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Lie algebras and equations of Korteweg-de Vries type

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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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Literature cited

  1. V. I. Arnol'd, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  2. N. Bourbaki, Lie Groups and Algebras [Russian translation], Mir, Moscow (1976).

    Google Scholar 

  3. N. Bourbaki, Lie Groups and Algebras. Lie Algebras. Free Lie Algebras [Russian translation], Mir, Moscow (1972).

    Google Scholar 

  4. N. Bourbaki, Lie Groups and Algebras. Cartan Subalgebras, Regular Elements, Decomposable Semisimple Lie Algebras [Russian translation], Mir, Moscow (1978).

    Google Scholar 

  5. I. M. Gel'fand and L. A. Dikii, “Fractional powers of operators and Hamiltonian systems,” Funkts. Anal. Prilozhen.,10, No. 4, 13–29 (1976).

    Google Scholar 

  6. I. M. Gel'fand and L. A. Dikii, “The resolvent and Hamiltonian systems,” Funkts. Anal. Prilozhen.,11, No. 2, 11–27 (1977).

    Google Scholar 

  7. I. M. Gel'fand and L. A. Dikii, “A family of Hamiltonian structures connected with integrable nonlinear differential equations,” Preprint No. 136, IPM AN SSSR, Moscow (1978).

    Google Scholar 

  8. I. M. Gel'fand, L. A. Dikii, and I. Ya. Dorfman, “Hamiltonian operators and algebraic structures connected with them,” Funkts. Anal. Prilozhen.,13, No. 4, 13–30 (1979).

    Google Scholar 

  9. I. M. Gel'fand and L. A. Dikii, “The Schoutten bracket and Hamiltonian operators,” Funkts. Anal. Prilozhen.,14, No. 3, 71–74 (1980).

    Google Scholar 

  10. I. M. Gel'fand and L. A. Dikii, “Hamiltonian operators and infinite-dimensional Lie algebras,” Funkts. Anal. Prilozhen.,15, No. 3, 23–40 (1981).

    Google Scholar 

  11. I. M. Gel'fand and L. A. Dikii, “Hamiltonian operators and the classical Yang-Baxter equation,” Funkts. Anal. Prilozhen.,16, No. 4, 1–9 (1982).

    Google Scholar 

  12. V. G. Drinfel'd and V. V. Sokolov, “Equations of Korteweg-de Vries type and simple Lie algebras,” Dokl. Akad. Nauk SSSR,258, No. 1, 11–16 (1981).

    Google Scholar 

  13. A. V. Zhiber and A. B. Shabat, “Klein-Gordon equations with a nontrivial group,” Dokl. Akad. Nauk SSSR,247, No. 5, 1103–1107 (1979).

    Google Scholar 

  14. V. E. Zakharov, “On the problem of stochastization of one-dimensional chains of nonlinear oscillators,” Zh. Eksp. Teor. Fiz.,65, No. 1, 219–225 (1973).

    Google Scholar 

  15. V. E. Zakharov and S. V. Manakov, “On the theory of resonance interaction of wave packets in nonlinear media,” Zh. Eksp. Teor. Fiz.,69, No. 5, 1654–1673 (1975).

    Google Scholar 

  16. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: the Method of the Inverse Problem [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  17. V. E. Zakharov and A. V. Mikhailov, “Relativistically invariant two-dimensional models of a field theory integrable by the method of the inverse problem,” Zh. Eksp. Teor. Fiz.,74, No. 6, 1953–1973 (1978).

    Google Scholar 

  18. V. E. Zakharov and L. A. Takhtadzhyan, “Equivalence of the nonlinear Schrödinger equation and Heisenberg's ferromagnetic equation,” Teor. Mat. Fiz.,38, No. 1, 26–35 (1979).

    Google Scholar 

  19. V. E. Zakharov and A. B. Shabat, “A scheme of integrating nonlinear equations of mathematical physics by the method of the inverse scattering problem. I,” Funkts. Anal. Prilozhen.,8, No. 3, 54–56 (1974).

    Google Scholar 

  20. V. E. Zakharov and A. B. Shabat, “Integration of nonlinear equations of mathematical physics by the method of the inverse scattering problem. II,” Funkts. Anal. Prilozhen.,13, No. 3, 13–22 (1979).

    Google Scholar 

  21. V. G. Kats, “Simple irreducible graded Lie algebras of finite growth,” Izv. Akad. Nauk SSSR, Ser. Mat.,32, No. 6, 1323–1367 (1968).

    Google Scholar 

  22. V. G. Kats, “Infinite Lie algebras and the Dedekind η-function,” Funkts. Anal. Prilozhen.,8, No. 2, 77–78 (1974).

    Google Scholar 

  23. V. G. Kats, “Automorphisms of finite order of semisimple Lie algebras,” Funkts. Anal. Prilozhen.,3 No. 3, 94–96 (1969).

    Google Scholar 

  24. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill (1955).

  25. V. G. Konopel'chenko, “The Hamiltonian structure of integrable equations under reduction,” Preprint Inst. Yad. Fiz. Sib. Otd. Akad. Nauk SSSR No. 80-223, Novosibirsk (1981).

