Capel, H. W., Nijhoff, F. W. and Papageorgiou, V. G. 1991. Complete integrability of Lagrangian mappings and lattices of KdV type. Phys. Lett. A, 155(6–7), 377–387.Google Scholar

Carmichael, R. D. 1912. The general theory of linear q-difference equations. Am. J. Math., 34, 147–168.Google Scholar

Chihara, T. S. 2014. An Introduction to Orthogonal Polynomials. New York: Dover Publications.

Choodnovsky, D. V. and Choodnovsky, G. V. 1977. Pole expansions of nonlinear partial differential equations. Il Nuovo Cimento B Series 11, 40(2), 339–353.Google Scholar

Clarkson, P. A., Hone, A. N. W. and Joshi, N. 2003. Hierarchies of difference equations and Bäcklund transformations. J. Nonlinear Math. Phys., 10, Supp 2, 13–26.Google Scholar

Conte, R. M. 1999. The Painlevé Property: One Century Later. CRM Series in Mathematical Physics. New York: Springer.

Conte, R. M. and Musette, M. 2008. The Painlevé Handbook. Netherlands: Springer.

Courant, R., Friedrichs, K. and Lewy, H. 1928. Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann., 100(1), 32–74.Google Scholar

Cresswell, C. and Joshi, N. 1999a. The discrete first, second and thirty-fourth Painlevé hierarchies. J. Phys. A: Math. Gen., 32(4), 655.Google Scholar

Cresswell, C. and Joshi, N. 1999b. The discrete Painlevé I hierarchy. In P., Clarkson and F. W., Nijhoff (eds), Symmetries and Integrability of Difference Equations. London Mathematical Society Lecture Note Series, vol. 255, Cambridge, UK: Cambridge University Press, 197–205.

Darboux, G. 1914. Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal. Première partie: Généralités. Coordonnées curvilignes. Surfaces minima. Paris: Gauthier-Villars.

Date, E., Kashiwara, M. and Miwa, T. 1981. Vertex operators an. τfunctions transformation groups for soliton equations, II. Proc. Japan Acad. Ser. A Math. Sci., 57(8), 387–392.Google Scholar

Date, E., Jimbo, M. and Miwa, T. 1982a. Method for generating discrete soliton equations. I. J. Phys. Soc. Japan, 51(12), 4116–4124.Google Scholar

Date, E., Jimnbo, M. and Miwa, T. 1982b. Method for generating discrete soliton equations. II. J. Phys. Soc. Japan, 51(12), 4125–4131.Google Scholar

Date, E., Jimbo, M. and Miwa, T. 1983a. Method for generating discrete soliton equations. III. J. Phys. Soc. Japan, 52(2), 388–393.Google Scholar

Date, E., Jimbo, M. and Miwa, T. 1983b. Method for generating discrete soliton equations. IV. J. Phys. Soc. Japan, 52(3), 761–765.Google Scholar

Date, E., Jimbo, M. and Miwa, T. 1983c. Method for generating discrete soliton equations. V. J. Phys. Soc. Japan, 52(3), 766–771.Google Scholar

David, Jr., E., E., et al. 1984. Renewing US mathematics: critical resource for the future. Not. Am. Math. Soc., 31(5), 435–466.Google Scholar

Deift, P., Li, L. C. and Tomei, C. 1989. Matrix factorizations and integrable systems. Commun. Pure Appl. Math., 42(4), 443–521.Google Scholar

Di Vizio, L., Ramis, J.-P., Sauloy, J. and Zhang, C. 2003. Équations au. q-différences. Gaz. Math., 96, 20–49.Google Scholar

Dickey, L. A. 1990. General Zakharov–Shabat equations, multi-time Hamiltonian formalism, and constants of motion. Commun. Math. Phys., 132(3), 485–497.Google Scholar

Dickey, L. A. 2003. Soliton Equations and Hamiltonian Systems, 2nd ed. Advanced Series in Mathematical Physics, vol. 26. River Edge, NJ:World Scientific Publishing Co. Inc.

Diller, J. and Favre, C. 2001. Dynamics of bimeromorphic maps of surfaces. Amer. J. Math. 123(6), 1135–1169.Google Scholar

DLMF. 2010. NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.9 of 201408-29. Online companion to Olver et al. (2010).

Doliwa, A. 2005. On τ -function of conjugate nets. J. Nonlinear Math. Phys., 12, Supp 1, 244–252.Google Scholar

Doliwa, A. 2009. Th. τ-function of the quadrilateral lattice. J. Phys. A: Math. Theor., 42(40), 404008, 9pp.Google Scholar

Doliwa, A. 2010. Desargues maps and the Hirota–Miwa equation. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci., 466(2116), 1177–1200.Google Scholar

Doliwa, A. and Santini, P. M. 1997. Multidimensional quadrilateral lattices are integrable. Phys. Lett. A, 233(4–6), 365–372.Google Scholar

Doliwa, A. and Santini, P. M. 1999. Planarity and integrability. In Degasperis, A. and Gaeta, G. (eds), Symmetry and Perturbation Theory (Rome, 1998). River Edge, NJ: World Scientific Publishing Co. Inc., 167–177.

Doliwa, A. and Santini, P. M. 2000. Integrable discrete geometry: the quadrilateral lattice, its transformations and reductions. In Levi, D. and Ragnisco, O. (eds), SIDE III – Symmetries and Integrability of Difference Equations (Sabaudia, 1998). CRM Proc. Lecture Notes, vol. 25. Providence, RI: American Mathematical Society, 101–119.

Doliwa, A., Mañas, M. and Alonso, L. M. 1999. Generating quadrilateral and circular lattices in KP theory. Phys. Lett. A, 262(4), 330–343.Google Scholar

Doliwa, A., Santini, P. M. and Mañas, M. 2000. Transformations of quadrilateral lattices. J. Math. Phys., 41(2), 944–990.Google Scholar

Doliwa, A., Alonso, L. M. and Santini, P. M. 2004. Geometric discretization of the Bianchi system. J. Geom. Phys., 52(3), 217–240.Google Scholar

Doliwa, A., Grinevich, P., Alonso, L. M. and Santini, P. M. 2007. Integrable lattices and their sublattices: from the discrete Moutard (discrete Cauchy–Riemann) 4-point equation to the self-adjoint 5-point scheme. J. Math. Phys., 48(1), 013513, 28.Google Scholar

Dorfman, I. Ya. and Nijhoff, F. W. 1991. On a (2 + 1)-dimensional version of the Krichever-Novikov equation. Phys. Lett. A, 157(2–3), 107–112.Google Scholar

Dragovíc, V. and Gajíc, B. 2008. Hirota-Kimura type discretization of the classical nonholonomic Suslov problem. Regul. Chaotic Dyn., 13(4), 250–256.Google Scholar

Dragovíc, V. and Radnovíc, M. 2011. Poncelet Porisms and Beyond: Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics. Frontiers in Mathematics. Basel: Springer.

Drazin, P. G. and Johnson, R. S. 1989. Solitons: An Introduction. Cambridge Texts in Applied Mathematics. Cambridge, UK: Cambridge University Press.

Drinfeld, V. G. 1992. On some unsolved problems in quantum group theory. In P. P., Kulish (ed), Quantum Groups. Lecture Notes in Mathematics, vol. 1510. Berlin Heidelberg: Springer, 1–8.

Drinfel'd, V. G. and Sokolov, V. V. 1985. Lie algebras and equations of Korteweg–de Vries type. J. Sov. Math., 30(2), 1975–2036.Google Scholar

Duistermaat, J. J. 2010. Discrete Integrable Systems: QRT and Elliptic Surfaces. New York: Springer.

