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In contrast to the well-established theory of differential equations, the theory of difference equations has not quite developed so far. The most recent advances in the theory of discrete integrable systems have brought a true revolution to the study of difference equations. Multidimensional consistency is a new concept appearing in the research of discrete integrable systems. This property, as an explanation to a type of discrete integrability, plays an important role in constructing the Bäcklund transformations, Lax pairs and exact solutions for discrete integrable system. In the present paper, the multidimensional consistency and its applications in the research of discrete integrable systems are reviewed.
[1] Zabusky N J, Kruskal M D 1965 Phys. Rev. Lett. 15 240
[2] Gardner C S, Greene J M, Kruskal M D, Miura R M 1967 Phys. Rev. Lett. 19 1095Google Scholar
[3] Lax P D 1968 Commun. Pure Appl. Math. 21 467Google Scholar
[4] Deift P 2019 arXiv: 1902.10267
[5] Case K M, Kac M 1973 J. Math. Phys. 14 594Google Scholar
[6] Ablowitz M J, Ladik J F 1975 J. Math. Phys. 16 598Google Scholar
[7] Ablowitz M J, Ladik J F 1976 J. Math. Phys. 17 1011Google Scholar
[8] Ablowitz M J, Ladik J F 1976 Stud. Appl. Math. 55 213Google Scholar
[9] Hirota R 1977 J. Phys. Soc. Jpn. 43 1424Google Scholar
[10] Hirota R 1977 J. Phys. Soc. Jpn. 43 2074Google Scholar
[11] Hirota R 1977 J. Phys. Soc. Jpn. 43 2079Google Scholar
[12] Date E, Jimbo M, Miwa T 1982 J. Phys. Soc. Jpn. 51 4116Google Scholar
[13] Date E, Jimbo M, Miwa T 1982 J. Phys. Soc. Jpn. 51 4125Google Scholar
[14] Date E, Jimbo M, Miwa T 1983 J. Phys. Soc. Jpn. 52 388Google Scholar
[15] Date E, Jimbo M, Miwa T 1983 J. Phys. Soc. Jpn. 52 761Google Scholar
[16] Date E, Jimbo M, Miwa T 1983 J. Phys. Soc. Jpn. 52 766Google Scholar
[17] Ueno K, Takasaki K 1984 In: Okamoto K (ed) Group Representations and Systems of Differential Equations Advanced Studies in Pure Mathematics (Vol. 4) (Tokyo: Kinokuniya) pp1−95
[18] Nijhoff F W, Quispel G R W, Capel H W 1983 Phys. Lett. A 97 125Google Scholar
[19] Nijhoff F W, Quispel G R W, Capel H W 1983 Phys. Lett. A 98 83Google Scholar
[20] Nijhoff F W, Capel H W, Wiersma G L, Quispel G R W 1984 Phys. Lett. A 105 267Google Scholar
[21] Nijhoff F W 1985 Lett. Math. Phys. 9 235Google Scholar
[22] Nijhoff F W, Capel H W, Wiersma G L 1985 In: Martini R (ed) Geometric Aspects of the Einstein Equations and Integrable Systems (Scheveningen 1984) Lecture Notes in Phys (Vol. 239) (Berlin: Springer) pp263−302
[23] Quispel G R W, Nijhoff F W, Capel H W, van ver Linden J 1984 Physica A 125 344Google Scholar
[24] Fokas A S, Ablowitz M 1981 Phys. Rev. Lett. 47 1096
[25] Levi D, Benguria R 1980 Proc. Natl. Acad. Sci. U.S.A. 77 5025Google Scholar
[26] Takahashi D, Satsuma J 1990 J. Phys. Soc. Jpn. 59 3514Google Scholar
