General information
Latest issue
Impact factor
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Funktsional. Anal. i Prilozhen.:

Personal entry:
Save password
Forgotten password?

Funktsional. Anal. i Prilozhen., 2009, Volume 43, Issue 1, Pages 3–21 (Mi faa2936)  

This article is cited in 81 scientific papers (total in 81 papers)

Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases

V. E. Adlera, A. I. Bobenkob, Yu. B. Surisc

a L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
b Institut für Mathematik, Technische Universität Berlin
c Zentrum Mathematik, Technische Universität München

Abstract: We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on $\mathbb{Z}^2$. The fields are associated with the vertices and an equation of the form $Q(x_1,x_2,x_3,x_4)=0$ relates four vertices of one cell. The integrability of equations is understood as 3D-consistency, which means that it is possible to impose equations of the same type on all faces of a three-dimensional cube so that the resulting system will be consistent. This allows one to extend these equations also to the multidimensional lattices $\mathbb{Z}^N$. We classify integrable equations with complex fields $x$ and polynomials $Q$ multiaffine in all variables. Our method is based on the analysis of singular solutions.

Keywords: integrability, quad-graph, multidimensional consistency, zero curvature representation, Bäcklund transformation, Bianchi permutability, Möbius transformation


Full text: PDF file (282 kB)
References: PDF file   HTML file

English version:
Functional Analysis and Its Applications, 2009, 43:1, 3–17

Bibliographic databases:

UDC: 517.962.24+517.965+517.957+517.958
Received: 04.06.2007

Citation: V. E. Adler, A. I. Bobenko, Yu. B. Suris, “Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases”, Funktsional. Anal. i Prilozhen., 43:1 (2009), 3–21; Funct. Anal. Appl., 43:1 (2009), 3–17

Citation in format AMSBIB
\by V.~E.~Adler, A.~I.~Bobenko, Yu.~B.~Suris
\paper Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases
\jour Funktsional. Anal. i Prilozhen.
\yr 2009
\vol 43
\issue 1
\pages 3--21
\jour Funct. Anal. Appl.
\yr 2009
\vol 43
\issue 1
\pages 3--17

