RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 TMF: Year: Volume: Issue: Page: Find

 TMF, 2000, Volume 125, Number 3, Pages 355–424 (Mi tmf675)

Symmetry approach to the integrability problem

V. E. Adlera, A. B. Shabatb, R. I. Yamilova

a Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences
b L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: We review the results of the twenty-year development of the symmetry approach to classifying integrable models in mathematical physics. The generalized Toda chains and the related equations of the nonlinear Schrödinger type, discrete transformations, and hyperbolic systems are central in this approach. Moreover, we consider equations of the Painlevé type, master symmetries, and the problem of integrability criteria for $(2+1)$-dimensional models. We present the list of canonical forms for $(1+1)$-dimensional integrable systems. We elaborate the effective tests for integrability and the algorithms for reduction to the canonical form.

DOI: https://doi.org/10.4213/tmf675

Full text: PDF file (612 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 2000, 125:3, 1603–1661

Bibliographic databases:

Citation: V. E. Adler, A. B. Shabat, R. I. Yamilov, “Symmetry approach to the integrability problem”, TMF, 125:3 (2000), 355–424; Theoret. and Math. Phys., 125:3 (2000), 1603–1661

Citation in format AMSBIB
\Bibitem{AdlShaYam00} \by V.~E.~Adler, A.~B.~Shabat, R.~I.~Yamilov \paper Symmetry approach to the integrability problem \jour TMF \yr 2000 \vol 125 \issue 3 \pages 355--424 \mathnet{http://mi.mathnet.ru/tmf675} \crossref{https://doi.org/10.4213/tmf675} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1839659} \zmath{https://zbmath.org/?q=an:1029.37041} \elib{http://elibrary.ru/item.asp?id=13358446} \transl \jour Theoret. and Math. Phys. \yr 2000 \vol 125 \issue 3 \pages 1603--1661 \crossref{https://doi.org/10.1023/A:1026602012111} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000167036300001} 

• http://mi.mathnet.ru/eng/tmf675
• https://doi.org/10.4213/tmf675
• http://mi.mathnet.ru/eng/tmf/v125/i3/p355

 SHARE:

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. V. G. Marikhin, “Coulomb Gas Representation for Rational Solutions of the Painlevé Equations”, Theoret. and Math. Phys., 127:2 (2001), 646–663
2. Svinin, AK, “A class of integrable lattices and KP hierarchy”, Journal of Physics A-Mathematical and General, 34:48 (2001), 10559
3. Levi, D, “On the integrability of a new discrete nonlinear Schrodinger equation”, Journal of Physics A-Mathematical and General, 34:41 (2001), L553
4. A. K. Svinin, “Integrable Chains and Hierarchies of Differential Evolution Equations”, Theoret. and Math. Phys., 130:1 (2002), 11–24
5. Meshkov, AG, “Integrable evolution equations on the N-dimensional sphere”, Communications in Mathematical Physics, 232:1 (2002), 1
6. Svinin, AK, “Extension of the discrete KP hierarchy”, Journal of Physics A-Mathematical and General, 35:8 (2002), 2045
7. Kudryashov, NA, “Fourth-order analogies to the Painlevé equations”, Journal of Physics A-Mathematical and General, 35:21 (2002), 4617
8. N. A. Kudryashov, “On the Fourth Painlevé Hierarchy”, Theoret. and Math. Phys., 134:1 (2003), 86–93
9. V. K. Mel'nikov, “Structure of Equations Solvable by the Inverse Scattering Transform for the Schrödinger Operator”, Theoret. and Math. Phys., 134:1 (2003), 94–106
10. N. A. Kudryashov, “Amalgamations of the Painlevé Equations”, Theoret. and Math. Phys., 137:3 (2003), 1703–1715
11. Willox, R, “Painlevé equations from Darboux chains: I. P-III-P-V”, Journal of Physics A-Mathematical and General, 36:42 (2003), 10615
12. Hickman, MS, “Computation of densities and fluxes of nonlinear differential-difference equations”, Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences, 459:2039 (2003), 2705
13. Tam, HW, “A special integrable differential-difference equation and its related systems: Bilinear forms soliton solutions and Lax pairs”, Journal of the Physical Society of Japan, 72:2 (2003), 265
14. Sergyeyev, A, “A remark on nonlocal symmetries for the Calogero-Degasperis-Ibragimov-Shabat equation”, Journal of Nonlinear Mathematical Physics, 10:1 (2003), 78
15. Kudryashov, NA, “Amalgamations of the Painlevé equations”, Journal of Mathematical Physics, 44:12 (2003), 6160
16. T. G. Kazakova, “Finite-Dimensional Discrete Systems Integrated in Quadratures”, Theoret. and Math. Phys., 138:3 (2004), 356–369
17. A. G. Meshkov, V. V. Sokolov, “Classification of Integrable Divergent $N$-Component Evolution Systems”, Theoret. and Math. Phys., 139:2 (2004), 609–622
18. R. I. Yamilov, “Relativistic Toda Chains and Schlesinger Transformations”, Theoret. and Math. Phys., 139:2 (2004), 623–635
19. I. T. Habibullin, E. V. Gudkova, “Boundary Conditions for Multidimensional Integrable Equations”, Funct. Anal. Appl., 38:2 (2004), 138–148
20. Baldwin, D, “Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations”, Computer Physics Communications, 162:3 (2004), 203
21. Yamilov, R, “Integrability conditions for n and t dependent dynamical lattice equations”, Journal of Nonlinear Mathematical Physics, 11:1 (2004), 75
22. Anatoly G. Meshkov, Maxim Ju. Balakhnev, “Integrable Anisotropic Evolution Equations on a Sphere”, SIGMA, 1 (2005), 027, 11 pp.
23. Tsuchida, T, “Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I”, Journal of Physics A-Mathematical and General, 38:35 (2005), 7691
24. Hydon, PE, “Multisymplectic conservation laws for differential and differential-difference equations”, Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 461:2058 (2005), 1627
25. Hereman W., Sanders J.A., Sayers J., Wang J.P., “Symbolic computation of polynomial conserved densities, generalized symmetries, and recursion operators for nonlinear differential-difference equations”, Group Theory and Numerical Analysis, CRM Proceedings & Lecture Notes, 39, 2005, 133–148
26. A. G. Meshkov, “On symmetry classification of third order evolutionary systems of divergent type”, J. Math. Sci., 151:4 (2008), 3167–3181
27. M. D. Vereschagin, S. D. Vereschagin, A. V. Yurov, “Trekhmernoe preobrazovanie Mutara”, Matem. modelirovanie, 18:5 (2006), 111–125
28. Vsevolod E. Adler, Alexey B. Shabat, “On the One Class of Hyperbolic Systems”, SIGMA, 2 (2006), 093, 17 pp.
29. Yamilov, R, “Symmetries as integrability criteria for differential difference equations”, Journal of Physics A-Mathematical and General, 39:45 (2006), R541
30. Levi, D, “Continuous symmetries of difference equations”, Journal of Physics A-Mathematical and General, 39:2 (2006), R1
31. Wang, Q, “New rational formal solutions for (1+1)-dimensional Toda equation and another Toda equation”, Chaos Solitons & Fractals, 29:4 (2006), 904
32. Svinin, AK, “Comment to: “Two hierarchies of lattice soliton equations associated with a new discrete eigenvalue problem and Darboux transformation””, Physics Letters A, 350:5–6 (2006), 419
