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Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh.:
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Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh., 1984, Volume 24, Pages 81–180 (Mi intd72)  

This article is cited in 54 scientific papers (total in 54 papers)

Lie algebras and equations of Korteweg–de Vries type

V. G. Drinfeld, V. V. Sokolov


Abstract: The survey contains a description of the connection between the infinite-dimensional Lie algebras of Kats–Moody and systems of differential equations generalizing the Korteweg–de Vries and sine-Gordon equations and integrable by the method of the inverse scattering problem. A survey of the theory of Kats–Moody algebras is also given.

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English version:
Journal of Soviet Mathematics, 1985, 30:2, 1975–2036

Bibliographic databases:

UDC: 515.168.3+517.957

Citation: V. G. Drinfeld, V. V. Sokolov, “Lie algebras and equations of Korteweg–de Vries type”, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh., 24, VINITI, Moscow, 1984, 81–180; J. Soviet Math., 30:2 (1985), 1975–2036

Citation in format AMSBIB
\Bibitem{DriSok84}
\by V.~G.~Drinfeld, V.~V.~Sokolov
\paper Lie algebras and equations of Korteweg--de~Vries type
\serial Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh.
\yr 1984
\vol 24
\pages 81--180
\publ VINITI
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/intd72}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=760998}
\zmath{https://zbmath.org/?q=an:0578.58040|0558.58027}
\transl
\jour J. Soviet Math.
\yr 1985
\vol 30
\issue 2
\pages 1975--2036
\crossref{https://doi.org/10.1007/BF02105860}


