Discrete integrable systems: Multidimensional consistency

Zhang Da-Jun

doi:10.7498/aps.69.20191647

Abstract

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.

Keywords:

discrete integrable systems /
multidimensional consistency

Authors and contacts

Zhang Da-Jun

Department of Mathematics, Shanghai University, Shanghai 200444, China

Corresponding author: Zhang Da-Jun, djzhang@staff.shu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11875040, 11631007)

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References

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[71]	Hietarinta J 2011 J. Phys. A: Math. Theor. 44 165204
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[84]	Zhang D J, Cheng J W, Sun Y Y 2013 J. Phys. A: Math. Theor. 46 265202 Google Scholar
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[89]	Atkinson J, Hietarinta J, Nijhoff F W 2007 J. Phys. A: Math. Theor. 40 F1 Google Scholar
[90]	Konopelchenko B G, Schief W K 2002 J. Phys. A: Math. Gen. 35 6125 Google Scholar

Cited By

图 1 $[x_0, x]$上的数值离散

Figure 1. Numerical discretisation on $[x_0, x]$

DownLoad: Full-Size Img PowerPoint

图 2 Bäcklund变换解的交换性质

Figure 2. Permutation property of Bäcklund transformation

DownLoad: Full-Size Img PowerPoint

图 3 变换与平面网格

Figure 3. Map and lattice

DownLoad: Full-Size Img PowerPoint

图 4 相容立方体

Figure 4. Consistent cube

DownLoad: Full-Size Img PowerPoint

图 5 离散sine-Gordon和势mKdV方程的相容立方体

Figure 5. The consistent cube for the discrete sine-Gordon equation and potential mKdV equation