Duistermaat, J. J. and Joshi, N. 2011. Okamoto's space for the first Painlevé equation in Boutroux coordinates. Arch. Ration. Mech. Anal., 202(Dec.), 707–785.Google Scholar

Etingof, P., Schedler, T. and Soloviev, A. 1999. Set-theoretical solutions to the quantum Yang–Baxter equation. Duke Math. J., 100(2), 169–209.Google Scholar

Euler, L. 1761. De integratione aequationis differentialis √mdx 1-x 4 = √ndy 1-y 4 . Nov. Comm. Acad. Sci. Petr., 6, 35–57.Google Scholar

Faddeev, L. D. and Takhtajan, L. A. 2007. Hamiltonian Methods in the Theory of Solitons. English ed. Classics in Mathematics. Berlin: Springer. Translated from the 1986 Russian original by A. G., Reyman.

Falqui, G. and Viallet, C.-M. 1993. Singularity, complexity, and quasi-integrability of rational mappings. Commun. Math. Phys., 154(1), 111–125.Google Scholar

Ferapontov, E. V., Novikov, V. S. and Roustemoglou, I. 2014. On the classification of discrete Hirota-type equations in 3D. Int. Math. Res. Not., 2015(13): 4933–4974.Google Scholar

Field, C. M. and Nijhoff, F. W. 2003. A note on modified Hamiltonians for numerical integrations admitting an exact invariant. Nonlinearity, 16(5), 1673.Google Scholar

Field, C. M., Joshi, N. and Nijhoff, F. W. 2008. q-Difference equations of KdV type and Chazy-type second-degree difference equations. J. Phys. A: Math. Theor., 41(33), 332005,13pp.Google Scholar

Flanders, H. 1963. Differential Forms with Applications to the Physical Sciences. New York: Dover Publications.

Flaschka, H. 1974. The Toda lattice. II. Existence of integrals. Phys. Rev. B, 9(Feb), 1924–1925.Google Scholar

Flaschka, H. and Newell, A. C. 1980. Monodromy- and spectrum-preserving deformations I. Commun. Math. Phys., 76(1), 65–116.Google Scholar

Flaschka, H. and Newell, A. C. 1981. Multiphase similarity solutions of integrable evolution equations. Physica D, 3(1), 203–221.Google Scholar

Fokas, A. S. and Ablowitz, M. J. 1981. Linearization of the Korteweg–de Vries and Painlevé II equations. Phys. Rev. Lett., 47(Oct), 1096–1100.Google Scholar

Fokas, A. S. and Ablowitz, M. J. 1982. On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys., 23(11), 2033–2042.Google Scholar

Fokas, A. S. and Ablowitz, M. J. 1983. The inverse scattering transform for multidimensional (2 + 1) problems. In Wolf, K. B. (ed), Nonlinear Phenomena (Oaxtepec, 1982). Lecture Notes in Physics, vol. 189. Berlin: Springer, 137–183.

Fokas, A. S., Its, A. R. and Zhou, X. 1992a. Continuous and discrete Painlevé equations. In Levi, D. and Winternitz, P. (eds), Painlevé Transcendents, Their Asymptotics and Physical Applications. New York: Springer Science+Business Media, 33–47.

Fokas, A. S., Its, A. R. and Kitaev, A. V. 1992b. The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys., 147(2), 395–430.Google Scholar

Fokas, A. S., Grammaticos, B. and Ramani, A. 1993. From continuous to discrete Painlevé equations. J. Math. Anal. Appl., 180(2), 342–360.Google Scholar

Fokas, A. S., Its, A. R., Kapaev, A. A. and Novokshenov, V. Yu. 2006. Painlevé Transcendents: A Riemann–Hilbert Approach. Providence, RI: American Mathematical Society.

Fomin, S. and Zelevinsky, A. 2002a. Cluster algebras I: foundations. J. Am. Math. Soc., 15(2), 497–529.Google Scholar

Fomin, S. and Zelevinsky, A. 2002b. The Laurent Phenomenon. Adv. Appl. Math., 28(2), 119–144.Google Scholar

Fomin, S. and Zelevinsky, A. 2003. Cluster algebras II: finite type classification. Invent. Math., 154(1), 63–121.Google Scholar

Fordy, A. P. and Hone, A. N. W. 2014. Discrete integrable systems and Poisson algebras from cluster maps. Commun. Math. Phys., 325(2), 527–584.Google Scholar

Fordy, A. P. and Kassotakis, P. G. 2006. Multidimensional maps of QRT type. J. Phys. A: Math. Gen., 39(34), 10773.Google Scholar

Fordy, A. P. and Marsh, R. J. 2011. Cluster mutation-periodic quivers and associated Laurent sequences. J. Alg. Comb., 34, 19–66.Google Scholar

Forrester, P. J. 2003. Growth models, random matrices and Painlevé transcendents. Nonlinearity, 16(6), R27.Google Scholar

Forrester, P. J. 2010. Log-Gases and Random Matrices (LMS-34). Princeton, NJ: Princeton University Press.

Forrester, P. J. and Witte, N. S. 2004. Discrete Painlevé equations, orthogonal polynomials on the unit circle, and N-recurrences for averages over U(N) - PIII_ and P. τ-functions. Int. Math. Res. Not., 2004(4), 159–183.Google Scholar

Forrester, P. J. and Witte, N. S. 2006. Random matrix theory and the sixth Painlevé equation. J. Phys. A: Math. Gen., 39(39), 12211.Google Scholar

Freeman, N. C. and Nimmo, J. J. C. 1983. Soliton solutions of the Korteweg–de Vries and Kadomtsev–Petviashvili equations: The wronskian technique. Phys. Lett. A, 95(1), 1–3.Google Scholar

Freud, G. 1976. On the coefficients in the recursion formulae of orthogonal polynomials. Proc. R. Irish Acad. Sec. A, 76, 1–6.Google Scholar

Frobenius, G. 1881. Ueber Relationen zwischen den Näherungsbrüchen von Potenzreihen. J. Reine Angew. Math., 90, 1–17.Google Scholar

Fuchs, R. 1905. Sur quelques équations différentielles linéaires du second ordre. C. R. . Acad. Sci. Prais, 141, 555–558.Google Scholar

Fuchs, R. 1907. Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen. Math Ann., 63, 301–321.Google Scholar

Fukuda, A., Ishiwata, E., Iwasaki, M. and Nakamura, Y. 2009. The discrete hungry Lotka–Volterra system and a new algorithm for computing matrix eigenvalues. Inverse Prob., 25(1), 015007, 17.Google Scholar

Fukuda, A., Ishiwata, E., Yamamoto, Y., Iwasaki, M. and Nakamura, Y. 2013. Integrable discrete hungry systems and their related matrix eigenvalues. Ann. Mat. Pura Appl., 192(3), 423–445.Google Scholar

Gaeta, G. 2012. Nonlinear Symmetries and Nonlinear Equations. Mathematics and Its Applications. Netherlands: Springer.

Gambier, B. 1906a. Sur les équations différentielles dont l'intégrale générale est uniforme, C.R. Acad. Sc. Paris, 142, 1403–1406.Google Scholar

Gambier, B. 1906b. Sur les équations différentielles du deuxième ordre et du premier degré dont l'intégrale générale est uniforme, C.R. Acad. Sc. Paris, 142, 1497–1500.Google Scholar

Gambier, B. 1910. Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes. Acta Math., 33(1), 1–55.Google Scholar

Garabedian, P. R. 1964. Partial Differential Equations. New York: JohnWiley & Sons Inc.