[27] Tokihiro T, Takahashi D, Matsukidaira J, Satsuma J 1996 Phys. Rev. Lett. 76 3247Google Scholar
[28] Grammaticos B, Ramani A, Papageorgiou V G 1991 Phys. Rev. Lett. 67 1825Google Scholar
[29] Ramani A, Grammaticos B, Hietarinta J 1991 Phys. Rev. Lett. 67 1829Google Scholar
[30] Hietarinta J, Viallet C 1998 Phys. Rev. Lett. 81 325Google Scholar
[31] Bellon M P, Viallet C 1999 Commun. Math. Phys. 204 425Google Scholar
[32] Sakai H 2001 Commun. Math. Phys. 220 165Google Scholar
[33] Bobenko A I, Suris Yu B 2002 Int. Math. Res. Not. 2002 573Google Scholar
[34] Adler V E, Bobenko A I, Suris Yu B 2003 Commun. Math. Phys. 233 513Google Scholar
[35] Nijhoff F W, Walker A J 2001 Glasg. Math. J. 43A 109
[36] Nijhoff F W 2002 Phys. Lett. A 297 49Google Scholar
[37] Nijhoff F W, Atkinson J, Hietarinta J 2009 J. Phys. A: Math. Theor. 42 404005Google Scholar
[38] Hietarinta J, Zhang D J 2008 J. Phys. A: Math. Theor. 42 404006Google Scholar
[39] Atkinson J, Nijhoff F W 2010 Commun. Math. Phys. 299 283Google Scholar
[40] Nijhoff F W, Atkinson J 2010 Int. Math. Res. Not. 2010 3837Google Scholar
[41] Butler S, Joshi N 2010 Inverse Prob. 26 115012Google Scholar
[42] Butler S 2012 Nonlinearity 25 1613Google Scholar
[43] Cao C W, Xu X X 2012 J. Phys. A: Math. Theor. 45 055213Google Scholar
[44] Cao C W, Zhang G Y 2012 J. Phys. A: Math. Theor. 45 095203Google Scholar
[45] Zhang D J, Zhao S L 2013 Stud. Appl. Math. 131 72Google Scholar
[46] Bobenko A I, Its A 2016 Duke Math. J. 165 2607Google Scholar
[47] Hietarinta J, Joshi N, Nijhoff F W 2016 (Cambridge: Camb. Univ. Press)
[48] Zhang D J, Chen S T 2010 Stud. Appl. Math. 125 393Google Scholar
[49] Zhang D J, Chen S T 2010 Stud. Appl. Math. 125 419Google Scholar
[50] Fu W, Qiao Z J, Sun J W, Zhang D J 2013 arXiv: 1307.3671
[51] Fu W, Qiao Z J, Sun J W, Zhang D J 2015 J. Nonlinear. Math. Phys. 22 321Google Scholar
[52] Wahlquist H D, Estabrook F B 1973 Phys. Rev. Lett. 31 1386Google Scholar
[53] Lamb JR G L 1971 Rev. Mod. Phys. 43 99Google Scholar
[54] Chen H H 1974 Phys. Rev. Lett. 33 925Google Scholar
[55] Orfanidis S J 1978 Phys. Rev. D 18 3828Google Scholar
[56] Bianchi L 1892 Rend. Lincei 5 2
[57] Bianchi L 1894 Lezioni di Geometria Differenziale (3rd Ed.) (Pisa: Enrico Spoerri)
[58] Konopelchenko B G 1982 Phys. Lett. A 87 445Google Scholar
[59] Levi D 1981 J. Phys. A: Math. Gen. 14 1083Google Scholar
[60] Adler V E, Yamilov R I 1994 J. Phys. A: Math. Gen. 27 477Google Scholar
[61] Merola I, Ragnisco O, Tu G Z 1994 Inverse Prob. 10 1315Google Scholar
[62] Zhang H W, Tu G Z, Oevel W, Fuchssteiner B 1991 J. Math. Phys. 32 1908Google Scholar
[63] Chen K, Deng X, Zhang D J 2017 J. Nonlinear. Math. Phys. 24(Suppl.1) 18
[64] Cao C W, Zhang G Y 2012 Chin. Phys. Lett. 29 050202Google Scholar
[65] Walker A J 2001 Ph.D. Thesis (Leeds: University of Leeds)
[66] Adler V E 1998 Int. Math. Res. Not. 1998 1Google Scholar
[67] Hietarinta J 2005 J. Nonlinear. Math. Phys. 12 223
