Symmetries of Differential Equations and the Problem of Integrability

  • A.V. Mikhailov
  • V.V. Sokolov
Part of the Lecture Notes in Physics book series (LNP, volume 767)


Formal Series High Symmetry Jordan Algebra Symbolic Representation Vries Equation 
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  1. 1.
    V.E. Adler, A.B. Shabat, and R.I. Yamilov, The symmetry approach to the problem of integrability, Theor. Math. Phys. 125(3), 355–424, 2000.CrossRefMathSciNetGoogle Scholar
  2. 2.
    M. Adler, On the trace functional for formal pseudodifferential operators and the symplectic structure of the KdV type equations, Inventiones Math. 50, 219–248, 1979.zbMATHCrossRefADSGoogle Scholar
  3. 3.
    C. Athorne and A. Fordy, Generalized KdV and MKdV equations associated with symmetric spaces, J. Phys. A. 20, 1377–1386, 1987.zbMATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    S.P. Balandin and V.V. Sokolov, On the Painlevé test for non-Abelian equations, Phys. Lett. A 246(3–4), 267–272, 1998.zbMATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    I.M. Bakirov and Popkov V.Yu., Completely integrable systems of brusselator type, Phys. Lett. A 141(5), 275–277, 1989.Google Scholar
  6. 6.
    M.Ju. Balakhnev, A class of integrable evolutionary vector equations. Theor. Math. Phys. 142(1), 8–14, 2005.Google Scholar
  7. 7.
    F. Beukers, J. Sanders, and Jing Ping Wang On Integrability of Systems of Evolution Equations, J. Differ. Equations 172, 396–408, 2001.Google Scholar
  8. 8.
    R. Camassa, D.D.D. Holm, An integrable shallow water equation with peaked solutions, Phys. Rev. Lett. 71, 1661–1664, 1993.zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    F. Calogero, Why Are Certain Nonlinear PDE’s Both Widely Applicable and Integrable?, in bookWhat is integrability?, Springer-Verlag (Springer Series in Nonlinear Dynamics), 1–62, 1991.Google Scholar
  10. 10.
    F. Calogero and A. Degasperis, Spectral transforms and solitons, North-Holland Publ. Co., Amsterdam-New York-Oxford, 1982.Google Scholar
  11. 11.
    A. Degasperis and M. Procesi, Asymptotic integrability, inSymmetry and Perturbation Theory, A. Degasperis and G. Gaeta (eds.), World Scientific, 23–37, 1999.Google Scholar
  12. 12.
    A. Degasperis, D.D. Holm, and A.N.W. Hone, A New Integrable Equation with Peakon Solutions, to appear in NEEDS 2001 Proceedings, Theoretical and Mathematical Physics, 2002.Google Scholar
  13. 13.
    I.Ya. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, John Wiley&Sons, Chichester, 1993.Google Scholar
  14. 14.
    V.G. Drinfeld and V.V. Sokolov, Lie algebras and equations of Korteweg de Vries type. J. Sov. Math. 30, 1975–2036, 1985.CrossRefGoogle Scholar
  15. 15.
    V.G. Drinfeld, S.I. Svinolupov, and Sokolov, V.V., Classification of fifth order evolution equations with infinite series of conservation laws, Doklady of Ukrainian Akademy, Section A 10, 7–10, 1985.Google Scholar
  16. 16.
    A.S. Fokas, Symmetries and integrability, Stud. Appl. Math. 77, 253–299, 1987.zbMATHMathSciNetGoogle Scholar
  17. 17.
    A.S. Fokas, A symmetry approach to exactly solvable evolution equations, J. Math. Phys. 21(6), 1318–1325, 1980.zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    A.P. Fordy and P. Kulish, Nonlinear Schrödinger equations and simple Lie algebras, Commun. Math. Phys. 89, 427–443, 1983.zbMATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    A.P. Fordy, Derivative nonlinear Schrödinger equations and Hermitian symmetric spaces, J. Phys. A.: Math. Gen. 17, 1235–1245, 1984.zbMATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    I.M. Gel’fand and L.A. Dickii, Asymptotic properties of the resolvent of Sturm-Lioville equations, and the algebra of Korteweg de Vries equations. Russian Math. Surveys 30, 77–113, 1975.zbMATHCrossRefADSGoogle Scholar
  21. 21.
    I.M. Gel’fand, Yu. I.Manin, and M.A. Shubin Poisson brackets and kernel of variational derivative in formal variational calculus. Funct. Anal. Appl. 10(4), 30–34, 1976.zbMATHGoogle Scholar
  22. 22.