Gardner, C. S., Greene, J. M., Kruskal, M. D. and Miura, R. M. 1967. Method for solving the Korteweg-deVries equation. Phys. Rev. Lett., 19(Nov), 1095–1097.Google Scholar

Garnier, R. 1912. Sur des équations différentielles du troisième ordre dont l'intégrale générale est uniforme et sur une classe d'équations nouvelles d'ordre supérieur dont l'intégrale générale a ses points critiques fixes. Ann. Sci. Éc. Norm. Supér. (3), 29, 1–126.CrossRefGoogle Scholar

Gasper, G. and Rahman, M. 2004. Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications. Cambridge, UK: Cambridge University Press.

Gelfand, I. M. and Fomin, S. V. 2000. Calculus of Variations. Dover Books on Mathematics. New York: Dover Publications.

Gelfand, I. M., Kapranov, M. and Zelevinsky, A. 2008. Discriminants, Resultants, and Multidimensional Determinants. New York: Springer Science & Business Media.

Gilson, C. R., Nimmo, J. J. C. and Ohta, Y. 2007. Quasideterminant solutions of a non-Abelian Hirota–Miwa equation. J. Phys. A: Math. Theor., 40(42), 12607.Google Scholar

Glasser, M. L., Papageorgiou, V. G. and Bountis, T. C. 1989. Mel'nikov's function for two-dimensional mappings. SIAM J. Appl. Math., 49(3), 692–703.Google Scholar

Goldstein, H. 1950. Classical Mechanics. Boston: Addison-Wesley.

Goncharenko, V. M. and Veselov, A. P. 2004. Yang-Baxter maps and matrix solitons. In Shabat, A.B., González-López, A., Mañas, M., Martínez Alonso, L. and Rodríguez, M.A. (eds), New Trends in Integrability and Partial Solvability. Nato Science Series II, vol. 132. US: Springer, 191–197.

Gragg, W. B. 1972. The Padé table and its relation to certain algorithms of numerical analysis. SIAM Rev., 14, 1–16.Google Scholar

Grammaticos, B. and Ramani, A. 1996. The Gambier mapping. Physica A, 223(1–2), 125–136.Google Scholar

Grammaticos, B. and Ramani, A. 1999. On a novel q-discrete analogue of the Painlevé VI equation. Phys. Lett. A, 257(5–6), 288–292.Google Scholar

Grammaticos, B. and Ramani, A. 2004. Discrete Painlevé equations: a review. In Grammaticos, B., Tamizhmani, T., and Kosmann-Schwarzbach, Y. (eds), Discrete Integrable Systems, Lecture Notes in Physics, Vol. 644, Berlin: Springer, 245–321.

Grammaticos, B., Ramani, A. and Papageorgiou, V. G. 1991. Do integrable mappings have the Painlevé property. Phys. Rev. Lett., 67(14), 1825–1828.Google Scholar

Grammaticos, B., Ramani, A. and Moreira, I. 1993. Delay-differential equations and the Painlevé transcendents. Physica A, 196(4), 574–590.Google Scholar

Grammaticos, B., Ramani, A. and Hietarinta, J. 1994. Multilinear operators: the natural extension of Hirota's bilinear formalism. Phys. Lett. A, 190(1), 65–70.Google Scholar

Grammaticos, B., Ohta, Y., Ramani, A. and Sakai, H. 1998. Degeneration through coalescence of the q-Painlevé VI equation. J. Phys. A: Math. Gen., 31(15), 3545.Google Scholar

Grammaticos, B., Nijhoff, F. W. and Ramani, A. 1999. Discrete Painlevé equations. In Conte, R. M. (ed), The Painlevé Property: One Century Later. CRM Series in Mathematical Physics. New York: Springer, 413–516.

Grammaticos, B., Ramani, A. and Tamizhmani, K. M. 2000. Deautonomisation of differential-difference equations and discrete Painlevé equations. Chaos, Solitons Fractals, 11(5), 757–764.Google Scholar

Grammaticos, B., Kosmann-Schwarzbach, Y. and Tamizhmani, T. 2004. Discrete Integrable Systems. Lecture Notes in Physics, vol. 644. Berlin Heidelberg: Springer.

Grammaticos, B., Ramani, A., Satsuma, J.,Willox, R. and Carstea, A. S. 2005. Reductions of integrable lattices. J. Nonlinear Math. Phys., 12, Supp 1, 363–371.Google Scholar

Grammaticos, B., Halburd, R. G., Ramani, A. and Viallet, C.-M. 2009. How to detect the integrability of discrete systems. J. Phys. A: Math. Theor., 42(45), 454002.Google Scholar

Grammaticos, B., Ramani, A., Satsuma, J. and Willox, R. 2012. Discretising the Painlevé equations à la Hirota–Mickens. J. Math. Phys., 53(2), 023506.Google Scholar

Grammaticos, B., Ramani, A., Willox, R., Mase, T. and Satsuma, J. 2015. Singularity confinement and full-deautonomisation: A discrete integrability criterion. Phys. D, 313, 11–25.Google Scholar

Graves-Morris, P. R. and Roberts, D. E. 1997. Problems and progress in vector Padé approximation. J. Comput. Appl. Math., 77(1–2), 173–200.Google Scholar

Griffiths, D. J. 2005. Introduction to Quantum Mechanics, 2nd ed. Harlow, UK: Pearson Education.

Gromak, V. I. and Tsegel'nik, V. V. 1994. Functional relations between solutions of equations of P-type. Differ. Uravn., 30(7), 1118–1124, 1284.Google Scholar

Gromak, V. I., Laine, I. and Shimomura, S. 2002. Painlevé Differential Equations in the Complex Plane. De Gruyter Studies in Mathematics. Berlin: De Gruyter.

Gutknecht, M. H. and Parlett, B. N. 2011. From qd to LR, or, how were the qd and LR algorithms discovered. IMA J. Numer. Anal., 31(3), 741–754.Google Scholar

Hairer, E., Lubich, C. and Wanner, G. 2006. Geometric Numerical Integration: Structure- Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics. Berlin Heidelberg: Springer.

Halburd, R. G. 2005. Diophantine integrability. J. Phys. A: Math. Gen., 38(16), L263–L269.Google Scholar

Halburd, R. G. and Korhonen, R. J. 2006. Existence of finite-order meromorphic solutions as a detector of integrability in difference equations. Phys. D, 218(2), 191–203.Google Scholar

Halburd, R. G. and Korhonen, R. J. 2007a. Finite-order meromorphic solutions and the discrete Painlevé equations. Proc. London Math. Soc. (3), 94(2), 443–474.Google Scholar

Halburd, R. G. and Korhonen, R. J. 2007b.Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations. J. Phys. A: Math. Theor., 40(6), R1–R38.Google Scholar

Hartshorne, R. 1977. Algebraic Geometry, Graduate texts in mathematics: 52. New York: Springer.