[68] Nijhoff F W, Papageorgiou V G, Capel H W, Quispel G R W 1992 Inverse Prob. 8 597Google Scholar
[69] Nijhoff F W 1997 In: Fokas A S, Gel’fand I M (eds) Algebraic Aspects of Integrable Systems: In memory of Irene Dorfman (Boston: Birkhauser) pp237−260
[70] Nijhoff F W 1999 In: Bobenko A I, Seiler R (eds) Discrete Integrable Geometry and Physics (Oxford: Clarendon Press) pp209−234
[71] Hietarinta J 2011 J. Phys. A: Math. Theor. 44 165204
[72] Hietarinta J, Zhang D J 2008 preprint
[73] Atkinson J 2008 J. Phys. A: Math. Theor. 41 135202Google Scholar
[74] Adler V E, Bobenko A I, Suris Yu B 2009 Funct. Anal. Appl. 43 3Google Scholar
[75] Boll R 2011 J. Nonlinear. Math. Phys. 18 337Google Scholar
[76] Boll R 2012 Ph.D Dissertation (Berlin: Technischen Universität Berlin)
[77] Adler V E, Bobenko A I, Suris Yu B 2012 Int. Math. Res. Not. 2012 1822Google Scholar
[78] Miwa T 1982 Proc. Jpn. Acad. 58A 9
[79] Konopelchenko B G, Schief W K 2002 Stud. Appl. Math. 109 89Google Scholar
[80] Hirota R 1981 J. Phys. Soc. Jpn. 50 3785Google Scholar
[81] Bianchi L 1885 Ann. Matem. 13 177Google Scholar
[82] Atkinson J, Nieszporski M 2014 Int. Math. Res. Not. 2014 4215Google Scholar
[83] Zhang D D, Zhang D J 2018 J. Nonlinear. Math. Phys. 25 34Google Scholar
[84] Zhang D J, Cheng J W, Sun Y Y 2013 J. Phys. A: Math. Theor. 46 265202Google Scholar
[85] Xenitidis P 2011 J. Phys. A: Math. Theor. 44 435201
[86] Bridgman T, Hereman W, Quispel G R W, van der Kamp P H 2013 Found. Comput. Math. 13 517Google Scholar
[87] Hietarinta J, Zhang D J 2010 J. Math. Phys. 51 033505
[88] Hietarinta J, Zhang D J 2011 SIGMA 7 061
[89] Atkinson J, Hietarinta J, Nijhoff F W 2007 J. Phys. A: Math. Theor. 40 F1Google Scholar
[90] Konopelchenko B G, Schief W K 2002 J. Phys. A: Math. Gen. 35 6125Google Scholar
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[1] Zabusky N J, Kruskal M D 1965 Phys. Rev. Lett. 15 240
[2] Gardner C S, Greene J M, Kruskal M D, Miura R M 1967 Phys. Rev. Lett. 19 1095Google Scholar
[3] Lax P D 1968 Commun. Pure Appl. Math. 21 467Google Scholar
[4] Deift P 2019 arXiv: 1902.10267
[5] Case K M, Kac M 1973 J. Math. Phys. 14 594Google Scholar
[6] Ablowitz M J, Ladik J F 1975 J. Math. Phys. 16 598Google Scholar
[7] Ablowitz M J, Ladik J F 1976 J. Math. Phys. 17 1011Google Scholar
[8] Ablowitz M J, Ladik J F 1976 Stud. Appl. Math. 55 213Google Scholar
[9] Hirota R 1977 J. Phys. Soc. Jpn. 43 1424Google Scholar
[10] Hirota R 1977 J. Phys. Soc. Jpn. 43 2074Google Scholar
[11] Hirota R 1977 J. Phys. Soc. Jpn. 43 2079Google Scholar
[12] Date E, Jimbo M, Miwa T 1982 J. Phys. Soc. Jpn. 51 4116Google Scholar
[13] Date E, Jimbo M, Miwa T 1982 J. Phys. Soc. Jpn. 51 4125Google Scholar
[14] Date E, Jimbo M, Miwa T 1983 J. Phys. Soc. Jpn. 52 388Google Scholar
[15] Date E, Jimbo M, Miwa T 1983 J. Phys. Soc. Jpn. 52 761Google Scholar
[16] Date E, Jimbo M, Miwa T 1983 J. Phys. Soc. Jpn. 52 766Google Scholar