    I.Z. Golubchik and V.V. Sokolov Multicomponent generalization of the hierarchy of the Landau-Lifshitz equation, Theor. Math. Phys. 124(1), 909–917, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    I.T. Habibullin, Phys. Lett. A 369, 1993.Google Scholar
  24. 24.
    I.T. Habibullin, V.V. Sokolov, and R.I. Yamilov, Multi-component integrable systems and non-associative structures, in Nonlinear Physics: theory and experiment, E. Alfinito, M. Boiti, L. Martina, F. Pempinelli (eds.), World Scientific Publisher: Singapore, 139–168, 1996.Google Scholar
  25. 25.
    R.H. Heredero, V.V. Sokolov, and S.I. Svinolupov Toward the classification of third order integrable evolution equations, J. Phys. A: Math. General 13, 4557–4568, 1994.CrossRefADSGoogle Scholar
  26. 26.
    E. Husson, Sur un thereme de H.Poincaré, relativement d’un solide pesant, Acta Math. 31, 71–88, 1908.CrossRefMathSciNetGoogle Scholar
  27. 27.
    N.Kh. Ibragimov and A.B. Shabat, Evolution equation with non-trivial Lie-Bäcklund group, Funct. Anal. Appl. 14(1), 25–36, 1980. [in Russian]MathSciNetGoogle Scholar
  28. 28.
    N.Kh. Ibragimov and A.B. Shabat, Infinite Lie-Bäcklund algebras, Funct. Anal. Appl. 14(4), 79–80, 1980. [in Russian]zbMATHMathSciNetGoogle Scholar
  29. 29.
    N. Jacobson, Structure and representations of Jordan algebras, Amer. Math. Soc. Colloq. Publ., Providence R.I. 39, 1968.Google Scholar
  30. 30.
    I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris, 1957.zbMATHGoogle Scholar
  31. 31.
    L. Martínez Alonso and A.B. Shabat, Towards a theory of differential constraints of a hydrodynamic hierarchy, J. Nonlin. Math. Phys. 10, 229–242, 2003.Google Scholar
  32. 32.
    A.G. Meshkov, On symmetry classification of third order evolutionary systems of divergent type, Fund. Appl. Math. 12(7), 141–161, 2006. [in Russian]Google Scholar
  33. 33.
    A.G. Meshkov and M.Ju. Balakhnev, Two-field integrable evolutionary systems of the third order and their differential substitutions. Symmetry, Integrability and Geometry: Methods and Applications. 4, 018, 29, 2008.MathSciNetGoogle Scholar
  34. 34.
    A.G. Meshkov and V.V. Sokolov, Integrable evolution equations on the N-dimensional sphere, Comm. Math. Phys. 232(1), 1–18, 2002.zbMATHCrossRefADSMathSciNetGoogle Scholar
  35. 35.
    A.G. Meshkov and V.V. Sokolov, Classification of integrable divergent N-component evolution systems, Theoret. Math. Phys. 139(2), 609–622, 2004.CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    A.V. Mikhailov and V.S. Novikov Perturbative Symmetry Approach, J. Phys. A 35, 4775–4790, 2002.zbMATHCrossRefADSMathSciNetGoogle Scholar
  37. 37.
    A.V. Mikhailov and V.S. Novikov Classification of Integrable Benjamin-Ono type equations, Moscow Math. J. 3(4), 1293–1305, 2003.zbMATHMathSciNetGoogle Scholar
  38. 38.
    A.V. Mikhailov, V.S. Novikov, and J.P. Wang. On classification of integrable non-evolutionary equations. Stud. Appl. Math. 118, 419–457, 2007.CrossRefMathSciNetGoogle Scholar
  39. 39.
    A.V. Mikhailov, V.S. Novikov, and J.P. Wang., Symbolic representation and classification of integrable systems, in Algebraic Theory of Differential Equations, M.A.H. MacCallum and A.V. Mikhailov (eds.), CUP, 2008 (to appear)Google Scholar
  40. 40.
    A.V. Mikhailov and A.B. Shabat, Integrability conditions for systems of two equationsut=A(u)uxx+B(u, ux). I, Theor. Math. Phys. 62(2), 163–185, 1985.CrossRefMathSciNetGoogle Scholar
  41. 41.
    A.V. Mikhailov and A.B. Shabat, Integrability conditions for systems of two equationsut=A(u)uxx+B(u, ux). II, Theor. Math. Phys. 66(1), 47–65, 1986Google Scholar
  42. 42.
    A.V. Mikhailov, A.B. Shabat, and R.I. Yamilov, The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems, Russian Math. Surveys 42(4), 1–63, 1987.CrossRefMathSciNetADSGoogle Scholar
  43. 43.