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    1. A. O. Radul, “A description of Poisson brackets on a space of nonlocal functionals”, Funct. Anal. Appl., 19:2 (1985), 153–156  mathnet  crossref  mathscinet  zmath  isi
    2. T. G. Khovanova, “Structure of Lie superalgebras on eigenfunctions and jets of the kernel of the resolvent near the diagonal for an $n$th-order differential operator”, Funct. Anal. Appl., 20:2 (1986), 162–164  mathnet  crossref  mathscinet  zmath  isi
    3. T. G. Khovanova, “The Gel'fand–Dikii Lie Algebras and the Virasoro algebra”, Funct. Anal. Appl., 20:4 (1986), 332–334  mathnet  crossref  mathscinet  zmath  isi
    4. A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, “The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems”, Russian Math. Surveys, 42:4 (1987), 1–63  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. P. I. Holod, “Hidden symmetry of the Landau–Lifshitz equation, hierarchy of higher equations, and the dual equation for an asymmetric chiral field”, Theoret. and Math. Phys., 70:1 (1987), 11–19  mathnet  crossref  mathscinet  isi
    6. D. R. Lebedev, A. O. Radul, “Periodic intermediate long wave equation: The undressing method”, Theoret. and Math. Phys., 70:2 (1987), 140–147  mathnet  crossref  mathscinet  zmath  isi
    7. T. G. Khovanova, “Korteweg–de Vries superequation related to the Lie superalgebra of Neveu-Schwarz-2 string theory”, Theoret. and Math. Phys., 72:2 (1987), 899–904  mathnet  crossref  mathscinet  zmath  isi
    8. V. V. Sokolov, “On the symmetries of evolution equations”, Russian Math. Surveys, 43:5 (1988), 165–204  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. S. L. Luk'yanov, “Quantization of the Gel'fand?Dikii brackets”, Funct. Anal. Appl., 22:4 (1988), 255–262  mathnet  crossref  mathscinet  zmath  isi
    10. B. A. Khesin, “Invariants of Hamiltonian KdV-structures”, Russian Math. Surveys, 45:1 (1990), 209–210  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    11. V. Yu. Ovsienko, “Contact analogues of the Virasoro algebra”, Funct. Anal. Appl., 24:4 (1990), 306–314  mathnet  crossref  mathscinet  zmath  isi
    12. V. Yu. Ovsienko, B. A. Khesin, “Symplectic leaves of the Gel'fand–Dikii brackets and homotopy classes of nondegenerate curves”, Funct. Anal. Appl., 24:1 (1990), 33–40  mathnet  crossref  mathscinet  zmath  isi
    13. O. S. Kravchenko, B. A. Khesin, “A central extension of the algebra of pseudodifferential symbols”, Funct. Anal. Appl., 25:2 (1991), 152–154  mathnet  crossref  mathscinet  zmath  isi
    14. S. I. Svinolupov, “Jordan algebras and generalized Korteweg–de Vries equations”, Theoret. and Math. Phys., 87:3 (1991), 611–620  mathnet  crossref  mathscinet  zmath  isi
    15. S. I. Svinolupov, “Jordan Algebras and Integrable Systems”, Funct. Anal. Appl., 27:4 (1993), 257–265  mathnet  crossref  mathscinet  zmath  isi
    16. Ya. P. Pugay, “Lattice $W$ algebras and quantum groups”, Theoret. and Math. Phys., 100:1 (1994), 900–911  mathnet  crossref  mathscinet  zmath  isi
    17. E. I. Bogdanov, “Spatially distributed classical Lagrangian mechanics”, Theoret. and Math. Phys., 101:3 (1994), 1419–1421  mathnet  crossref  mathscinet  zmath  isi
    18. I. M. Krichever, “General Rational Reductions of the KP Hierarchy and Their Symmetries”, Funct. Anal. Appl., 29:2 (1995), 75–80  mathnet  crossref  mathscinet  zmath  isi
    19. V. I. Vakulenko, “Solution of Virasoro constraints for DKP hierarhy”, Theoret. and Math. Phys., 107:1 (1996), 435–440  mathnet  crossref  crossref  mathscinet  zmath  isi
    20. I. Z. Golubchik, V. V. Sokolov, “Integrable Systems Generated by a Constant Solution of the Yang–Baxter Equation”, Funct. Anal. Appl., 30:4 (1996), 275–277  mathnet  crossref  crossref  mathscinet  zmath  isi
    21. I. Z. Golubchik, V. V. Sokolov, “Integrable equations on $\mathbb Z$-graded Lie algebras”, Theoret. and Math. Phys., 112:3 (1997), 1097–1103  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    22. I. Z. Golubchik, V. V. Sokolov, “Multicomponent generalization of the hierarchy of the Landau–Lifshitz equation”, Theoret. and Math. Phys., 124:1 (2000), 909–917  mathnet  crossref  crossref  mathscinet  zmath  isi
    23. I. Z. Golubchik, V. V. Sokolov, “One More Kind of the Classical Yang–Baxter Equation”, Funct. Anal. Appl., 34:4 (2000), 296–298  mathnet  crossref  crossref  mathscinet  zmath  isi
    24. A. A. Bormisov, F. Kh. Mukminov, “Symmetries of Systems of the Hyperbolic Riccati Type”, Theoret. and Math. Phys., 127:1 (2001), 446–459  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    25. I. Z. Golubchik, V. V. Sokolov, “Compatible Lie Brackets and Integrable Equations of the Principal Chiral Model Type”, Funct. Anal. Appl., 36:3 (2002), 172–181  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    26. A. V. Kiselev, “Methods of geometry of differential equations in analysis of integrable models of field theory”, J. Math. Sci., 136:6 (2006), 4295–4377  mathnet  crossref  mathscinet  zmath  elib  elib
    27. E. V. Frenkel, “Opers on the projective line, flag manifolds and Bethe ansatz”, Mosc. Math. J., 4:3 (2004), 655–705  mathnet  crossref  mathscinet  zmath