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Grammaticos B., Halburd R.G., Ramani A., Viallet C.-M., “How to detect the integrability of discrete systems”, J. Phys. A, 42:45 (2009), 454002, 30 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Atkinson J., “Linear quadrilateral lattice equations and multidimensional consistency”, J. Phys. A, 42:45 (2009), 454005, 7 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Lobb S., Nijhoff F., “Lagrangian multiforms and multidimensional consistency”, J. Phys. A, 42:45 (2009), 454013, 18 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. Tsarev S.P., Wolf Th., “Hyperdeterminants as integrable discrete systems”, J. Phys. A, 42:45 (2009), 454023, 9 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Xenitidis P.D., Papageorgiou V.G., “Symmetries and integrability of discrete equations defined on a black-white lattice”, J. Phys. A, 42:45 (2009), 454025, 13 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Nijhoff F., Atkinson J., Hietarinta J., “Soliton solutions for ABS lattice equations: I. Cauchy matrix approach”, J. Phys. A, 42:40 (2009), 404005, 34 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    7. Ramani A., Grammaticos B., Satsuma J., Willox R., “On two (not so) new integrable partial difference equations”, J. Phys. A, 42:28 (2009), 282002, 6 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    8. Tsoubelis D., Xenitidis P., “Continuous symmetric reductions of the Adler-Bobenko-Suris equations”, J. Phys. A, 42:16 (2009), 165203, 29 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. Bobenko A.I., Suris Yu.B., “On the Lagrangian structure of integrable quad-equations”, Lett. Math. Phys., 92:1 (2010), 17–31  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    10. van der Kamp P.H., Quispel G.R.W., “The staircase method: integrals for periodic reductions of integrable lattice equations”, J. Phys. A, 43:46 (2010), 465207, 34 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    11. Nijhoff F., Atkinson J., “Elliptic $N$-soliton solutions of ABS lattice equations”, Int. Math. Res. Not., 2010, no. 20, 3837–3895  mathscinet  zmath  isi  elib
    12. Tsuchida T., “A systematic method for constructing time discretizations of integrable lattice systems: local equations of motion”, J. Phys. A, 43:41 (2010), 415202, 22 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    13. Atkinson J., Nijhoff F., “A Constructive Approach to the Soliton Solutions of Integrable Quadrilateral Lattice Equations”, Comm. Math. Phys., 299:2 (2010), 283–304  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    14. Hydon P.E., Viallet C.-M., “Asymmetric integrable quad-graph equations”, Appl. Anal., 89:4 (2010), 493–506  crossref  mathscinet  zmath  isi  elib  scopus
    15. Boll R., Suris Yu.B., “Non-symmetric discrete Toda systems from quad-graphs”, Appl. Anal., 89:4 (2010), 547–569  crossref  mathscinet  zmath  isi  elib  scopus
    16. Levi D., Yamilov R.I., “Integrability test for discrete equations via generalized symmetries”, Symmetries in nature, AIP Conf. Proc., 1323, 2010, 203–214  crossref  isi  scopus
    17. Zhang Da-jun, Hietarinta J., “Generalized solutions for the H1 model in ABS List of lattice equations”, Nonlinear and modern mathematical physics, AIP Conf. Proc., 1212, Amer. Inst. Phys., Melville, NY, 2010, 154–161  crossref  mathscinet  zmath  isi  scopus
    18. A. V. Mikhailov, J. P. Wang, P. Xenitidis, “Recursion operators, conservation laws, and integrability conditions for difference equations”, Theoret. and Math. Phys., 167:1 (2011), 421–443  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    19. Levi D., Yamilov R.I., “Generalized symmetry integrability test for discrete equations on the square lattice”, J. Phys. A, 44:14 (2011), 145207, 22 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    20. James Atkinson, “Singularities of Type-Q ABS Equations”, SIGMA, 7 (2011), 073, 14 pp.  mathnet  crossref  mathscinet
    21. Oleg I. Mokhov, “On Initial Data in the Problem of Consistency on Cubic Lattices for $3\times3$ Determinants”, SIGMA, 7 (2011), 075, 19 pp.  mathnet  crossref  mathscinet