[17] Ueno K, Takasaki K 1984 In: Okamoto K (ed) Group Representations and Systems of Differential Equations Advanced Studies in Pure Mathematics (Vol. 4) (Tokyo: Kinokuniya) pp1−95
[18] Nijhoff F W, Quispel G R W, Capel H W 1983 Phys. Lett. A 97 125Google Scholar
[19] Nijhoff F W, Quispel G R W, Capel H W 1983 Phys. Lett. A 98 83Google Scholar
[20] Nijhoff F W, Capel H W, Wiersma G L, Quispel G R W 1984 Phys. Lett. A 105 267Google Scholar
[21] Nijhoff F W 1985 Lett. Math. Phys. 9 235Google Scholar
[22] Nijhoff F W, Capel H W, Wiersma G L 1985 In: Martini R (ed) Geometric Aspects of the Einstein Equations and Integrable Systems (Scheveningen 1984) Lecture Notes in Phys (Vol. 239) (Berlin: Springer) pp263−302
[23] Quispel G R W, Nijhoff F W, Capel H W, van ver Linden J 1984 Physica A 125 344Google Scholar
[24] Fokas A S, Ablowitz M 1981 Phys. Rev. Lett. 47 1096
[25] Levi D, Benguria R 1980 Proc. Natl. Acad. Sci. U.S.A. 77 5025Google Scholar
[26] Takahashi D, Satsuma J 1990 J. Phys. Soc. Jpn. 59 3514Google Scholar
[27] Tokihiro T, Takahashi D, Matsukidaira J, Satsuma J 1996 Phys. Rev. Lett. 76 3247Google Scholar
[28] Grammaticos B, Ramani A, Papageorgiou V G 1991 Phys. Rev. Lett. 67 1825Google Scholar
[29] Ramani A, Grammaticos B, Hietarinta J 1991 Phys. Rev. Lett. 67 1829Google Scholar
[30] Hietarinta J, Viallet C 1998 Phys. Rev. Lett. 81 325Google Scholar
[31] Bellon M P, Viallet C 1999 Commun. Math. Phys. 204 425Google Scholar
[32] Sakai H 2001 Commun. Math. Phys. 220 165Google Scholar
[33] Bobenko A I, Suris Yu B 2002 Int. Math. Res. Not. 2002 573Google Scholar
[34] Adler V E, Bobenko A I, Suris Yu B 2003 Commun. Math. Phys. 233 513Google Scholar
[35] Nijhoff F W, Walker A J 2001 Glasg. Math. J. 43A 109
[36] Nijhoff F W 2002 Phys. Lett. A 297 49Google Scholar
[37] Nijhoff F W, Atkinson J, Hietarinta J 2009 J. Phys. A: Math. Theor. 42 404005Google Scholar
[38] Hietarinta J, Zhang D J 2008 J. Phys. A: Math. Theor. 42 404006Google Scholar
[39] Atkinson J, Nijhoff F W 2010 Commun. Math. Phys. 299 283Google Scholar
[40] Nijhoff F W, Atkinson J 2010 Int. Math. Res. Not. 2010 3837Google Scholar
[41] Butler S, Joshi N 2010 Inverse Prob. 26 115012Google Scholar
[42] Butler S 2012 Nonlinearity 25 1613Google Scholar
[43] Cao C W, Xu X X 2012 J. Phys. A: Math. Theor. 45 055213Google Scholar
[44] Cao C W, Zhang G Y 2012 J. Phys. A: Math. Theor. 45 095203Google Scholar
[45] Zhang D J, Zhao S L 2013 Stud. Appl. Math. 131 72Google Scholar
[46] Bobenko A I, Its A 2016 Duke Math. J. 165 2607Google Scholar
[47] Hietarinta J, Joshi N, Nijhoff F W 2016 (Cambridge: Camb. Univ. Press)
[48] Zhang D J, Chen S T 2010 Stud. Appl. Math. 125 393Google Scholar
[49] Zhang D J, Chen S T 2010 Stud. Appl. Math. 125 419Google Scholar
[50] Fu W, Qiao Z J, Sun J W, Zhang D J 2013 arXiv: 1307.3671
[51] Fu W, Qiao Z J, Sun J W, Zhang D J 2015 J. Nonlinear. Math. Phys. 22 321Google Scholar