    28. A. V. Ovchinnikov, “On the solution of Toda systems associated with simple Lie algebras”, J. Math. Sci., 141:1 (2007), 1031–1040  mathnet  crossref  mathscinet  zmath
    29. O. V. Il'in, “On $W$-geometry of Toda systems”, J. Math. Sci., 141:1 (2007), 1041–1047  mathnet  crossref  mathscinet  zmath
    30. T. V. Skrypnik, “Quasigraded lie algebras, Kostant–Adler scheme, and integrable hierarchies”, Theoret. and Math. Phys., 142:2 (2005), 275–288  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    31. A. G. Meshkov, “On symmetry classification of third order evolutionary systems of divergent type”, J. Math. Sci., 151:4 (2008), 3167–3181  mathnet  crossref  mathscinet  zmath
    32. Taras V. Skrypnyk, “Quasigraded Lie Algebras and Modified Toda Field Equations”, SIGMA, 2 (2006), 043, 14 pp.  mathnet  crossref  mathscinet  zmath
    33. A. V. Kiselev, “Algebraic properties of Gardner's deformations for integrable systems”, Theoret. and Math. Phys., 152:1 (2007), 963–976  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    34. V. A. Andreev, “System of equations for stimulated combination scattering and the related double periodic $A_n^{(1)}$ Toda chains”, Theoret. and Math. Phys., 156:1 (2008), 1020–1027  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    35. A. G. Meshkov, “Nonlocal symmetries in two-field divergent evolutionary systems”, Theoret. and Math. Phys., 156:3 (2008), 1268–1279  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    36. Anatoly G.G. Meshkov, Maxim Ju. Balakhnev, “Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions”, SIGMA, 4 (2008), 018, 29 pp.  mathnet  crossref  mathscinet  zmath
    37. Gloria Marí Beffa, “Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems”, SIGMA, 4 (2008), 034, 23 pp.  mathnet  crossref  mathscinet  zmath
    38. A. V. Kiselev, J. W. van de Leur, “Symmetry algebras of Lagrangian Liouville-type systems”, Theoret. and Math. Phys., 162:2 (2010), 149–162  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    39. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    40. A. A. Gerasimov, D. R. Lebedev, S. V. Oblezin, “New integral representations of Whittaker functions for classical Lie groups”, Russian Math. Surveys, 67:1 (2012), 1–92  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    41. V. S. Gerdjikov, “Two-dimensional Toda field equations related to the exceptional algebra $\mathfrak g_2$: Spectral properties of the Lax operators”, Theoret. and Math. Phys., 172:2 (2012), 1085–1096  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    42. A. V. Domrin, “On holomorphic solutions of equations of Korteweg–de Vries type”, Trans. Moscow Math. Soc., 73 (2012), 193–206  mathnet  crossref  mathscinet  zmath  elib
    43. A. G. Meshkov, V. V. Sokolov, “Integriruemye evolyutsionnye uravneniya s postoyannoi separantoi”, Ufimsk. matem. zhurn., 4:3 (2012), 104–154  mathnet
    44. I. T. Habibullin, M. V. Yangubaeva, “Formal diagonalization of a discrete Lax operator and conservation laws and symmetries of dynamical systems”, Theoret. and Math. Phys., 177:3 (2013), 1655–1679  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    45. Johan van de Leur, “The $(n,1)$-Reduced DKP Hierarchy, the String Equation and $W$ Constraints”, SIGMA, 10 (2014), 007, 19 pp.  mathnet  crossref  mathscinet
    46. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    47. A. V. Mikhailov, “Formal diagonalisation of Lax–Darboux schemes”, Model. i analiz inform. sistem, 22:6 (2015), 795–817  mathnet  crossref  mathscinet  elib
    48. O. K. Sheinman, “Lax operator algebras and integrable systems”, Russian Math. Surveys, 71:1 (2016), 109–156  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    49. B. L. Feigin, “Integrable systems, shuffle algebras, and Bethe equations”, Trans. Moscow Math. Soc., 77 (2016), 203–246  mathnet  crossref  elib
    50. I. T. Habibullin, A. R. Khakimova, “Invariant manifolds and Lax pairs for integrable nonlinear chains”, Theoret. and Math. Phys., 191:3 (2017), 793–810  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    51. V. S. Gerdjikov, “Kulish–Sklyanin-type models: Integrability and reductions”, Theoret. and Math. Phys., 192:2 (2017), 1097–1114  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    52. G. G. Grahovski, A. J. Mustafa, H. Susanto, “Nonlocal reductions of the multicomponent nonlinear Schrödinger equation on symmetric spaces”, Theoret. and Math. Phys., 197:1 (2018), 1430–1450  mathnet  crossref  crossref  adsnasa  isi  elib
    53. V. S. Gerdjikov, A. A. Stefanov, I. D. Iliev, G. P. Boyadjiev, A. O. Smirnov, V. B. Matveev, M. V. Pavlov, “Recursion operators and hierarchies of $mKdV$ equations related to the Kac–Moody algebras $D_4^{(1)}$, $D_4^{(2)}$, and $D_4^{(3)}$”, Theoret. and Math. Phys., 204:3 (2020), 1110–1129  mathnet  crossref  crossref  isi
    54. I. V. Volovich, “On Integrability of Dynamical Systems”, Proc. Steklov Inst. Math., 310 (2020), 70–77  mathnet  crossref  crossref  isi
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