Abstract
In this chapter we shall summarize and generalize our experience in describing integrable models gained from the study of particular examples. The principal entities of the inverse scattering method and its Hamiltonian interpretation were the auxiliary linear problem operator L = d/dx − U(x, λ) and the fundamental Poisson brackets for U(x, λ) involving the r-matrix. Similar objects were introduced for lattice models. We will show that these notions have a simple geometric interpretation.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Keywords
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.
References
Adler, M.: On a trace functional for formal pseudodifferential operators and symplectic structure of the Korteweg-de Vries type equations. Invent. Math. 50, 219–248 (1979)
Adler, M., van Moerbeke, P.: Completely integrable systems, Euclidean Lie algebras and curves. Adv. Math. 38, 267–317 (1980)
Bäcklund, A. V.: Zur Theorie der Flächentransformationen. Math. Ann. 19, 387–422 (1882)
Beresin, F. A.: Several remarks on the associative envelope of a Lie algebra. Funk. Anal. Priloi. 1 (2), 1–14 (1967) [Russian]
Bogoyavlensky, O. I.: On perturbations of the periodic Toda lattice. Comm. Math. Phys. 51, 201–209 (1976)
Bogoyavlensky, O. I.: Integrable equations on Lie algebras arising in problems of mathematical physics. Izv. Akad. Nauk SSSR, Ser. Mat. 48, 883–938 (1984) [Russian]
Belavin, A. A., Drinfeld, V. G.: Solutions of the classical Yang-Baxter equation for simple Lie algebras. Funk. Anal. Priloz. 16 (3), 1–29 (1982) [Russian]; English transl. in Funct. Anal. Appl. 16, 159–180 (1982)
Belavin, A. A.: Discrete groups and the integrability of quantum systems. Funk. Anal. Priloz. 14 (4), 18–26 (1980) [Russian]; English transl. in Funct. Anal. Appl. 14, 260–267 (1980)
Bogoyavlensky, O. I., Novikov, S. P.: The connection between the Hamiltonian formalisms of stationary and non-stationary problems. Funk. Anal. Priloz. 10 (1), 9–13 (1976) [Russian]; English transl. in Funct. Anal. Appl. 10, 8–11 (1976)
Cherednik, I. V.: On the definition of i-function for generalized affine Lie algebras. Funk. Anal. Priloz. 17 (3), 93–95 (1983) [Russian]
Dirac, P. A. M.: Lectures on quantum mechanics. Belfer Grad. School of Science, Yeshiva University, N-Y 1964
Drinfeld, V. G.: Hamiltonian structures on Lie groups, Lie bialgebras and the geometrical meaning of classical Yang-Baxter equations. Dokl. Akad. Nauk SSSR 268, 285–287 (1983) [Russian]; English transl. in Sov. Math. Dokl. 27, 68–71 (1983)
Dolan, L.: Kac-Moody algebras and exact solvability in hadronic physics. Phys. Rep. 109 (1), 1–94 (1984)
Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Operator approach to the Kadomtsev-Petviashvili equation. Transformation groups for soliton equations. III. J. Phys. Soc. Japan 50, 3806–3812 (1981)
Dubrovin, B. A., Matveev, V. B., Novikov, S. P.: Nonlinear equations of Korteweg-de Vries type, finite-zone linear operators and abelian varieties. Uspekhi Mat. Nauk 31 (1), 55–136 (1976) [Russian]; English transl. in Russian Math. Surveys 31 (1), 59–146 (1976)
Drinfeld, V. G., Sokolov, V. V.: Equations of Korteweg-de Vries type and simple Lie algebras. Dokl. Akad. Nauk SSSR 258, 11–16 (1981) [Russian]; English transl. in Sov. Math. Dokl. 23, 457–461 (1981)
Drinfeld, V. G., Sokolov, V. V.: Lie algebras and equations of Korteweg-de Vries type. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24,81–180 Moscow, VINITI 1984 [Russian]
Flaschka, H., Newell, A. C., Ratiu, T.: Kac-Moody Lie algebras and soliton equations. II. Lax equations associated with A3“. Physica D 9, 303–323 (1983)
Flaschka, H., Newell, A. C., Ratiu, T.: Kac-Moody Lie algebras and soli-ton equations. III. Stationary equations associated with AY). Physica D9, 324–332 (1983)
Flaschka, H., Newell, A. C., Ratiu, T.: Kac-Moody Lie algebras and soli-ton equations. IV. Physica D9, 333–345 (1983)
Gelfand, I. M., Dikii, L. A.: Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations. Usp. Mat. Nauk 30 (5), 67–100 (1975) [Russian]; English transi. in Russian Math. Surveys 30 (5), 77–113 (1975)
Gelfand, I. M., Dikii, L. A.: A family of Hamiltonian structures connected with integrable nonlinear differential equations. Preprint No 136 Inst. Applied. Math., Moscow 1978