    22. Mikhailov A.V., Wang J.P., “A new recursion operator for Adler's equation in the Viallet form”, Phys. Lett. A, 375:45 (2011), 3960–3963  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    23. Xenitidis P., “Symmetries and conservation laws of the ABS equations and corresponding differential-difference equations of Volterra type”, J. Phys. A, 44:43 (2011), 435201, 22 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    24. Boll R., “Classification of 3D consistent quad-equations”, J. Nonlinear Math. Phys., 18:3 (2011), 337–365  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    25. Xenitidis P., Nijhoff F., Lobb S., “On the Lagrangian formulation of multidimensionally consistent systems”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467:2135 (2011), 3295–3317  crossref  mathscinet  zmath  adsnasa  isi  scopus
    26. Hietarinta J., Viallet C., “Integrable lattice equations with vertex and bond variables”, J. Phys. A, 44:38 (2011), 385201, 13 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    27. Mikhailov A.V., Wang J.P., Xenitidis P., “Cosymmetries and Nijenhuis recursion operators for difference equations”, Nonlinearity, 24:7 (2011), 2079–2097  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    28. J. Atkinson, S. B. Lobb, F. W. Nijhoff, “An integrable multicomponent quad-equation and its Lagrangian formulation”, Theoret. and Math. Phys., 173:3 (2012), 1644–1653  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    29. Adler V.E., Shabat A.B., “a theory of integrable hyperbolic equations of third order”, J. Phys. A, 45:39 (2012), 395207, 17 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    30. Dragović V., Radnović M., “Billiard algebra, integrable line congruences, and double reflection nets”, J. Nonlinear Math. Phys., 19:3 (2012), 1250019, 18 pp.  crossref  mathscinet  zmath  isi  scopus
    31. Hietarinta J., Viallet C., “Weak Lax pairs for lattice equations”, Nonlinearity, 25:7 (2012), 1955–1966  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    32. Atkinson J., Joshi N., “Schwarzian variable associated with discrete KdV-type equations”, Nonlinearity, 25:6 (2012), 1851–1866  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    33. Boll R., Suris Yu.B., “On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations”, J. Phys. A, 45:11 (2012), 115201, 18 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    34. Xenitidis P., “Reductions on the lattice and Painlevé equations $P_2$, $P_5$, $P_6$”, Painlevé equations and related topics (2012), Degruyter Proceedings in Mathematics, eds. Bruno A., Batkhin A., Walter de Gruyter & Co, 2012, 267–270  crossref  mathscinet  isi
    35. Dragović V., “-geometric approach to the Yang–Baxter equation and related topics”, Publ. Inst. Math. (Beograd) (N.S.), 91:105 (2012), 25–48  crossref  mathscinet  zmath  isi  scopus
    36. Adler V.E., Bobenko A.I., Suris Yu.B., “Classification of integrable discrete equations of octahedron type”, Int. Math. Res. Not. IMRN, 2012, no. 8, 1822–1889  crossref  mathscinet  zmath  isi  elib  scopus
    37. Feng W., Zhao S., “Generalized Cauchy matrix approach for lattice KP-type equations”, Commun. Nonlinear Sci. Numer. Simul., 18:7 (2013), 1652–1664  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    38. Ormerod C.M., van der Kamp P.H., Quispel G.R.W., “Discrete Painlevé equations and their Lax pairs as reductions of integrable lattice equations”, J. Phys. A, 46:9 (2013), 095204  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    39. F. Khanizadeh, A. V. Mikhailov, Jing Ping Wang, “Darboux transformations and recursion operators for differential–difference equations”, Theoret. and Math. Phys., 177:3 (2013), 1606–1654  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    40. Boll R., “On Bianchi Permutability of Backlund Transformations for Asymmetric Quad-Equations”, J. Nonlinear Math. Phys., 20:4 (2013), 577–605  crossref  mathscinet  isi  scopus