    A.V. Mikhailov, A.B. Shabat, and R.I. Yamilov, Extension of the module of invertible transformations. Classification of integrable systems, Commun. Math. Phys. 115, 1–19, 1988.CrossRefADSMathSciNetGoogle Scholar
  44. 44.
    A.V. Mikhailov and V.V. Sokolov, Integrable ODEs on Associative Algebras, Comm. Math. Phys. 211(1), 231–251, 2000.zbMATHCrossRefADSMathSciNetGoogle Scholar
  45. 45.
    A.V. Mikhailov, V.V. Sokolov, A.B. Shabat, The symmetry approach to classification of integrable equations, in What is Integrability? V.E. Zakharov (ed.), Springer series in Nonlinear Dynamics, 115–184, 1991.Google Scholar
  46. 46.
    A.V. Mikhailov, R.I. Yamilov, Towards classification of (2+1)– dimensional integrable equations. Integrability conditions I., J. Phys. A: Math. Gen. 31, 6707–6715, 1998.zbMATHCrossRefADSMathSciNetGoogle Scholar
  47. 47.
    R.M. Miura, Korteweg-de Vries equation and generalization. I. A remarkable explicit nonlinear transformation, J. Math. Phys. 9, 1202–1204, 1968.zbMATHCrossRefADSMathSciNetGoogle Scholar
  48. 48.
    V.S. Novikov and J.P. Wang. Symmetry structure of integrable nonevolutionary equations. Stud. Appl. Math. 119(4):393–428, 2007.CrossRefMathSciNetGoogle Scholar
  49. 49.
    P.J. Olver, Applications of Lie groups to differential equations, Volume 107 of Graduate texts in Mathematics, Springer Verlag, New York, 1993.Google Scholar
  50. 50.
    P.J. Olver and V.V. Sokolov, Integrable evolution equations on associative algebras, Comm. Math. Phys. 193(2), 245–268, 1998.zbMATHCrossRefADSMathSciNetGoogle Scholar
  51. 51.
    P.J. Olver and V.V. Sokolov, Non-abelian integrable systems of the derivative nonlinear Schrödinger type, Inverse Problems 14(6), L5–L8, 1998.zbMATHCrossRefADSMathSciNetGoogle Scholar
  52. 52.
    P. Olver, J.P. Wang, Classification of integrable one-component systems on associative algebras, Proc. London Math. Soc. 81(3), 566–586, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    J. Sanders, J.P. Wang, On the Integrability of homogeneous scalar evolution equations, J. Differ. Equations 147, 410–434, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    A.B. Shabat and R.I. Yamilov On a complete list of integrable systems of the formiut=uxx+f(u, v, ux, vx), -ivt=vxx+g(u, v, ux, vx), Preprint BFAN, Ufa, 28 pages, 1985.Google Scholar
  55. 55.
    V.V. Sokolov and A.B. Shabat, Classification of Integrable Evolution Equations, Soviet Sci. Rev., Section C 4, 221–280, 1984.MathSciNetGoogle Scholar
  56. 56.
    V.V. Sokolov, On the symmetries of evolution equations, Russian Math. Surveys 43(5), 165–204, 1988.zbMATHCrossRefMathSciNetADSGoogle Scholar
  57. 57.
    V.V. Sokolov, A new integrable case for the Kirchhoff equation, Theoret. Math. Phys. 129(1), 1335–1340, 2001.zbMATHCrossRefGoogle Scholar
  58. 58.
    V.V. Sokolov and S.I. Svinolupov, Vector-matrix generalizations of classical integrable equations, Theor. Math. Phys. 100(2), 959–962, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    V.V. Sokolov and S.I. Svinolupov, Weak nonlocalities in evolution equations, Math. Notes 48(5–6), 1234–1239, 1991.Google Scholar
  60. 60.
    V.V. Sokolov and T. Wolf, A symmetry test for quasilinear coupled systems, Inverse Problems 15, L5–L11, 1999zbMATHCrossRefADSMathSciNetGoogle Scholar
  61. 61.
    V.V. Sokolov and S.I. Svinolupov, Deformation of nonassociative algebras and integrable differential equations, Acta Applicandae Mathematica, 41(1–2), 323–339, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    V.V. Sokolov, T. Wolf, Classification of integrable polynomial vector evolution equations, J. Phys. A 2001, 34, 11139–11148.zbMATHCrossRefADSMathSciNetGoogle Scholar
  63. 63.
    V.A. Steklov On the motion of a rigid body in a fluid, Kharkov, 234 pages, 1893.Google Scholar
  64. 64.