Hasselblatt, B. and Propp, J. 2007. Degree-growth of monomial maps. Ergod. Theor. Dyn. Syst., 27(05), 1375–1397.Google Scholar

Hay, M., Hietarinta, J., Joshi, N. and Nijhoff, F. W. 2007. A Lax pair for a lattice modified KdV equation, reductions t. q-Painlevé equations and associated Lax pairs. J. Phys. A: Math. Theor., 40(2), F61–F73.Google Scholar

Hay, M., Kajiwara, K. and Masuda, T. 2011. Bilinearization and special solutions to the discrete Schwarzian KdV equation. Journal of Math-for-Industry, 3 (2011A-5), 53–62.Google Scholar

He, Y., Hu, X.-B., Sun, J. Q. and Weniger, E. J. 2011. Convergence acceleration algorithm via an equation related to the lattice boussinesq equation. SIAM J. Sci. Comput., 33(3), 1234–1245.Google Scholar

Herbst, B. M. and Ablowitz, M. J. 1989. Numerically induced chaos in the nonlinear Schrödinger equation. Phys. Rev. Lett., 62(18), 2065.Google Scholar

Herbst, B M., Varadi, F. and Ablowitz, M. J. 1994. Symplectic methods for the nonlinear Schrödinger equation. Math. Comput. Simul., 37(4), 353–369.Google Scholar

Hertrich-Jeromin, U., McIntosh, I., Norman, P. and Pedit, F. 2001. Periodic discrete conformal maps. J. Reine Angew Math., 534, 129–153.Google Scholar

Hietarinta, J. 1987a. A search for bilinear equations passing Hirota's three-soliton condition. I. KdV-type bilinear equations. J. Math. Phys., 28(8), 1732–1742.Google Scholar

Hietarinta, J. 1987b. A search for bilinear equations passing Hirota's three-soliton condition. II. mKdV-type bilinear equations. J. Math. Phys., 28(9), 2094–2101.Google Scholar

Hietarinta, J. 1987c. A search for bilinear equations passing Hirota's three-soliton condition. III. Sine-Gordon-type bilinear equations. J. Math. Phys., 28(11), 2586–2592.Google Scholar

Hietarinta, J. 1997. Permutation-type solutions to the Yang-Baxter and othe. n-simplex equations. J. Phys. A: Math. Gen., 30(13), 4757–4771.Google Scholar

Hietarinta, J. 2004. A new two-dimensional lattice model that is ‘consistent around a cube’. J. Phys. A: Math. Gen., 37(6), L67–L73.Google Scholar

Hietarinta, J. 2005. Searching for CAC-maps. J. Nonlinear Math. Phys., 12, Supp 2, 223–230.Google Scholar

Hietarinta, J. 2011. Boussinesq-like multi-component lattice equations and multi-dimensional consistency. J. Phys. A: Math. Theor., 44(16), 165204, 22pp.Google Scholar

Hietarinta, J. and Kruskal, M. D. 1992. Hirota forms for the six Painlevé equations from singularity analysis. In Levi, D. and Winternitz, P. (eds), Painlevé Transcendents, Their Asymptotics and Physical Applications. New York: Springer Science+Business Media, 175–185.

Hietarinta, J. and Viallet, C. 1998. Singularity confinement and chaos in discrete systems. Phys. Rev. Lett., 81(Jul), 325–328.Google Scholar

Hietarinta, J. and Viallet, C. 2000. Discrete Painlevé I and singularity confinement in projective space. Chaos, Solitons Fractals, 11(1–3), 29–32.Google Scholar

Hietarinta, J. and Viallet, C. 2012. Weak Lax pairs for lattice equations. Nonlinearity, 25(7), 1955–1966.Google Scholar

Hietarinta, J. and Zhang, D. J. 2009. Soliton solutions for ABS lattice equations. II. Casoratians and bilinearization. J. Phys. A: Math. Theor., 42(40), 404006, 30pp.Google Scholar

Hietarinta, J. and Zhang, D. J. 2011. Soliton taxonomy for a modification of the lattice Boussinesq equation. SIGMA 7, 061, 14pp.Google Scholar

Hietarinta, J. and Zhang, D. J. 2013. Hirota's method and the search for integrable partial difference equations. 1. Equations on a 3x3 stencil. J. Differ. Equ. Appl., 19(8), 1292–1316.Google Scholar

Hildebrand, F. B. 1968. Finite-Difference Equations and Simulations. Englewood Cliffs, NJ: Prentice-Hall Inc.

Hille, E. 1976. Ordinary Differential Equations in the Complex Domain. NewYork: Dover.

Hirota, R. 1971. Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett., 27(Nov), 1192–1194.Google Scholar

Hirota, R. 1973. Exact three-soliton solution of the two-dimensional sine-gordon equation. J. Phys. Soc. Japan, 35(5), 1566–1566.Google Scholar

Hirota, R. 1974. A new form of Bäcklund transformations and its relation to the inverse scattering problem. Prog. Theor. Phys., 52(5), 1498–1512.Google Scholar

Hirota, R. 1977a. Nonlinear partial difference equations. I. A difference analogue of the Korteweg-de Vries equation. J. Phys. Soc. Japan, 43(4), 1424–1433.Google Scholar

Hirota, R. 1977b. Nonlinear partial difference equations. II. Discrete-time Toda equation. J. Phys. Soc. Japan, 43(6), 2074–2078.Google Scholar

Hirota, R. 1977c. Nonlinear partial difference equations. III. Discrete sine-Gordon equation. J. Phys. Soc. Japan, 43(6), 2079–2086.Google Scholar

Hirota, R. 1981. Discrete analogue of a generalized Toda equation. J. Phys. Soc. Japan, 50(11), 3785–3791.Google Scholar

Hirota, R. 1986. Reduction of soliton equations in bilinear form. Physica D, 18(1–3), 161–170.Google Scholar

Hirota, R. 2004. The Direct Method in Soliton Theory. Cambridge Tracts in Mathematics, vol. 155. Cambridge, UK: Cambridge University Press. Translated from the 1992 Japanese original and edited by A., Nagai, J., Nimmo and C., Gilson.

Hirota, R. and Ito, M. 1981. A direct approach to multi-periodic wave solutions to nonlinear evolution equations. J. Phys. Soc. Japan, 50(1), 338–342.Google Scholar

Hirota, R. and Kimura, K. 2000. Discretization of the Euler top. J. Phys. Soc. Japan, 69(3), 627–630.Google Scholar

Hirota, R. and Satsuma, J. 1976. A variety of nonlinear network equations generated from the Bäcklund transformation for the toda lattice. Prog. Theor. Phys. Supp., 59, 64–100.Google Scholar

Hirota, R. Tsujimoto, S. and Imai, T. 1993. Difference scheme of soliton equations. In Christiansen, P. L., Eilbeck, J. C., and Parmentier, R. D. (eds), Future Directions of Nonlinear Dynamics in Physical and Biological Systems. NATO ASI Series, vol. 312. US: Springer, 7–15.

Hirota, R., Kimura, K. and Yahagi, H. 2001. How to find the conserved quantities of nonlinear discrete equations. J. Phys. A: Math. Gen., 34(48), 10377–10386.Google Scholar

Hone, A. N. W. 2007. Sigma function solution of the initial value problem for Somos 5 sequences. Trans. Am. Math. Soc., 359(10), 5019–5034.Google Scholar

Hone, A. N. W. and Petrera, M. 2009. Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras. J. Geom. Mech., 1(1), 55–85.Google Scholar

Hone, A. N. W. and Swart, C. 2008. Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences. Math. Proc. Cambridge Philos. Soc., 145(7), 65–85.Google Scholar

Humphreys, J. E. 1992. Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics. Cambridge, UK: Cambridge University Press.

Hydon, P. E. 2014. Difference Equations by Differential Equations Methods. Cambridge Monographs on Applied and Computational Mathematics, vol. 149. Cambridge, UK: Cambridge University Press.

Hydon, P. E. and Viallet, C.-M. 2010. Asymmetric integrable quad-graph equations. Appl. Anal., 89(4), 493–506.Google Scholar

Iatrou, A. and Roberts, J. A. G. 2001. Integrable mappings of the plane preserving biquadratic invariant curves. J. Phys. A: Math. Gen., 34(34), 6617–6636.Google Scholar

Ince, E. L. 1956. Ordinary Differential Equations. Dover Books on Science. New York: Dover Publications Incorporated.