    41. Ormerod Ch.M., “Tropical Geometric Interpretation of Ultradiscrete Singularity Confinement”, J. Phys. A-Math. Theor., 46:30 (2013), 305204  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    42. Lou Sen-Yue, Li Yu-Qi, Tang Xiao-Yan, “From Nothing to Something: Discrete Integrable Systems”, Chin. Phys. Lett., 30:8 (2013), 080202  crossref  adsnasa  isi  elib  scopus
    43. Grammaticos B., Ramani A., Viallet C.-M., “From Yang–Baxter Maps to Integrable Recurrences”, J. Nonlinear Math. Phys., 20:2 (2013), 260–270  crossref  mathscinet  isi  elib  scopus
    44. Atkinson J., Joshi N., “Singular-Boundary Reductions of Type-Q Abs Equations”, Int. Math. Res. Notices, 2013, no. 7, 1451–1481  crossref  mathscinet  zmath  isi  elib  scopus
    45. Sergey Ya. Startsev, “Non-Point Invertible Transformations and Integrability of Partial Difference Equations”, SIGMA, 10 (2014), 066, 13 pp.  mathnet  crossref  mathscinet
    46. N. Joshi, N. Nakazono, Y. Shi, “Geometric reductions of ABS equations on an $n$-cube to discrete Painlevé systems”, J. Phys. A, 47:50 (2014), 505201  crossref  mathscinet  zmath  isi  scopus
    47. V. Dragović, K. Kukić, “Discriminantly separable polynomials and quad-equations”, J. Geom. Mech., 6:3 (2014), 319–333  crossref  mathscinet  zmath  isi  scopus
    48. Ch.M. Ormerod, P.H. van der Kamp, J. Hietarinta, G.R.W. Quispel, “Twisted reductions of integrable lattice equations, and their Lax representations”, Nonlinearity, 27:6 (2014), 1367–1390  crossref  mathscinet  zmath  adsnasa  isi  scopus
    49. S.Ya. Startsev, “Darboux integrable discrete equations possessing an autonomous first-order integral”, J. Phys. A, 47:10 (2014), 105204, 16 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    50. J. Atkinson, M. Nieszporski, “Multi-quadratic quad equations: integrable cases from a factorized-discriminant hypothesis”, Int. Math. Res. Notices, 2014, no. 15, 4215–4240  crossref  mathscinet  zmath  isi  scopus
    51. D.T. Tran, “A Lax pair of a lattice equation whose entropy vanishes”, J. Phys. A, 48:11 (2015), 115202, 11 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    52. N. Delice, F.W. Nijhoff, S. Yoo-Kong, “On elliptic Lax systems on the lattice and a compound theorem for hyperdeterminants”, J. Phys. A, 48:3 (2015), 035206, 27 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    53. J. Atkinson, “On the lattice-geometry and birational group of the six-point multi-ratio equation”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 470:2173 (2015), 20140612  crossref  mathscinet  isi  scopus
    54. Joshi N., Nakazono N., Shi Ya., “Lattice Equations Arising From Discrete Painlevé Systems. i. (a(2) + a(1))((1)) and (a(1) + a'(1))((1)) Cases”, J. Math. Phys., 56:9 (2015), 092705  crossref  mathscinet  zmath  isi  elib  scopus
    55. Viallet C.-M., “on the Algebraic Structure of Rational Discrete Dynamical Systems”, J. Phys. A-Math. Theor., 48:16 (2015), 16FT01  crossref  mathscinet  zmath  isi  elib  scopus
    56. van der Kamp P.H., “Initial Value Problems For Quad Equations”, J. Phys. A-Math. Theor., 48:6 (2015), 065204  crossref  mathscinet  zmath  isi  scopus
    57. S. V. Dmitriev, E. A. Korznikova, Yu. A. Baimova, M. G. Velarde, “Discrete breathers in crystals”, Phys. Usp., 59:5 (2016), 446–461  mathnet  crossref  crossref  adsnasa  isi  elib
    58. Nobutaka Nakazono, “Hypergeometric $\tau$ Functions of the $q$-Painlevé Systems of Types $A_4^{(1)}$ and $(A_1+A_1')^{(1)}$”, SIGMA, 12 (2016), 051, 23 pp.  mathnet  crossref
    59. G. Gubbiotti, D. Levi, Ch. Scimiterna, “Linearizability and a fake Lax pair for a nonlinear nonautonomous quad-graph equation consistent around the cube”, Theoret. and Math. Phys., 189:1 (2016), 1459–1471  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    60. Christopher M. Ormerod, Eric M. Rains, “Commutation Relations and Discrete Garnier Systems”, SIGMA, 12 (2016), 110, 50 pp.  mathnet  crossref
    61. Joshi N., Nakazono N., Shi Ya., “Lattice equations arising from discrete Painlevé systems: II. ${A}_{4}^{(1)}$ case”, J. Phys. A-Math. Theor., 49:49 (2016), 495201  crossref  mathscinet  zmath  isi  scopus
    62. Joshi N., Nakazono N., “Lax pairs of discrete Painlevé equations: ( A _{2} + A _{1} ) ^{(1)} case”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 472:2196 (2016), 20160696  crossref  mathscinet  zmath  isi  scopus
