zbMATH — the first resource for mathematics

Lie algebras and equations of Korteweg-de Vries type. (English) Zbl 0578.58040
Translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 81–180 (Russian) (1984; Zbl 0558.58027).

37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
35Q53 KdV equations (Korteweg-de Vries equations)
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI
[1] V. I. Arnol’d, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1979).
[2] N. Bourbaki, Lie Groups and Algebras [Russian translation], Mir, Moscow (1976).
[3] N. Bourbaki, Lie Groups and Algebras. Lie Algebras. Free Lie Algebras [Russian translation], Mir, Moscow (1972).
[4] N. Bourbaki, Lie Groups and Algebras. Cartan Subalgebras, Regular Elements, Decomposable Semisimple Lie Algebras [Russian translation], Mir, Moscow (1978).
[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).
[6] I. M. Gel’fand and L. A. Dikii, ”The resolvent and Hamiltonian systems,” Funkts. Anal. Prilozhen.,11, No. 2, 11–27 (1977).
[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).
[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).
[9] I. M. Gel’fand and L. A. Dikii, ”The Schoutten bracket and Hamiltonian operators,” Funkts. Anal. Prilozhen.,14, No. 3, 71–74 (1980). · Zbl 0648.47035 · doi:10.1007/BF01078432
[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).
[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). · Zbl 0498.32010 · doi:10.1007/BF01081801
[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).
[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).
[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).
[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).
[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). · Zbl 0598.35002
[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).
[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). · doi:10.1007/BF01030253
[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). · Zbl 0301.54027 · doi:10.1007/BF02028308
[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).
[21] V. G. Kats, ”Simple irreducible graded Lie algebras of finite growth,” Izv. Akad. Nauk SSSR, Ser. Mat.,32, No. 6, 1323–1367 (1968).
[22] V. G. Kats, ”Infinite Lie algebras and the Dedekind \(\eta\)-function,” Funkts. Anal. Prilozhen.,8, No. 2, 77–78 (1974). · Zbl 0298.57019 · doi:10.1007/BF02028317
[23] V. G. Kats, ”Automorphisms of finite order of semisimple Lie algebras,” Funkts. Anal. Prilozhen.,3 No. 3, 94–96 (1969).
[24] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill (1955). · Zbl 0064.33002
[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). · Zbl 0346.35028
[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).
[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). · Zbl 0445.35032 · doi:10.1007/BF01018624
[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).
[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). · Zbl 0536.58014 · doi:10.1007/BF01037979
[31] I. G. Macdonald, ”Affine root systems and the Dedekind \(\eta\)-function,” Matematika,16, No. 4, 3–49 (1972).
[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). · Zbl 0412.35062 · doi:10.1007/BF01029027
[33] S. V. Manakov, ”On complete integrability and stochastization in discrete dynamical systems,” Zh. Eksp. Teor. Fiz.,67, 543–555 (1974).
[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.
[35] Yu. I. Manin, ”Matrix solitons and bundles over curves with singularities,” Funkts. Anal. Prilozhen.,12, No. 4, 53–67 (1978).
[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).
[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.
[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).
[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).
[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). · Zbl 0447.52011 · doi:10.1007/BF01086547
[43] I. V. Cherednik, ”Differential equations for Baker-Akhiezer functions of algebraic curves,” Funkts. Anal. Prilozhen.,12, No. 3, 45–54 (1978). · Zbl 0385.35019
[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). · Zbl 0393.35058 · doi:10.1007/BF01410079
[45] O. I. Bogojavlensky, ”On perturbations of the periodic Toda lattice,” Commun. Math. Phys.,51, 201–209 (1976). · doi:10.1007/BF01617919
[46] S. A. Bulgadaev, ”Two-dimensional integrable field theories connected with simple Lie algebras,” Phys. Lett.,96B, 151–153 (1980). · doi:10.1016/0370-2693(80)90233-6
[47] E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, ”Transformation groups for soliton equations,” Preprint RIMS-394, Kyoto Univ. (1982). · Zbl 0571.35103
[48] E. Date, M. Kashiwara, and T. Miwa, ”Vertex operators and \(\tau\)-functions. Transformation groups for soliton equations. II,” Proc. Jpn. Acad.,A57, No. 8, 387–392 (1981). · Zbl 0538.35066 · doi:10.3792/pjaa.57.387
[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). · Zbl 0376.35009 · doi:10.1098/rspa.1978.0011
[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). · Zbl 0942.37504 · doi:10.1103/PhysRevB.9.1924
[53] H. Flaschka, ”Toda lattice II,” Progr. Theor. Phys.,51, 703–716 (1974). · Zbl 0942.37505 · doi:10.1143/PTP.51.703
[54] A. P. Fordy and J. D. Gibbons, ”Factorization of operators. I. Miura transformations,” J. Math. Phys.,21, 2508–2510 (1980). · Zbl 0456.35079 · doi:10.1063/1.524357
[55] A. P. Fordy and J. D. Gibbons, ”Integrable nonlinear Klein-Gordon equations and Toda lattice,” Commun. Math. Phys.,77, 21–30 (1980). · Zbl 0456.35080 · doi:10.1007/BF01205037
[56] I. B. Frenkel and V. G. Kac, ”Basic representation of affine Lie algebras and dual resonance models,” Invent. Math.,62, 23–66 (1980). · Zbl 0493.17010 · doi:10.1007/BF01391662
[57] V. G. Kac, ”Infinite-dimensional algebras, Dedekind’s \(\eta\)-functions, classical Möbius function and the very strange formula,” Adv. Math.,30, No. 2, 85–136 (1978). · Zbl 0391.17010 · doi:10.1016/0001-8708(78)90033-6
[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). · Zbl 0099.25603 · doi:10.2307/2372999
[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). · Zbl 0487.35078 · doi:10.1007/BF01208894
[60] B. A. Kupershmidt and G. Wilson, ”Modifying Lax equations and the second Hamiltonian structure,” Invent. Math.,62, 403–436 (1981). · Zbl 0464.35024 · doi:10.1007/BF01394252
[61] A. N. Leznov and M. V. Saveliev, ”Representation of zero curvature for the system of nonlinear partial differential equations X\(\alpha\)’zz=exp (KX)\(\alpha\) and its integrability,” Lett. Math. Phys.,3, No. 6, 489–494 (1979). · Zbl 0415.35017 · doi:10.1007/BF00401930
[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). · Zbl 0383.35065 · doi:10.1063/1.523777
[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).
[65] A. V. Mikhailov, M. A. Olshanetsky, and A. M. Perelomov, ”Two-dimensional generalized Toda lattice,” Commun. Math. Phys.,79, 473–488 (1981). · Zbl 0491.35061 · doi:10.1007/BF01209308
[66] R. V. Moody, ”Macdonald identities and Euclidean Lie algebras,” Proc. Am. Math. Soc.,48, No. 1, 43–52 (1975). · Zbl 0315.17003 · doi:10.1090/S0002-9939-1975-0442048-2
[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.
[70] G. Wilson, ”Commuting flows and conservation laws for Lax equations,” Math. Proc. Cambr. Philos. Soc.,86, No. 1, 131–143 (1979). · Zbl 0427.35024 · doi:10.1017/S0305004100000700
[71] G. Wilson, ”On two constructions of conservation laws for Lax equations,” Q. J. Math. Oxford,32, No. 128, 491–512 (1981). · Zbl 0477.35079 · doi:10.1093/qmath/32.4.491
[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). · Zbl 0495.58008 · doi:10.1017/S0143385700001292
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.