Ismail, M. 2005. Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and Its Applications. vol. 98. Cambridge, UK: Cambridge University Press.

Its, A. R., Kitaev, A. V. and Fokas, A. S. 1990. The isomonodromy approach in the theory of two-dimensional quantum gravitation. Russ. Math. Surv., 45(6), 155–157.Google Scholar

Iwasaki, K., Kimura, H., Shimemura, S. and Yoshida, M. 1991. From Gauss to Painlevé: A Modern Theory of Special Functions. Aspects of Mathematics. Braunschweig: Vieweg+Teubner Verlag.

Jacobi, C. G. J. 1846. Uber die Darstellung einer Reihe gegebener Werthe durch eine gebrochene rationale Function. J. Reine Angew. Math., 30, 127–156.Google Scholar

Jimbo, M. and Miwa, T. 1981. Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. Physica D, 2, 407–448.Google Scholar

Jimbo, M. and Miwa, T. 1983. Solitons and infinite-dimensional Lie algebras. Publ. Res. Inst. Math. Sci., 19(3), 943–1001.Google Scholar

Jimbo, M. and Sakai, H. 1996. q-analog of the sixth Painlevé equation. Lett. Math. Phys., 38(2), 145–154.Google Scholar

Joshi, N. 1997. A local asymptotic analysis of the discrete first Painlevé equation as the discrete independent variable approaches infinity. Methods Appl. Anal., 4(2), 124–133.Google Scholar

Joshi, N. 2009. Direct “delay” reductions of the Toda equation. J. Phys. A: Math. Theor., 42(2), 022001.Google Scholar

Joshi, N. 2015. Quicksilver solutions of. q-difference first Painlevé equation. Stud. Appl. Math, 134(2), 233–251.Google Scholar

Joshi, N. and Kruskal, M. D. 1994. A direct proof that solutions of the six Painlevé equations have no movable singularities except poles. Stud. Appl. Math., 93(3), 187–207.Google Scholar

Joshi, N. and Lobb, S. B. 2016. Singular dynamics of. q-difference Painlevé equation in its initial-value space. J. Phys. A: Math. Theor, 49(1), 014002.Google Scholar

Joshi, N. and Lustri, C. J. 2015. Stokes phenomena in discrete Painlevé I. Proc. R. Soc. London A Math. Phys. Eng. Sci., 471(2177), 20140874.Google Scholar

Joshi, N. and Shi, Y. 2011. Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions. Proc. R. Soc. London A Math. Phys. Eng. Sci., rspa20110167.

Joshi, N. and Shi, Y. 2012. Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem. II. Hypergeometric solutions. Proc. R. Soc. London A Math. Phys. Eng. Sci., 468(2146), 3247–3264.Google Scholar

Joshi, N., Burtonclay, D. and Halburd, R. G. 1992. Nonlinear nonautonomous discrete dynamical systems from a general discrete isomonodromy problem. Lett. Math. Phys., 26(2), 123–131.Google Scholar

Joshi, N., Ramani, A. and Grammaticos, B. 1998. A bilinear approach to discrete Miura transformations. Phys. Lett. A, 249(1–2), 59–62.Google Scholar

Joshi, N., Grammaticos, B., Tamizhmani, T. and Ramani, A. 2006. From integrable lattices to non-QRT mappings. Lett. Math. Phys., 78(1), 27–37.Google Scholar

Joshi, N., Nakazono, N. and Shi, Y. 2014. Geometric reductions of ABS equations on an n-cube to discrete Painlevé system. J. Phys A: Math. Theor., 47(50), 505201.Google Scholar

Joshi, N., Nakazono, N., and Shi, Y. 2015. Lattice equations arising from discrete Painlevé systems. I. (A2 + A(1) and (A1 + A1)(1) cases. J. Math. Phys., 56(9), 092705.Google Scholar

Kac, M. and van Moerbeke, P. 1975. On an explictly soluble system fo nonlinear differential equations related to certain Toda lattices. Adv. Math., 16, 160–169.Google Scholar

Kac, V. G. 1994. Infinite-Dimensional Lie Algebras. Cambridge, UK: Cambridge university Press.

Kahan, W. 1993. Unconventional numerical methods for trajectory calculations. Lecture notes, CS Division, Department of EECS.

Kajiwara, K. and Ohta, Y. 2008. Bilinearization and casorati determinant solution to the non-autonomous discrete KdV equation. J. Phys. Soc. Japan, 77(5), 054004.Google Scholar

Kajiwara, K., Maruno, K. and Oikawa, M. 2000. Bilinearization of discrete soliton equations through the singularity confinement test. Chaos, Solitons & Fractals, 11(1–3), 33–39.Google Scholar

Kajiwara, K., Noumi, M. and Yamada, Y. 2001. A study on the fourt. q-Painlevé equation. J. Phys. A: Math. Gen., 34(41), 8563–8581.Google Scholar

Kajiwara, K., Noumi, M. and Yamada, Y. 2002. q-Painlevé systems arising from q-KP hierarchy. Lett. Math. Phys., 62(3), 259–268.Google Scholar

Kajiwara, K., Masuda, T. Noumi, M. Ohta, Y. and Yamada, Y. 2003. 10E9 solution to the elliptic Painlevé equation. J. Phys. A: Math. Gen., 36(17), L263–L272.Google Scholar

Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y. and Yamada, Y. 2004. Hypergeometric solutions to th. q-Painlevé equations. Int. Math. Res. Not., 2004(47), 2497–2521.Google Scholar

Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y. and Yamada, Y. 2005a. Construction of hypergeometric solutions to the q-Painlevé equations. Int. Math. Res. Not., 2005(24), 1439–1463.Google Scholar

Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y. and Yamada, Y. 2005b. A geometric description of the elliptic Painleve equation. Rokko Lect. Math., 18, 43–48.Google Scholar

Kajiwara, K.,Masuda, T., Noumi, M., Ohta, Y. and Yamada, Y. 2006. Point configurations, Cremona transformations and the elliptic difference Painlevé equation. In Delabaere, E. and Lodauy-Richaud, M. (ed), Théories asymptotiques et équations de Painlevé, Seminaires et Congres Sémin. Congr, vol. 14. Paris: SMF, 169–198.

Kajiwara, K., Noumi, M. and Yamada, Y. 2015. Geometric aspects of Painlevé equations. arXiv:1509.08186, 168pp.Google Scholar

Kakei, S., Nimmo, J. J. C. and Willox, R. 2009. Yang–Baxter maps and the discrete KP hierarchy. Glasg. Math. J., 51(2), 107–119.Google Scholar

Kakei, S., Nimmo, J. J. C. and Willox, R. 2010. Yang–Baxter maps from the discrete BKP equation. SIGMA 6, 028, 11pp.Google Scholar

Kanki, M. 2013. Studies on the Discrete Integrable Equations over Finite Fields. Ph.D. thesis, University of Tokyo.

Kassotakis, P. and Alonso, L. M. 2011. Families of integrable equations. SIGMA 7, 100, 14pp.Google Scholar

Kassotakis, P. and Alonso, L. M. 2012. On non-multiaffine consistent-around-the-cube lattice equations. Phys. Lett. A, 376(45), 3135–3140.Google Scholar

Kay, I. and Moses, H. E. 1956. Reflectionless transmission through dielectrics and scattering potentials. J. Appl. Phys., 27(12), 1503–1508.Google Scholar

Kelley, W. G., and Peterson, A. C. 2001. Difference Equations: An Introduction with Applications. San Diego: Harcourt/Academic Press.