  26. I. M. Krichever, “Integration of nonlinear equations by methods of algebraic geometry,” Funkts. Anal. Prilozhen.,11, No. 1, 15–31 (1977).

    Google Scholar 

  27. D. R. Lebedev and Yu. I. Manin, “The Hamiltonian operator of Gel'fand-Dikii and the coadjoint representation of the Volterra group,” Funkts. Anal. Prilozhen.,13, No. 4, 40–46 (1979).

    Google Scholar 

  28. A. N. Leznov, “On complete integrability of a nonlinear system of partial differential equations in two-dimensional space,” Teor. Mat. Fiz.,42, No. 3, 343–349 (1980).

    Google Scholar 

  29. A. N. Leznov and M. V. Savel'ev, “Exact cylindrically symmetric solutions of the classical equations of gauge theories for arbitrary compact Lie groups,” Fiz. El. Chastits At. Yad.,11, No. 1, 40–91 (1980).

    Google Scholar 

  30. A. N. Leznov, M. V. Savel'ev, and V. G. Smirnov, “Theory of group representations and integration of nonlinear dynamical systems,” Teor. Mat. Fiz.,48, No. 1, 3–12 (1981).

    Google Scholar 

  31. I. G. Macdonald, “Affine root systems and the Dedekind η-function,” Matematika,16, No. 4, 3–49 (1972).

    Google Scholar 

  32. S. V. Manakov, “An example of a completely integrable nonlinear wave field with nontrivial dynamics (the Lie model),” Teor. Mat. Fiz.,28, No. 2, 172–179 (1976).

    Google Scholar 

  33. S. V. Manakov, “On complete integrability and stochastization in discrete dynamical systems,” Zh. Eksp. Teor. Fiz.,67, 543–555 (1974).

    Google Scholar 

  34. Yu. I. Manin, “Algebraic aspects of nonlinear differential equations,” in: Sov. Probl. Mat. Tom 11 (Itogi Nauki i Tekhniki VINITI AN SSSR), Moscow (1978), pp. 5–112.

    Google Scholar 

  35. Yu. I. Manin, “Matrix solitons and bundles over curves with singularities,” Funkts. Anal. Prilozhen.,12, No. 4, 53–67 (1978).

    Google Scholar 

  36. A. V. Mikhailov, “On the integrability of a two-dimensional generalization of the Toda lattice,” Pis'ma Zh. Eksp. Teor. Fiz.,30, No. 7, 443–448 (1979).

    Google Scholar 

  37. A. G. Reiman, “Integrable Hamiltonian systems connected with graded Lie algebras,” in: Differents. Geometriya, Gruppy Li i Mekhanika. III, Zap. Nauchn. Sem., LOMI, Vol. 95, Nauka, Leningrad (1980), pp. 3–54.

    Google Scholar 

  38. A. G. Reiman and M. A. Semenov-Tyan-Shanskii, “Algebras of flows and nonlinear partial differential equations,” Dokl. Akad. Nauk SSSR,251, No. 6, 1310–1314 (1980).

    Google Scholar 

  39. A. G. Reiman and M. A. Semenov-Tyan-Shanskii, “A family of Hamiltonian structures, a hierarchy of Hamiltonians, and reduction for matrix differential operators of first order,” Funkts. Anal. Prilozhen.,14, No. 2, 77–78 (1980).

    Google Scholar 

  40. J.-P. Serre, Lie Algebras and Lie Groups, W. A. Benjamin (1965).

  41. V. V. Sokolov, “Quasisoliton solutions of Lax equations,” Dissertation, Sverdlovsk (1981).

  42. V. V. Sokolov and A. B. Shabat, “(L-A) pairs and substitution of Ricatti type,” Funkts. Anal. Prilozhen.,14, No. 2, 79–80 (1980).

    Google Scholar 

  43. I. V. Cherednik, “Differential equations for Baker-Akhiezer functions of algebraic curves,” Funkts. Anal. Prilozhen.,12, No. 3, 45–54 (1978).

    Google Scholar 

  44. M. Adler, “On a trace functional for formal pseudodifferential operators and the symplectic structure of Korteweg-de Vries type equations,” Inventiones Math.,50, 219–248 (1979).

    Article  Google Scholar 

  45. O. I. Bogojavlensky, “On perturbations of the periodic Toda lattice,” Commun. Math. Phys.,51, 201–209 (1976).

    Article  Google Scholar 

  46. S. A. Bulgadaev, “Two-dimensional integrable field theories connected with simple Lie algebras,” Phys. Lett.,96B, 151–153 (1980).

    Google Scholar 

  47. E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, “Transformation groups for soliton equations,” Preprint RIMS-394, Kyoto Univ. (1982).

  48. E. Date, M. Kashiwara, and T. Miwa, “Vertex operators and τ-functions. Transformation groups for soliton equations. II,” Proc. Jpn. Acad.,A57, No. 8, 387–392 (1981).

    Google Scholar 

  49. R. K. Dodd and J. D. Gibbons, “The prolongation structure of higher-order Korteweg-de Vries equation,” Proc. R. Soc. London,358, Ser. A, 287–296 (1977).