[52] Wahlquist H D, Estabrook F B 1973 Phys. Rev. Lett. 31 1386Google Scholar
[53] Lamb JR G L 1971 Rev. Mod. Phys. 43 99Google Scholar
[54] Chen H H 1974 Phys. Rev. Lett. 33 925Google Scholar
[55] Orfanidis S J 1978 Phys. Rev. D 18 3828Google Scholar
[56] Bianchi L 1892 Rend. Lincei 5 2
[57] Bianchi L 1894 Lezioni di Geometria Differenziale (3rd Ed.) (Pisa: Enrico Spoerri)
[58] Konopelchenko B G 1982 Phys. Lett. A 87 445Google Scholar
[59] Levi D 1981 J. Phys. A: Math. Gen. 14 1083Google Scholar
[60] Adler V E, Yamilov R I 1994 J. Phys. A: Math. Gen. 27 477Google Scholar
[61] Merola I, Ragnisco O, Tu G Z 1994 Inverse Prob. 10 1315Google Scholar
[62] Zhang H W, Tu G Z, Oevel W, Fuchssteiner B 1991 J. Math. Phys. 32 1908Google Scholar
[63] Chen K, Deng X, Zhang D J 2017 J. Nonlinear. Math. Phys. 24(Suppl.1) 18
[64] Cao C W, Zhang G Y 2012 Chin. Phys. Lett. 29 050202Google Scholar
[65] Walker A J 2001 Ph.D. Thesis (Leeds: University of Leeds)
[66] Adler V E 1998 Int. Math. Res. Not. 1998 1Google Scholar
[67] Hietarinta J 2005 J. Nonlinear. Math. Phys. 12 223
[68] Nijhoff F W, Papageorgiou V G, Capel H W, Quispel G R W 1992 Inverse Prob. 8 597Google Scholar
[69] Nijhoff F W 1997 In: Fokas A S, Gel’fand I M (eds) Algebraic Aspects of Integrable Systems: In memory of Irene Dorfman (Boston: Birkhauser) pp237−260
[70] Nijhoff F W 1999 In: Bobenko A I, Seiler R (eds) Discrete Integrable Geometry and Physics (Oxford: Clarendon Press) pp209−234
[71] Hietarinta J 2011 J. Phys. A: Math. Theor. 44 165204
[72] Hietarinta J, Zhang D J 2008 preprint
[73] Atkinson J 2008 J. Phys. A: Math. Theor. 41 135202Google Scholar
[74] Adler V E, Bobenko A I, Suris Yu B 2009 Funct. Anal. Appl. 43 3Google Scholar
[75] Boll R 2011 J. Nonlinear. Math. Phys. 18 337Google Scholar
[76] Boll R 2012 Ph.D Dissertation (Berlin: Technischen Universität Berlin)
[77] Adler V E, Bobenko A I, Suris Yu B 2012 Int. Math. Res. Not. 2012 1822Google Scholar
[78] Miwa T 1982 Proc. Jpn. Acad. 58A 9
[79] Konopelchenko B G, Schief W K 2002 Stud. Appl. Math. 109 89Google Scholar
[80] Hirota R 1981 J. Phys. Soc. Jpn. 50 3785Google Scholar
[81] Bianchi L 1885 Ann. Matem. 13 177Google Scholar
[82] Atkinson J, Nieszporski M 2014 Int. Math. Res. Not. 2014 4215Google Scholar
[83] Zhang D D, Zhang D J 2018 J. Nonlinear. Math. Phys. 25 34Google Scholar
[84] Zhang D J, Cheng J W, Sun Y Y 2013 J. Phys. A: Math. Theor. 46 265202Google Scholar
[85] Xenitidis P 2011 J. Phys. A: Math. Theor. 44 435201
[86] Bridgman T, Hereman W, Quispel G R W, van der Kamp P H 2013 Found. Comput. Math. 13 517Google Scholar
[87] Hietarinta J, Zhang D J 2010 J. Math. Phys. 51 033505
[88] Hietarinta J, Zhang D J 2011 SIGMA 7 061
[89] Atkinson J, Hietarinta J, Nijhoff F W 2007 J. Phys. A: Math. Theor. 40 F1Google Scholar
[90] Konopelchenko B G, Schief W K 2002 J. Phys. A: Math. Gen. 35 6125Google Scholar
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