Kimura, K. and Hirota, R. 2000. Discretization of the Lagrange Top. J. Phys. Soc. Japan, 69(10), 3193–3199.Google Scholar

Kimura, K., Yahagi, H., Hirota, R., Ramani, A., Grammaticos, B. and Ohta, Y. 2002. A new class of integrable discrete systems, J. Phys. A: Math. Gen., 35(43), 9205–3212.Google Scholar

King, A. D. and Schief, W. K. 2003. Tetrahedra, octahedra and cubo-octahedra: integrable geometry of multi-ratios. J. Phys. A: Math. Gen., 36(3), 785–802.Google Scholar

King, A. D. and Schief, W. K. 2006. Application of an incidence theorem for conics: Cauchy problem and integrability of the dCKP equation. J. Phys. A: Math. Gen., 39(8), 1899–1913.Google Scholar

Koekoek, R., Lesky, P. A. and Swarttouw, R. F. 2010. Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer Monographs in Mathematics. Berlin: Springer-Verlag.

Koenigs, G. 1891. Sur les systèmes conjugués à invariants égaux. C. R. Acad. Sci. Paris., 113, 1022–1024.Google Scholar

Koenigs, G. 1892. Sur les réseaux plans à invariants égaux et les lignes asymptotiques. C. R. Acad. Sci. Paris., 114, 55–57.Google Scholar

Kolchin, E. R. 1973. Differential Algebra & Algebraic Groups. Pure and Applied Mathematics. New York and London: Academic Press.

Konopelchenko, B. G. and Schief, W. K. 1998. Three-dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality. R. Soc. London Proc. Ser. A Math. Phys. Eng. Sci., 454(1980), 3075–3104.Google Scholar

Konopelchenko, B. G. and Schief, W. K. 2002a. Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy. J. Phys. A: Math. Gen., 35(29), 6125–6144.Google Scholar

Konopelchenko, B. G. and Schief, W. K. 2002b. Reciprocal figures, graphical statics, and inversive geometry of the Schwarzian BKP hierarchy. Stud. Appl. Math., 109(2), 89–124.Google Scholar

Konstantinou-Rizos, S. 2014. Darboux transformations, discrete integrable systems and related Yang–Baxter maps. arXiv preprint arXiv:1410.5013.

Konstantinou-Rizos, S., and Mikhailov, A. V. 2013. Darboux transformations, finite reduction groups and related Yang–Baxter maps. J. Phys. A: Math. Theor., 46(42), 425201.Google Scholar

Konstantinou-Rizos, S., Mikhailov, A. V. and Xenitidis, P. 2015. Reduction groups and related integrable difference systems of NLS type. arXiv:1503.06406.

Korepin, V. E., Bogoliubov, N. M. and Izergin, A.G. 1997. Quantum Inverse Scattering Method and Correlation Functions. Cambridge, UK: Cambridge University Press.

Korteweg, D. J. and de Vries, G. 1895. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. Ser. 5, 39(240), 422–443.Google Scholar

Kouloukas, T. E. and Papageorgiou, V. G. 2009. Yang–Baxter maps with first-degree-polynomial 2× 2 Lax matrices. J. Phys. A:Math. Theor., 42(40), 404012.Google Scholar

Kouloukas, T. E., and Papageorgiou, V. G. 2012. 3D compatible ternary systems and Yang–Baxter maps. J. Phys. A: Math. Theor., 45(34), 345204.Google Scholar

Kowalevski, S. 1889. Sur le probleme de la rotation d'un corps solide autour d'un point fixe. Acta Math., 12(1), 177–232.Google Scholar

Kowalevski, S. 1890. Sur une propriété du système d'équations différentielles qui définit la rotation d'un corps solide autour d'un point fixe. Acta Math., 14(1), 81–93.Google Scholar

Krichever, I. M. 1978. Rational solutions of the Kadomtsev–Petviashvili equation and integrable systems of N particles on a line. Funct. Anal. Appl., 12(1), 59–61.Google Scholar

Krichever, I. M. 1980. Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles. Funct. Anal. Appl., 14(4), 282–290.Google Scholar

Krichever, I. M. and Novikov, S. P. 1979. Holomorphic fiberings and nonlinear equations. Finite zone solutions of rank 2. Sov. Math. Dokl, 20(4), 650–654.Google Scholar

Krichever, I. M. and Novikov, S. P. 1980. Holomorphic bundles over algebraic curves and non-linear equations. Russ. Math. Surv., 35(6), 53–79.Google Scholar

Krichever, I. M., Wiegmann, P. and Zabrodin, A. 1998. Elliptic solutions to difference non-linear equations and related many-body problems. Commun. Math. Phys., 193(2), 373–396.Google Scholar

Kruskal, M. D. and Clarkson, P. A. 1992. The Painlevé-Kowalevski and poly-Painlevé tests for integrability. Stud. Appl. Math., 86(2), 87–165.Google Scholar

Kupershmidt, B. A. 2000. KP or mKP: Noncommutative Mathematics of Lagrangian, Hamiltonian, and Integrable Systems. Mathematical surveys and monographs. Providence, RI: American Mathematical Society.

Lafortune, S., Ramani, A., Ohta, Y., Grammaticos, B., and Tamizhmani, K. M. 2001. Blending two discrete integrability criteria: singularity confinement and algebraic entropy. In Coley, A. A., Levi, D., Milson, R., Rogers, C. and Winternitz, P. (eds) Bäcklund and Darboux Transformations. The Geometry of Solitons. CRM Proceedings & Lecture Notes, vol. 29. Providence, RI: American Mathematical Society, 299–311.

Lamb, G. L. 1980. Elements of Soliton Theory. Pure and Applied Mathematics. New York: John Wiley & Sons.

Lax, P. D. 1968. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Applied Math. 21, 467–490.Google Scholar

LeCaine, J. 1943. The linear q-difference equation of the second order. Am. J. Math., 65, 585–600.Google Scholar

Levi, D. 1981. Nonlinear differential difference equations as Backlund transformations. J. . Phys. A: Math. Gen., 14(5), 1083.Google Scholar

Levi, D. and Benguria, R. 1980. Bäcklund transformations and nonlinear differential difference equations. Proc. Natl. Acad. Sci. U.S.A., 77(9), 5025–5027.Google Scholar

Levi, D. and Petrera, M. 2007. Continuous symmetries of the lattice potential KdV equation. J. Phys. A: Math. Theor., 40(15), 4141.Google Scholar

Levi, D. and Winternitz, P. 1993. Symmetries and conditional symmetries of differential-difference equations., J. Math. Phys. 34, 3713–3730.Google Scholar

Levi, D., andWinternitz, P. 2006. Continuous symmetries of difference equations. J. Phys. A: Math. Gen., 39(2), R1–R63.Google Scholar

Levi, D. and Yamilov, R. 1997. Conditions for the existence of higher symmetries of evolutionary equations on the lattice. J. Math. Phys., 38, 6648.Google Scholar

Levi, D. and Yamilov, R. I. 2009a. The generalized symmetry method for discrete equations. J. Phys. A: Math. Theor., 42(45), 454012, 18pp.Google Scholar

Levi, D. and Yamilov, R. I. 2009b. On a nonlinear integrable difference equation on the square. Ufimsk. Mat. Zh., 1, 101–105.Google Scholar

Levi, D., and Yamilov, R. I. 2011. Generalized symmetry integrability test for discrete equations on the square lattice. J. Phys. A: Math. Theor., 44(14), 145207, 22pp.Google Scholar

Levi, D. Pilloni, L. and Santini, P. M. 1981. Integrable three-dimensional lattices. J. Phys. A: Math. Gen., 14(7), 1567.Google Scholar

Levi, D. Ragnisco, O. and Rodriguez, M. A. 1992. On non-isospectral flows, Painlevé equations, and symmetries of differential and difference equations. Theor. Math. Phys., 93(3), 1409–1414.Google Scholar

Levi, D., Petrera, M. and Scimiterna, C. 2007. The lattice Schwarzian KdV equation and its symmetries. J. Phys. A: Math. Theor., 40(42), 12753.Google Scholar

Levi, D., Olver, P., Thomova, Z. and Winternitz, P. 2011. Symmetries and Integrability of Difference Equations. London Mathematical Society Lecture Note Series, vol. 381. Cambridge, UK: Cambridge University Press.