DownLoad: Full-Size Img PowerPoint

图 6 定义3维方程的6面体以及8面体

Figure 6. Cube and octahedron for 3D equations

DownLoad: Full-Size Img PowerPoint

图 7 围绕超立方体的4D相容性

Figure 7. 4D consistency around the hyper cube

DownLoad: Full-Size Img PowerPoint

图 8 相容立方体

Figure 8. The consistent cube

DownLoad: Full-Size Img PowerPoint

map

[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 1095 Google Scholar
[3]	Lax P D 1968 Commun. Pure Appl. Math. 21 467 Google Scholar
[4]	Deift P 2019 arXiv: 1902.10267
[5]	Case K M, Kac M 1973 J. Math. Phys. 14 594 Google Scholar
[6]	Ablowitz M J, Ladik J F 1975 J. Math. Phys. 16 598 Google Scholar
[7]	Ablowitz M J, Ladik J F 1976 J. Math. Phys. 17 1011 Google Scholar
[8]	Ablowitz M J, Ladik J F 1976 Stud. Appl. Math. 55 213 Google Scholar
[9]	Hirota R 1977 J. Phys. Soc. Jpn. 43 1424 Google Scholar
[10]	Hirota R 1977 J. Phys. Soc. Jpn. 43 2074 Google Scholar
[11]	Hirota R 1977 J. Phys. Soc. Jpn. 43 2079 Google Scholar
[12]	Date E, Jimbo M, Miwa T 1982 J. Phys. Soc. Jpn. 51 4116 Google Scholar
[13]	Date E, Jimbo M, Miwa T 1982 J. Phys. Soc. Jpn. 51 4125 Google Scholar
[14]	Date E, Jimbo M, Miwa T 1983 J. Phys. Soc. Jpn. 52 388 Google Scholar
[15]	Date E, Jimbo M, Miwa T 1983 J. Phys. Soc. Jpn. 52 761 Google Scholar
[16]	Date E, Jimbo M, Miwa T 1983 J. Phys. Soc. Jpn. 52 766 Google 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 97 125 Google Scholar
[19]	Nijhoff F W, Quispel G R W, Capel H W 1983 Phys. Lett. A 98 83 Google Scholar
[20]	Nijhoff F W, Capel H W, Wiersma G L, Quispel G R W 1984 Phys. Lett. A 105 267 Google Scholar
[21]	Nijhoff F W 1985 Lett. Math. Phys. 9 235 Google 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 125 344 Google 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 5025 Google Scholar
[26]	Takahashi D, Satsuma J 1990 J. Phys. Soc. Jpn. 59 3514 Google Scholar
[27]	Tokihiro T, Takahashi D, Matsukidaira J, Satsuma J 1996 Phys. Rev. Lett. 76 3247 Google Scholar
[28]	Grammaticos B, Ramani A, Papageorgiou V G 1991 Phys. Rev. Lett. 67 1825 Google Scholar
[29]	Ramani A, Grammaticos B, Hietarinta J 1991 Phys. Rev. Lett. 67 1829 Google Scholar
[30]	Hietarinta J, Viallet C 1998 Phys. Rev. Lett. 81 325 Google Scholar
[31]	Bellon M P, Viallet C 1999 Commun. Math. Phys. 204 425 Google Scholar
[32]	Sakai H 2001 Commun. Math. Phys. 220 165 Google Scholar
[33]	Bobenko A I, Suris Yu B 2002 Int. Math. Res. Not. 2002 573 Google Scholar
[34]	Adler V E, Bobenko A I, Suris Yu B 2003 Commun. Math. Phys. 233 513 Google Scholar
[35]	Nijhoff F W, Walker A J 2001 Glasg. Math. J. 43A 109
[36]	Nijhoff F W 2002 Phys. Lett. A 297 49 Google Scholar
[37]	Nijhoff F W, Atkinson J, Hietarinta J 2009 J. Phys. A: Math. Theor. 42 404005 Google Scholar
[38]	Hietarinta J, Zhang D J 2008 J. Phys. A: Math. Theor. 42 404006 Google Scholar
[39]	Atkinson J, Nijhoff F W 2010 Commun. Math. Phys. 299 283 Google Scholar
[40]	Nijhoff F W, Atkinson J 2010 Int. Math. Res. Not. 2010 3837 Google Scholar
[41]	Butler S, Joshi N 2010 Inverse Prob. 26 115012 Google Scholar
[42]	Butler S 2012 Nonlinearity 25 1613 Google Scholar
[43]	Cao C W, Xu X X 2012 J. Phys. A: Math. Theor. 45 055213 Google Scholar
[44]	Cao C W, Zhang G Y 2012 J. Phys. A: Math. Theor. 45 095203 Google Scholar
[45]	Zhang D J, Zhao S L 2013 Stud. Appl. Math. 131 72 Google Scholar
[46]	Bobenko A I, Its A 2016 Duke Math. J. 165 2607 Google 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. 125 393 Google Scholar
[49]	Zhang D J, Chen S T 2010 Stud. Appl. Math. 125 419 Google 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. 22 321 Google Scholar
[52]	Wahlquist H D, Estabrook F B 1973 Phys. Rev. Lett. 31 1386 Google Scholar
[53]	Lamb JR G L 1971 Rev. Mod. Phys. 43 99 Google Scholar
[54]	Chen H H 1974 Phys. Rev. Lett. 33 925 Google Scholar
[55]	Orfanidis S J 1978 Phys. Rev. D 18 3828 Google 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 87 445 Google Scholar
[59]	Levi D 1981 J. Phys. A: Math. Gen. 14 1083 Google Scholar
[60]	Adler V E, Yamilov R I 1994 J. Phys. A: Math. Gen. 27 477 Google Scholar
[61]	Merola I, Ragnisco O, Tu G Z 1994 Inverse Prob. 10 1315 Google Scholar
[62]	Zhang H W, Tu G Z, Oevel W, Fuchssteiner B 1991 J. Math. Phys. 32 1908 Google 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 050202 Google Scholar
[65]	Walker A J 2001 Ph.D. Thesis (Leeds: University of Leeds)
[66]	Adler V E 1998 Int. Math. Res. Not. 1998 1 Google 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. 8 597 Google 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 135202 Google Scholar
[74]	Adler V E, Bobenko A I, Suris Yu B 2009 Funct. Anal. Appl. 43 3 Google Scholar
[75]	Boll R 2011 J. Nonlinear. Math. Phys. 18 337 Google 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 1822 Google Scholar
[78]	Miwa T 1982 Proc. Jpn. Acad. 58A 9
[79]	Konopelchenko B G, Schief W K 2002 Stud. Appl. Math. 109 89 Google Scholar
[80]	Hirota R 1981 J. Phys. Soc. Jpn. 50 3785 Google Scholar
[81]	Bianchi L 1885 Ann. Matem. 13 177 Google Scholar
[82]	Atkinson J, Nieszporski M 2014 Int. Math. Res. Not. 2014 4215 Google Scholar
[83]	Zhang D D, Zhang D J 2018 J. Nonlinear. Math. Phys. 25 34 Google Scholar
[84]	Zhang D J, Cheng J W, Sun Y Y 2013 J. Phys. A: Math. Theor. 46 265202 Google 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. 13 517 Google 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 F1 Google Scholar
[90]	Konopelchenko B G, Schief W K 2002 J. Phys. A: Math. Gen. 35 6125 Google Scholar

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