    S.I. Svinolupov, Second-order evolution equations with symmetries, Uspehi Mat. Nauk 40(5), 263, 1985.MathSciNetGoogle Scholar
  65. 65.
    S.I. Svinolupov, On the analogues of the Burgers equation, Phys. Lett. A 135(1), 32–36, 1989.CrossRefADSMathSciNetGoogle Scholar
  66. 66.
    S.I. Svinolupov, Generalized Schrödinger equations and Jordan pairs, Comm. Math. Phys. 143(1), 559–575, 1992.zbMATHCrossRefADSMathSciNetGoogle Scholar
  67. 67.
    S.I. Svinolupov, Jordan algebras and generalized Korteweg-de Vries equations, Theor. Math. Phys. 87(3), 391–403, 1991.CrossRefMathSciNetGoogle Scholar
  68. 68.
    S.I. Svinolupov and V.V. Sokolov, Deformations of Jordan triple systems and integrable equations, Theor. Math. Phys. 108(3), 1160–1163, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    S.I. Svinolupov and V.V. Sokolov, Evolution equations with nontrivial conservation laws, Func. analiz i pril. 16(4), 86–87, 1982. [in Russian],MathSciNetGoogle Scholar
  70. 70.
    S.I. Svinolupov and V.V. Sokolov, On conservation laws for equations with nontrivial Lie-Bäcklund algebra, in Integrable systems, A.B. Shabat (ed.), Ufa, BFAN SSSR 53–67, 1982. [in Russian].Google Scholar
  71. 71.
    S.I. Svinolupov and V.V. Sokolov, Factorization of evolution equations, Russian Math. Surveys 47(3), 127–162, 1992.zbMATHCrossRefMathSciNetADSGoogle Scholar
  72. 72.
    S.I. Svinolupov and V.V. Sokolov, Deformations of Jordan triple systems and integrable equations, Theoret. Math. Phys. 1996, 108(3), 1160–1163, 1997.CrossRefMathSciNetGoogle Scholar
  73. 73.
    S.I. Svinolupov, V.V. Sokolov, and R.I. Yamilov, Bäcklund transformations for integrable evolution equations, Dokl. Akad. Nauk SSSR 271(4), 802–805, 1983.MathSciNetGoogle Scholar
  74. 74.
    T. Tsuchida, M. Wadati, New integrable systems of derivative nonlinear Schrödinger equations with multiple components, Phys. Lett. A 257, 53–64, 1999.zbMATHCrossRefADSMathSciNetGoogle Scholar
  75. 75.
    T. Tsuchida, T. Wolf, Classification of polynomial integrable systems of mixed scalar and vector evolution equations, J. Phys. A: Math. Gen. 38, 7691–7733, 2005.zbMATHCrossRefADSMathSciNetGoogle Scholar
  76. 76.
    Jing Ping Wang, Symmetries and Conservation Laws of Evolution Equations, PhD thesis, published by Thomas Stieltjes Institute for Mathematics, Amsterdam, 1998.Google Scholar
  77. 77.
    V.E. Zakharov, E.I. Schulman, Integrability of Nonlinear Systems and Perturbation Theory, in What is Integrability? V.E. Zakharov (ed.), Springer series in Nonlinear Dynamics, 185–250, 1991.Google Scholar
  78. 78.
    A.V. Zhiber and A.B. Shabat, Klein-Gordon equations with a nontrivial group, Sov. Phys. Dokl. 247(5), 1103–1107, 1979.MathSciNetGoogle Scholar
  79. 79.
    A.V. Zhiber and A.B. Shabat, Systems of equations ux=p(u, v), vy=q(u, v) possessing symmetries, Sov. Math. Dokl. 30, 23–26, 1984.zbMATHGoogle Scholar
  80. 80.
    A.V. Zhiber and V.V. Sokolov Exactly integrable hyperbolic equations of Liouville type, Russian Math. Surveys 56(1), 63–106, 2001.CrossRefMathSciNetADSGoogle Scholar
  81. 81.
    A.V. Zhiber, V.V. Sokolov, and Startsev S. Ya, On nonlinear Darbouxintegrable hyperbolic equations, Doklady RAN 343(6), 746–748, 1995.MathSciNetGoogle Scholar
  82. 82.
    S.L. Ziglin The branching of solutions and non-existing of first integrals in Hamiltonian mechanics. I, II, Funct. Anal. Appl. 16(3), 30–41, 1982; 17(1), 8–23, 1983.MathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • A.V. Mikhailov
    • 1
  • V.V. Sokolov
  1. 1.School of MathematicsUniversity of LeedsLeedsUK

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