Levin, A. 2008. Difference Algebra. Algebra and Applications, vol. 8. Berlin: Springer.

Lobb, S. B. 2010. Lagrangian Structures and Multidimensional Consistency. Ph.D. thesis, University of Leeds.

Lobb, S. B. and Nijhoff, F. W. 2009. Lagrangian multiforms and multidimensional consistency. J. Phys. A: Math. Theor., 42(45), 454013.Google Scholar

Lobb, S. B. and Nijhoff, F. W. 2010. Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy. J. Phys. A: Math. Theor., 43(7), 072003, 11pp.Google Scholar

Lobb, S. B. and Nijhoff, F.W. 2013. A variational principle for discrete integrable systems. arXiv preprint arXiv:1312.1440.

Lobb, S. B., Nijhoff, F. W. and Quispel, G. R. W. 2009. Lagrangian multiform structure for the lattice KP system. J. Phys. A: Math. Theor., 42(47), 472002.Google Scholar

Logan, J. D. 1973. First integrals in the discrete variational calculus. Aequationes Math., 9, 210–220.Google Scholar

Maeda, S. 1981. Extension of discrete Noether theorem. Math. Japon., 26(1), 85–90.Google Scholar

Magnus, A. P. 1995. Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. J. Comput. Appl. Math., 57(1), 215–237.Google Scholar

Magnus, A. P. 2009. Rational interpolation to solutions of Riccati difference equations on elliptic lattices. J. Comput. Appl. Math., 233(3), 793–801.Google Scholar

Maillet, J.-M. and Nijhoff, F. W. 1989. Integrability for multidimensional lattice models. Phys. Lett. B, 224(4), 389–396.Google Scholar

Mano, T. 2010. Asymptotic behaviour around a boundary point of the q-Painlevé VI equation and its connection problem. Nonlinearity, 23(7), 1585.Google Scholar

Marsh, R. J. 2013. Lecture Notes on Cluster Algebras. Zurich Lectures in Advanced Mathematics. European Mathematical Society.

Maruno, K. and Quispel, G. R. W. 2006. Construction of integrals of higher-order mappings. J. Phys. Soc. Jpn., 75, 123001–123005.Google Scholar

Maruno, K., Kajiwara, K., Nakao, S. and Oikawa, M. 1997. Bilinearization of discrete soliton equations and singularity confinement. Phys. Lett. A, 229(3), 173–182.Google Scholar

Matveev, V. B. and Salle, M. A. 1991. Darboux Transformations and Solitons. Nonlinear Dynamics Series. Berlin: Springer.

McMillan, E. M. 1971. A problem in the stability of periodic systems. Brittin, W. E. and Odabasi, H. (eds), Topics in Modern Physics. A Tribute to E.U. Condon. Colorado Associated Univ. Press, 219–44.

McMullen, C. T. 2007. Dynamics on blowups of the projective plane. Publ. Math. IHES 105(1), 49–89.Google Scholar

Mercat, C. 2001. Discrete Riemann surfaces and the Ising model. Commun. Math. Phys., 218(1), 177–216.Google Scholar

Mercat, C. 2004. Exponentials form a basis of discrete holomorphic functions on a compact. Bull. Soc. Math. France, 132(2), 305–326.Google Scholar

Mercat, C. 2008. Discrete Riemann surfaces, linear and non-linear. arXiv preprint arXiv:0802.1448.

Mikhailov, A. V., Wang, J. P. and Xenitidis, P. 2011a. Cosymmetries and Nijenhuis recursion operators for difference equations. Nonlinearity, 24(7), 2079–2097.Google Scholar

Mikhailov, A. V., Wang, J. P. and Xenitidis, P. 2011b. Recursion operators, conservation laws, and integrability conditions for difference equations. Theor. Math. Phys., 167(1), 421–443.Google Scholar

Milne-Thomson, L. M. 2000. The Calculus of Finite Differences. Providence, RI: AMS Chelsea Publishing (reprint of the 1933 edition).

Miura, R. M. 1968. Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys., 9, 1202–1204.Google Scholar

Miura, R. M., Gardner, C. S. and Kruskal, M. D. 1968. Korteweg–de vries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys., 9(8), 1204–1209.Google Scholar

Miwa, T. 1982. On Hirota's difference equations. Proc. Japan Acad. Ser. A Math. Sci., 58(1), 9–12.Google Scholar

Miwa, T., Jimbo, M. and Date, E. 2000. Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras. Cambridge Tracts in Mathematics, vol. 135. Cambridge, UK: Cambridge University Press. Translated from the 1993 Japanese original by M., Reid.

Moser, J. 1975. Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math., 16(2), 197–220.Google Scholar

Moser, J. and Veselov, A. P. 1991. Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys., 139(2), 217–243.Google Scholar

Muĝan, U., and Santini, P. M. 2011. On the solvability of the discrete second Painlevé equation. J. Phys. A: Math. Theor., 44(18), 185204.Google Scholar

Muir, T. 1890. The Theory of Determinants in the Historical Order of Its Development. vol. I. Determinants in General. Leibniz (1693) to Cayley (1841).London: Macmillan and Co.

Murata, M. 2009. Lax forms of the q-Painlevé equations. J. Phys. A: Math. Theor., 42(11), 115201.Google Scholar

Murata, M., Sakai, H. and Yoneda, J. 2003. Riccati solutions of discrete Painlevé equations with Weyl group symmetry of type E(1) 8. J. Math. Phys., 44(3), 1396–1414.Google Scholar

Nagai, A. and Satsuma, J. 1995. Discrete soliton equations and convergence acceleration algorithms. Phys. Lett. A, 209(5–6), 305–312.Google Scholar

Nagai, A., Tokihiro, T. and Satsuma, J. 1998. The Toda molecule equation and th.-algorithm. Math. Comp., 67(224), 1565–1575.Google Scholar

Nakamura, A. 1979. A direct method of calculating periodic wave solutions to nonlinear evolution equations. I. Exact two-periodic wave solution. J. Phys. Soc. Japan, 47(5), 1701–1705.Google Scholar

Nevai, P. 1986. Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory, 48(1), 3–167.Google Scholar

Newell, A. C. 1985. Solitons in Mathematics and Physics. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 48. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).Google Scholar

Alonso, L. M. 2007. Darboux transformations for a 6-point scheme. J. Phys. A: Math. Theor., 40(15), 4193.Google Scholar

Alonso, L. M. and Santini, P. M. 2005. The self-adjoint 5-point and 7-point difference operators, the associated Dirichlet problems, Darboux transformations and Lelieuvre formulae. Glasg. Math. J., 47(A), 133–147.Google Scholar

Nijhoff, F. W. 1985. Theory of integrable three-dimensional nonlinear lattice equations. Lett. Math. Phys., 9(3), 235–241.Google Scholar

Nijhoff, F. W. 1987. Integrable hierarches, Lagrangian structures and non-commuting Flows. In M., Ablowitz, B., Fuchssteiner and M. D., Kruskal (eds), Topics in Soliton Theory and Exactly Solvable Nonlinear Equations. World Scientific, 150–181.