33. R. I. Yamilov, “Integrability conditions for an analogue of the relativistic Toda chain”, Theoret. and Math. Phys., 151:1 (2007), 492–504
34. M. V. Demina, N. A. Kudryashov, “Special polynomials and rational solutions of the hierarchy of the second Painlevé equation”, Theoret. and Math. Phys., 153:1 (2007), 1398–1406
35. Tam, HW, “(2+1)-dimensional integrable lattice hierarchies related to discrete fourth-order nonisospectral problems”, Journal of Physics A-Mathematical and Theoretical, 40:43 (2007), 13031
36. Novikov, VS, “Symmetry structure of integrable nonevolutionary equations”, Studies in Applied Mathematics, 119:4 (2007), 393
37. Yu, YX, “Rational formal solutions of differential-difference equations”, Chaos Solitons & Fractals, 33:5 (2007), 1642
38. Xie, FD, “Some solutions of discrete sine-Gordon equation”, Chaos Solitons & Fractals, 33:5 (2007), 1791
39. Yu, YX, “Rational formal solutions of hybrid lattice equation”, Applied Mathematics and Computation, 186:1 (2007), 474
40. Hereman, W, “Continuous and discrete homotopy operators: A theoretical approach made concrete”, Mathematics and Computers in Simulation, 74:4–5 (2007), 352
41. B. I. Suleimanov, ““Quantizations” of the second Painlevé equation and the problem of the equivalence of its $L$$A$ pairs”, Theoret. and Math. Phys., 156:3 (2008), 1280–1291
42. V. V. Zharinov, “Evolution systems on a lattice”, Theoret. and Math. Phys., 157:3 (2008), 1694–1706
43. Kudryashov, NA, “The generalized Yablonskii-Vorob'ev polynomials and their properties”, Physics Letters A, 372:29 (2008), 4885
44. Demskoi, DK, “On recursion operators for elliptic models”, Nonlinearity, 21:6 (2008), 1253
45. Adler, VE, “Classification of integrable Volterra-type lattices on the sphere: isotropic case”, Journal of Physics A-Mathematical and Theoretical, 41:14 (2008), 145201
46. Decio Levi, Matteo Petrera, Christian Scimiterna, Ravil Yamilov, “On Miura Transformations and Volterra-Type Equations Associated with the Adler–Bobenko–Suris Equations”, SIGMA, 4 (2008), 077, 14 pp.
47. JETP Letters, 88:3 (2008), 164–166
48. Xu, XX, “A 2-parameter hierarchy of integrable lattice equations”, Modern Physics Letters B, 22:14 (2008), 1389
49. JETP Letters, 87:5 (2008), 266–270
50. V. M. Zhuravlev, “The method of generalized Cole–Hopf substitutions and new examples of linearizable nonlinear evolution equations”, Theoret. and Math. Phys., 158:1 (2009), 48–60
51. Hao Hong-hai, Zhang Da-jun, Deng Shu-fang, “The Kadomtsev–Petviashvili equation with self-consistent sources in nonuniform media”, Theoret. and Math. Phys., 158:2 (2009), 151–166
52. V. V. Zharinov, “Green's formula for difference operators”, Theoret. and Math. Phys., 161:2 (2009), 1445–1450
53. Levi, D, “The generalized symmetry method for discrete equations”, Journal of Physics A-Mathematical and Theoretical, 42:45 (2009), 454012
54. Huang, WH, “Jacobi elliptic function solutions of the Ablowitz-Ladik discrete nonlinear Schrodinger system”, Chaos Solitons & Fractals, 40:2 (2009), 786
55. V. V. Zharinov, “A differential-difference bicomplex”, Theoret. and Math. Phys., 165:2 (2010), 1401–1420
56. Levi D., Winternitz P., Yamilov R.I., “Lie point symmetries of differential-difference equations”, J. Phys. A: Math. Theor., 43:29 (2010), 292002
57. Gordoa P.R., Pickering A., Zhu Z.-N., “Matrix semidiscrete Ablowitz-Ladik equation hierarchy and a matrix discrete second Painlevé equation”, J Math Phys, 51:5 (2010), 053505
58. Zhuravlev V.M., Zinov'ev D.A., “The application of generalized Cole-Hopf substitutions in compressible-fluid hydrodynamics”, Physics of Wave Phenomena, 18:4 (2010), 245–250
59. 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
60. Balakhnev M.J., “New examples of the auto-Backlund transformations and nonlinear superposition formulas for vector evolution systems”, Phys Lett A, 375:3 (2011), 529–536
61. V. S. Gerdjikov, G. G. Grahovski, A. V. Mikhailov, T. I. Valchev, “Rational bundles and recursion operators for integrable equations on A.III-type symmetric spaces”, Theoret. and Math. Phys., 167:3 (2011), 740–750
62. V. V. Zharinov, “Symmetries and conservation laws of difference equations”, Theoret. and Math. Phys., 168:2 (2011), 1019–1034
63. Vladimir S. Gerdjikov, Georgi G. Grahovski, Alexander V. Mikhailov, Tihomir I. Valchev, “Polynomial Bundles and Generalised Fourier Transforms for Integrable Equations on A.III-type Symmetric Spaces”, SIGMA, 7 (2011), 096, 48 pp.
64. Decio Levi, Pavel Winternitz, Ravil I. Yamilov, “Symmetries of the Continuous and Discrete Krichever–Novikov Equation”, SIGMA, 7 (2011), 097, 16 pp.
65. Xenitidis P., “Symmetries and conservation laws of the ABS equations and corresponding differential-difference equations of Volterra type”, J. Phys. A: Math. Theor., 44:43 (2011), 435201
66. Mikhailov A.V., Wang J.P., Xenitidis P., “Cosymmetries and Nijenhuis recursion operators for difference equations”, Nonlinearity, 24:7 (2011), 2079–2097