    Google Scholar 

  50. B. L. Feigin and A. V. Zelevinsky, “Representations of contragradient Lie algebras and the Kac-Macdonald identities,” Proc. of the Summer School of Bolyai Math. Soc. Akademiai Kiado, Budapest (to appear).

  51. H. Flashcka, “Construction of conservation laws for Lax equations. Comments on a paper of G. Wilson,” Q. J. Math., Oxford (to appear).

  52. H. Flaschka, “Toda lattice I,” Phys. Rev.,B9, 1924–1925 (1974).

    Google Scholar 

  53. H. Flaschka, “Toda lattice II,” Progr. Theor. Phys.,51, 703–716 (1974).

    Google Scholar 

  54. A. P. Fordy and J. D. Gibbons, “Factorization of operators. I. Miura transformations,” J. Math. Phys.,21, 2508–2510 (1980).

    Article  Google Scholar 

  55. A. P. Fordy and J. D. Gibbons, “Integrable nonlinear Klein-Gordon equations and Toda lattice,” Commun. Math. Phys.,77, 21–30 (1980).

    Article  Google Scholar 

  56. I. B. Frenkel and V. G. Kac, “Basic representation of affine Lie algebras and dual resonance models,” Invent. Math.,62, 23–66 (1980).

    Article  Google Scholar 

  57. V. G. Kac, “Infinite-dimensional algebras, Dedekind's η-functions, classical Möbius function and the very strange formula,” Adv. Math.,30, No. 2, 85–136 (1978).

    Article  Google Scholar 

  58. B. Konstant, “The principle three-dimensional subgroup and the Bettin numbers of a complex simple Lie group,” Am. J. Math.,131, 973–1032 (1959).

    Google Scholar 

  59. B. A. Kupershmidt and G. Wilson, “Conservation laws and symmetries of generalized sine-Gordon equations,” Commun. Math. Phys.,81, No. 2, 189–202 (1981).

    Article  Google Scholar 

  60. B. A. Kupershmidt and G. Wilson, “Modifying Lax equations and the second Hamiltonian structure,” Invent. Math.,62, 403–436 (1981).

    Article  Google Scholar 

  61. A. N. Leznov and M. V. Saveliev, “Representation of zero curvature for the system of nonlinear partial differential equations Xα′zz=exp (KX)α and its integrability,” Lett. Math. Phys.,3, No. 6, 489–494 (1979).

    Article  Google Scholar 

  62. I. G. Macdonald, G. Segal, and G. Wilson, Kac-Moody Lie Algebras, Oxford Univ. Press (to appear).

  63. F. Magri, “A simple model of the integrable Hamiltonian equation,” J. Math. Phys.,19, No. 5, 1156–1162 (1978).

    Article  Google Scholar 

  64. A. V. Mikhailov, “The reduction problem and the inverse scattering method,” in: Proceedings of Soviet-American Symposium on Soliton Theory (Kiev, September 1979), Physica,3, Nos. 1–2, 73–117 (1981).

    Google Scholar 

  65. A. V. Mikhailov, M. A. Olshanetsky, and A. M. Perelomov, “Two-dimensional generalized Toda lattice,” Commun. Math. Phys.,79, 473–488 (1981).

    Article  Google Scholar 

  66. R. V. Moody, “Macdonald identities and Euclidean Lie algebras,” Proc. Am. Math. Soc.,48, No. 1, 43–52 (1975).

    Google Scholar 

  67. D. Mumford, “An algebrogeometric construction of commuting operators and of solution of the Toda lattice equation,” Korteweg-de Vries Equation and Related Equations. Proceedings of the International Symposium on Algebraic Geometry, Kyoto (1977), pp. 115–153.

  68. J. L. Verdier, “Les representations des algebres de Lie affines: applications a quelques problemes de physique,” Seminaire N. Bourbaki, Vol. 1981–1982, juin, expose No. 596 (1982).

  69. J. L. Verdier, “Equations differentielles algebriques,” Sem. Bourbaki 1977–1978, expose 512, Lect. Notes Math., No. 710, Springer-Verlag, Berlin (1979), pp. 101–122.

    Google Scholar 

  70. G. Wilson, “Commuting flows and conservation laws for Lax equations,” Math. Proc. Cambr. Philos. Soc.,86, No. 1, 131–143 (1979).

    Google Scholar 

  71. G. Wilson, “On two constructions of conservation laws for Lax equations,” Q. J. Math. Oxford,32, No. 128, 491–512 (1981).

    Google Scholar 

  72. G. Wilson, “The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras,” Ergodic Theory and Dynamical Systems,1, 361–380 (1981).

    Google Scholar 

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Authors
  1. V. G. Drinfel'd
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  2. V. V. Sokolov
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya), Vol. 24, pp. 81–180, 1984.

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Drinfel'd, V.G., Sokolov, V.V. Lie algebras and equations of Korteweg-de Vries type. J Math Sci 30, 1975–2036 (1985). https://doi.org/10.1007/BF02105860

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