Nijhoff, F. W. 1997. On some “Schwarzian” equations and their discrete analogues. In Fokas, A. S. and Gelfand, I. M. (eds), Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman. Progress in nonlinear differential equations and their applications, vol. 26. Boston, MA: Birkhäuser, 237–260.

Nijhoff, F. W. 1999. Discrete Painlevé equations and symmetry reduction on the lattice. In Bobenko, A. I. and Seiler, R. (eds), Discrete Integrable Geometry and Physics (Vienna, 1996). Oxford Lecture Ser. Math. Appl., vol. 16. New York: Oxford University Press, 209–234.

Nijhoff, F. W. 2002. Lax pair for the Adler (lattice Krichever-Novikov) system. Phys. Lett. A, 297(1–2), 49–58.Google Scholar

Nijhoff, F. W. and Atkinson, J. 2010. Elliptic N-soliton solutions of ABS lattice equations. Int. Math. Res. Not., 2010(20), 3837–3895.Google Scholar

Nijhoff, F. W. and Capel, H. W. 1990. The direct linearisation approach to hierarchies of integrable PDEs in 2 + 1 dimensions: I. Lattice equations and the differential-difference hierarchies. Inverse Prob., 6(4), 567.CrossRefGoogle Scholar

Nijhoff, F. W. and Capel, H. 1995. The discrete Korteweg-de Vries equation. Acta Appl. Math., 39(1–3), 133–158.Google Scholar

Nijhoff, F.W. and Enolskii, V. Z. 1999. Integrable mappings of KdV type and hyperelliptic addition theorems. In Clarkson, P. A. and Nijhoff, F. W. (eds), Symmetries and Integrability of Difference Equations. LMS Lecture Note Series 255, 64–78.

Nijhoff, F. W. and Pang, G. D. 1994. A time-discretized version of the Calogero-Moser model. Phys. Lett. A, 191(1–2), 101–107.Google Scholar

Nijhoff, F. W. and Pang, G. D. 1996. Discrete-time Calogero-Moser model and lattice KP equations. In Levi, D., Vinet, L. andWinternitz, P. (eds), Symmetries and Integrability of Difference Equations. CRM Proceedings & Lecture Notes, vol. 9. Providence, RI: American Mathematical Society, 253–264.

Nijhoff, F. W. and Papageorgiou, V. G. 1991. Similarity reductions of integrable lattices and discrete analogues of the Painlevé II equation. Phys. Lett. A, 153(6–7), 337–344.Google Scholar

Nijhoff, F. W. and Puttock, S. E. 2003. On a two-parameter extension of the lattice KdV system associated with an elliptic curve. J. Nonlinear Math. Phys., 10, Supp 1, 107–123.Google Scholar

Nijhoff, F. W. and Walke, A. J. 2001. The discrete and continuous Painlevé VI hierarchy and the Garnier systems. Glasg. Math. J., 43A, 109–123.Google Scholar

Nijhoff, F.W., Quispel, G. R.W. and Capel, H.W. 1983a. Direct linearization of nonlinear difference-difference equations. Phys. Lett. A, 97(4), 125–128.Google Scholar

Nijhoff, F. W., Capel, H. W. and Quispel, G. R. W. 1983b. Integrable lattice version of the massive Thirring model and its linearization. Phys. Lett. A, 98(3), 83–86.Google Scholar

Nijhoff, F. W., Capel, H. W., Wiersma, G. L. and Quispel, G. R. W. 1984. Bäcklund transformations and three-dimensional lattice equations. Phys. Lett. A, 105(6), 267–272.Google Scholar

Nijhoff, F. W., Capel, H. W. and Wiersma, G. L. 1985. Integrable lattice systems in two and three dimensions. In Martini, R. (ed), Geometric Aspects of the Einstein Equations and Integrable Systems (Scheveningen, 1984). Lecture Notes in Phys., vol. 239. Berlin: Springer, 263–302.

Nijhoff, F. W., Papageorgiou, V. G., Capel, H. W. and Quispel, G. R. W. 1992. The lattice Gel'fand-Dikii hierarchy. Inverse Prob., 8(4), 597.Google Scholar

Nijhoff, F. W., Ragnisco, O. and Kuznetsov, V. B. 1996. Integrable time-discretisation of the Ruijsenaars-Schneider model. Commun. Math. Phys., 176(3), 681–700.Google Scholar

Nijhoff, F. W., Hone, A. N. W. and Joshi, N. 2000a. On a Schwarzian PDE associated with the KdV hierarchy. Phys. Lett. A, 267(2–3), 147–156.Google Scholar

Nijhoff, F. W., Joshi, N. and Hone, A. N. W. 2000b. On the discrete and continuous Miura chain associated with the sixth Painlevé equation. Phys. Lett. A, 264(5), 396–406.Google Scholar

Nijhoff, F. W., Ramani, A., Grammaticos, B. and Ohta, Y. 2001. On discrete Painlevé equations associated with the lattice KdV systems and the Painlevé VI equation. Stud. Appl. Math., 106(3), 261–314.Google Scholar

Nijhoff, F. W. Atkinson, J. and Hietarinta, J. 2009. Soliton solutions for ABS lattice equations. I. Cauchy matrix approach. J. Phys. A:Math. Theor., 42(40), 404005, 34pp.Google Scholar

Nimmo, J. J. C. 2006. On a non-Abelian Hirota–Miwa equation. J. Phys. A: Math. Gen., 39(18), 5053.Google Scholar

Nimmo, J. J. C. and Schief, W. K. 1997. Superposition principles associated with the Moutard transformation: an integrable discretization of (2 + 1)-dimensional sine-Gordon system. Proc. R. Soc. London Ser. A, 453(1957), 255–279.Google Scholar

Nishioka, S. 2010. Transcendence of solutions of q-Painlevé equation of type A(1)7. Aequa. Math., 79(1–2), 1–12.Google Scholar

Nishioka, S. 2012. Irreducibility o. q-Painlevé equation of type A(1)6 in the sense of order. J. Differ. Equ. Appl., 18(2), 313–333.Google Scholar

Noether, E. 1918. Variationsprobleme. Nachr. Kong. Gesell. Wissensch. (Göttingen), Math.-Phys. Kl., 2, 235–257.Google Scholar

Nörlund, N. E. 1954. Vorlesungen über Differenzenrechnung. Providence, RI: AMS Chelsea Publishing (reprint of the 1924 edition from Springer, Berlin).

Noumi, M. 2004. Painlevé Equations Through Symmetry. Translations of Mathematical Monographs. Providence, RI: American Mathematical Society.

Noumi, M., Tsujimoto, S. and Yamada, Y. 2013. Padé interpolation for elliptic Painlevé equation. In Iohara, K.,Morier-Genoud, S. and Rémy, B.(eds), Symmetries, Integrable Systems and Representations. Springer, 463–482.

Ohta, Y., Satsuma, J., Takahashi, D. and Tokihiro, T. 1988. An elementary introduction to Sato theory. Prog. Theor. Phys. Suppl., 94, 210–241.Google Scholar

Ohta, Y. Hirota, R. Tsujimoto, S. and Imai, T. 1993. Casorati and discrete Gram type determinant representations of solutions to the discrete KP hierarchy. J. Phys. Soc. Japan, 62(6), 1872–1886.Google Scholar