# Lie algebras and equations of Korteweg-de Vries type

Article

- 1k Downloads
- 549 Citations

## 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.

## Keywords

Differential Equation Inverse Scattering Scattering Problem Inverse Scattering Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## 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).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 41.V. V. Sokolov, “Quasisoliton solutions of Lax equations,” Dissertation, Sverdlovsk (1981).Google Scholar
- 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).CrossRefGoogle Scholar
- 45.O. I. Bogojavlensky, “On perturbations of the periodic Toda lattice,” Commun. Math. Phys.,51, 201–209 (1976).CrossRefGoogle 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).Google Scholar
- 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).Google Scholar
- 51.H. Flashcka, “Construction of conservation laws for Lax equations. Comments on a paper of G. Wilson,” Q. J. Math., Oxford (to appear).Google Scholar
- 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).CrossRefGoogle Scholar
- 55.A. P. Fordy and J. D. Gibbons, “Integrable nonlinear Klein-Gordon equations and Toda lattice,” Commun. Math. Phys.,77, 21–30 (1980).CrossRefGoogle 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).CrossRefGoogle 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).CrossRefGoogle 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).CrossRefGoogle Scholar
- 60.B. A. Kupershmidt and G. Wilson, “Modifying Lax equations and the second Hamiltonian structure,” Invent. Math.,62, 403–436 (1981).CrossRefGoogle Scholar
- 61.A. N. Leznov and M. V. Saveliev, “Representation of zero curvature for the system of nonlinear partial differential equations X
_{α′zz}=exp (KX)_{α}and its integrability,” Lett. Math. Phys.,3, No. 6, 489–494 (1979).CrossRefGoogle Scholar - 62.I. G. Macdonald, G. Segal, and G. Wilson, Kac-Moody Lie Algebras, Oxford Univ. Press (to appear).Google Scholar
- 63.F. Magri, “A simple model of the integrable Hamiltonian equation,” J. Math. Phys.,19, No. 5, 1156–1162 (1978).CrossRefGoogle 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).CrossRefGoogle 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.Google Scholar
- 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).Google Scholar
- 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

## Copyright information

© Plenum Publishing Corporation 1985