67. Tsuchida T., “Systematic method of generating new integrable systems via inverse Miura maps”, J Math Phys, 52:5 (2011), 053503
68. Balakhnev M.J., “The Vector Ito-Drienfel'd-Sokolov System: Bilinear Backlund Transformation and Lax pair”, J Phys Soc Japan, 80:4 (2011), 045002
69. Levi D., Yamilov R.I., “Generalized symmetry integrability test for discrete equations on the square lattice”, J. Phys. A: Math. Theor., 44:14 (2011), 145207
70. B. I. Suleimanov, ““Kvantovaya” linearizatsiya uravnenii Penleve kak komponenta ikh $L,A$ par”, Ufimsk. matem. zhurn., 4:2 (2012), 127–135
71. A. V. Zhiber, R. D. Murtazina, I. T. Khabibullin, A. B. Shabat, “Kharakteristicheskie koltsa Li i integriruemye modeli matematicheskoi fiziki”, Ufimsk. matem. zhurn., 4:3 (2012), 17–85
72. Demskoi D.K., Viallet C.-M., “Algebraic Entropy for Semi-Discrete Equations”, J. Phys. A-Math. Theor., 45:35 (2012), 352001
73. Tsuda T., “From KP/Uc Hierarchies to Painlevé Equations”, Int. J. Math., 23:5 (2012), 1250010
74. Garifullin R.N., Yamilov R.I., “Generalized Symmetry Classification of Discrete Equations of a Class Depending on Twelve Parameters”, J. Phys. A-Math. Theor., 45:34 (2012), 345205
75. 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
76. Ferapontov E.V., Novikov V.S., Roustemoglou I., “Towards the Classification of Integrable Differential-Difference Equations in 2+1 Dimensions”, J. Phys. A-Math. Theor., 46:24 (2013), 245207
77. R. N. Garifullin, A. V. Mikhailov, R. I. Yamilov, “Discrete equation on a square lattice with a nonstandard structure of generalized symmetries”, Theoret. and Math. Phys., 180:1 (2014), 765–780
78. Scimiterna Ch., Hay M., Levi D., “On the Integrability of a New Lattice Equation Found By Multiple Scale Analysis”, J. Phys. A-Math. Theor., 47:26 (2014), 265204
79. Demskoi D.K., “Quad-Equations and Auto-Backlund Transformations of NLS-Type Systems”, J. Phys. A-Math. Theor., 47:16 (2014), 165204
80. A. B. Shabat, “Scattering theory for delta-type potentials”, Theoret. and Math. Phys., 183:1 (2015), 540–552
81. J. P. Wang, “Representations of $\mathfrak{sl}(2,\mathbb{C})$ in category $\mathcal O$ and master symmetries”, Theoret. and Math. Phys., 184:2 (2015), 1078–1105
82. Adler V.E., “Integrability Test For Evolutionary Lattice Equations of Higher Order”, J. Symbolic Comput., 74 (2016), 125–139
83. Arnaudon A., “On a Lagrangian Reduction and a Deformation of Completely Integrable Systems”, J. Nonlinear Sci., 26:5 (2016), 1133–1160
84. Garifullin R.N. Yamilov R.I. Levi D., “Non-invertible transformations of differential–difference equations”, J. Phys. A-Math. Theor., 49:37 (2016), 37LT01
85. Talati D., Turhan R., “Two-component integrable generalizations of Burgers equations with nondiagonal linearity”, J. Math. Phys., 57:4 (2016), 041502
86. Garifullin R.N. Yamilov R.I. Levi D., “Classification of five-point differential-difference equations”, J. Phys. A-Math. Theor., 50:12 (2017), 125201
87. Talati D., Wazwaz A.-M., “Some new integrable systems of two-component fifth-order equations”, Nonlinear Dyn., 87:2 (2017), 1111–1120
88. Ismagil Habibullin, Mariya Poptsova, “Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings”, SIGMA, 13 (2017), 073, 26 pp.
89. Gubbiotti G., Scimiterna C., Levi D., “The Non-Autonomous Ydkn Equation and Generalized Symmetries of Boll Equations”, J. Math. Phys., 58:5 (2017), 053507
90. Tian K., Wang J.P., “Symbolic Representation and Classification of N=1 Supersymmetric Evolutionary Equations”, Stud. Appl. Math., 138:4 (2017), 467–498
91. Talati D., Wazwaz A.-M., “Some Classification of Non-Commutative Integrable Systems”, Nonlinear Dyn., 88:2 (2017), 1487–1492
92. 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
93. Sergyeyev A., “New Integrable (3+1)-Dimensional Systems and Contact Geometry”, Lett. Math. Phys., 108:2 (2018), 359–376
94. Perepelkin E.E., Sadovnikov B.I., Inozemtseva N.G., “Solutions of Nonlinear Equations of Divergence Type in Domains Having Corner Points”, J. Elliptic Parabol. Equat., 4:1 (2018), 107–139
95. M. N. Poptsova, I. T. Habibullin, “Algebraic properties of quasilinear two-dimensional lattices connected with integrability”, Ufa Math. J., 10:3 (2018), 86–105
96. M. N. Poptsova, “Simmetrii odnoi periodicheskoi tsepochki”, Kompleksnyi analiz. Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 162, VINITI RAN, M., 2019, 80–84
97. Gubbiotti G., “Algebraic Entropy of a Class of Five-Point Differential-Difference Equations”, Symmetry-Basel, 11:3 (2019), 432
98. Perepelkin E.E., Kovalenko A.D., Tarelkin A.A., Polyakova R.V., Sadovnikov B.I., Inozemtseva N.G., Sysoev P.N., Sadovnikova M.B., “Simulation of Magnetic Systems in the Domain With a Corner”, Phys. Part. Nuclei, 50:3 (2019), 341–394
•  Number of views: This page: 1022 Full text: 371 References: 68 First page: 3