    63. Garifullin R.N., Yamilov R.I., Levi D., “Non-invertible transformations of differential–difference equations”, J. Phys. A-Math. Theor., 49:37 (2016), 37LT01  crossref  mathscinet  zmath  isi  elib  scopus
    64. Atkinson J., Howes Ph., Joshi N., Nakazono N., “Geometry of an elliptic difference equation related to Q4”, J. Lond. Math. Soc.-Second Ser., 93:3 (2016), 763–784  crossref  mathscinet  zmath  isi  scopus
    65. Gubbiotti G., Scimiterna C., Levi D., “Algebraic entropy, symmetries and linearization of quad equations consistent on the cube”, J. Nonlinear Math. Phys., 23:4 (2016), 507–543  crossref  mathscinet  isi  elib  scopus
    66. Hietarinta J., Joshi N., Nijhoff F., “Discrete Systems and Integrability”, Discrete Systems and Integrability, Cambridge Texts in Applied Mathematics, Cambridge Univ Press, 2016, 1–445  mathscinet  zmath  isi
    67. Garifullin R.N. Yamilov R.I. Levi D., “Classification of five-point differential-difference equations”, J. Phys. A-Math. Theor., 50:12 (2017), 125201  crossref  mathscinet  zmath  isi  scopus
    68. Danda Zhang, Zhang Da-Jun, “Rational Solutions to the ABS List: Transformation Approach”, SIGMA, 13 (2017), 078, 24 pp.  mathnet  crossref
    69. Joshi N., Kajiwara K., Masuda T., Nakazono N., Shi Ya., “Geometric Description of a Discrete Power Function Associated With the Sixth Painlevé Equation”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 473:2207 (2017), 20170312  crossref  mathscinet  isi  scopus
    70. Petrera M., Suris Yu.B., “On the Classification of Multidimensionally Consistent 3D Maps”, Lett. Math. Phys., 107:11 (2017), 2013–2027  crossref  mathscinet  zmath  isi  scopus
    71. Gubbiotti G., Scimiterna C., Levi D., “The Non-Autonomous Ydkn Equation and Generalized Symmetries of Boll Equations”, J. Math. Phys., 58:5 (2017), 053507  crossref  mathscinet  zmath  isi  scopus
    72. Gubbiotti G., Yamilov R.I., “Darboux Integrability of Trapezoidal H-4 and H-4 Families of Lattice Equations i: First Integrals”, J. Phys. A-Math. Theor., 50:34 (2017), 345205, 1–26  crossref  mathscinet  isi  scopus
    73. Giorgio Gubbiotti, Christian Scimiterna, “Reconstructing a Lattice Equation: a Non-Autonomous Approach to the Hietarinta Equation”, SIGMA, 14 (2018), 004, 21 pp.  mathnet  crossref
    74. Giorgio Gubbiotti, Christian Scimiterna, Ravil I. Yamilov, “Darboux Integrability of Trapezoidal $H^{4}$ and $H^{6}$ Families of Lattice Equations II: General Solutions”, SIGMA, 14 (2018), 008, 51 pp.  mathnet  crossref
    75. Garifullin R.N. Yamilov R.I. Levi D., “Classification of Five-Point Differential-Difference Equations II”, J. Phys. A-Math. Theor., 51:6 (2018), 065204  crossref  mathscinet  zmath  isi  scopus
    76. Zhang D., Zhang D.-j., “On Decomposition of the Abs Lattice Equations and Related Backlund Transformations”, J. Nonlinear Math. Phys., 25:1 (2018), 34–53  crossref  mathscinet  isi
    77. Kassotakis P., Nieszporski M., “Difference Systems in Bond and Face Variables and Non-Potential Versions of Discrete Integrable Systems”, J. Phys. A-Math. Theor., 51:38 (2018), 385203  crossref  isi  scopus
    78. V. E. Adler, “Integrable seven-point discrete equations and second-order evolution chains”, Theoret. and Math. Phys., 195:1 (2018), 513–528  mathnet  crossref  crossref  adsnasa  isi  elib
    79. Roberts J.A.G., Tran D.T., “Algebraic Entropy of (Integrable) Lattice Equations and Their Reductions”, Nonlinearity, 32:2 (2019)  crossref  mathscinet  zmath  isi
    80. Cheng Zhang, Linyu Peng, Da-jun Zhang, “Discrete Crum's theorems and lattice KdV-type equations”, Theoret. and Math. Phys., 202:2 (2020), 165–182  mathnet  crossref  crossref  isi  elib
    81. Dan-Da Zhang, Peter H. van der Kamp, Da-Jun Zhang, “Multi-Component Extension of CAC Systems”, SIGMA, 16 (2020), 060, 30 pp.  mathnet  crossref
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
    Number of views:
    This page:809
    Full text:248
    First page:23

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2021