Gelfand, I. M., Dorfman, I. Ya.: Hamiltonian operators and algebraic structures associated with them. Funk. Anal. Priloz. 13 (4), 13–30 (1979) [Russian]; English transi. in Funct. Anal. Appl. 13, 248–262 (1979)
Gelfand, I. M., Dorfman, I. Ya.: Hamiltonian operators and infinite-dimensional Lie algebras. Funk. Anal. Priloz. 15 (3), 23–40 (1981) [Russian]; English transi. in Funct. Anal. Appl. 15, 173–187 (1981)
Gelfand, I. M., Fuks, D. B.: Cohomology of the Lie algebra of vector fields on the circle. Funk. Anal. Priloz. 2 (4), 92–93 (1968) [Russian]
Gerdjikov, V. S., Yanovski, A. B.: Gauge covariant formulation of the generating operator. 1. The Zakharov-Shabat system. Phys. Lett. 103A, 232–236 (1984)
Hirota, R.: Direct method of finding exact solutions of nonlinear evolution equations. In: Miura R. (ed.) Bäcklund transformations. Lecture Notes in Mathematics, Vol. 515. Berlin-Heidelberg-New York, Springer 1976
Kostant, B.: Quantization and unitary representations. I. Prequantization. Lecture Notes in Mathematics, Vol. 170, 87–208. Berlin-Heidelberg-New York, Springer 1970
Kostant, B.: Quantization and representation theory. Proc. of Symposium on Representations of Lie groups. Oxford 1977, London Math. Soc. Lect. Notes Ser. 34, 287–316 (1979)
Kostant, B.: The solution to a generalized Toda lattice and representation theory. Adv. Math. 34, 195–338 (1979)
Kac, V. G.: Infinite dimensional Lie algebras. Progress in Mathematics, v. 44. Boston, Birkhäuser 1983
Kirillov, A. A.: Elements of the Theory of Representations. Moscow: Nauka 1972 [Russian]; English transl. Berlin-Heidelberg-New York, Springer 1976
Kulish, P. P., Reyman, A. G.: A hierarchy of symplectic forms for the Schrödinger and the Dirac equations on the line. In: Problems in quantum field theory and statistical physics. I. Zapiski Nauchn. Semin. LOMI 77, 134–147 (1978) [Russian]; English transi. in J. Soy. Math. 22, 1627–1637 (1983)
Kulish, P. P., Reyman, A. G.: Hamiltonian structure of polynomial bundles. In: Differential geometry, Lie groups and mechanics. V. Zap. Nauchn. Semin. LOMI 123, 67–76 (1983) [Russian]
Lie, S. Dunter Mitwirkung von F. Engel): Theorie der Transformationsgruppen. Bd. 1–3. Leipzig, Teubner, 1888, 1890, 1893
Lebedev, D. R., Manin, Yu. I.: Gelfand-Dikii Hamiltonian operator and coadjoint representation of Volterra group. Funk. Anal. Priloz. 13 (4), 4046 (1979) [Russian]
Miura, R. M.: Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys. 9, 1202–1204 (1968)
Miura, R. (ed.): Bäcklund transformations. Lecture Notes in Mathematics, Vol. 515. Berlin-Heidelberg-New York, Springer 1976
Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19, 1156–1162 (1978)
Mikhailov, A. V.: Integrability of the two-dimensional generalization of the Toda chain. Pisma Zh. Exp. Teor. Fiz. 30, 443–448 (1979) [Russian]; English transi. in Sov. Phys. JETP Letters 30, 414–418 (1979)
Moerbeke, P. van, Mumford, D.: The spectrum of difference operators and algebraic curves. Acta Math. 143, 93–154 (1979)
Mikhailov, A. V., Shabat, A. B.: Integrability conditions for a system of two equations of the form u,=A(u)u xx +F(u, u i ). I. Teor. Mat. Fiz. 62, 163–185 (1985) [Russian]
Novikov, S. P.: The periodic problem for the Korteweg-de Vries equation. Funk. Anal. Priloi. 8 (3), 54–66 (1974) [Russian]; English transi. in Funct. Anal. Appl. 8, 236–246 (1974)
Reyman, A. G.: Integrable Hamiltonian systems connected with graded Lie algebras. In: Differential geometry, Lie groups and mechanics. III. Zap. Nauchn. Semin. LOMI 95, 3–54 (1980) [Russian]; English transi. in J. Sov. Math. 19, 1507–1545 (1982)
Reyman, A. G.: A unified Hamiltonian system on polynomial bundles and the structure of stationary problems. In: Problems in quantum field theory and statistical physics. 4. Zap. Nauchn. Semin. LOMI 131, 118–127 (1983) [Russian]; English transi. in J. Soy. Math. 30, 2319–2325 (1985)
Reshetikhin, N. Yu., Faddeev, L. D.: Hamiltonian structures for integrable models of field theory. Teor. Mat. Fiz. 56, 323–343 (1983) [Russian]; English transi. in Theor. Math. Phys. 56, 847–862 (1983)
Reyman, A. G., Semenov-Tian-Shansky, M. A.: Reduction of Hamiltonian systems, affine Lie algebras and Lax equations. I. Invent. Math. 54, 81–100 (1979)
Reyman, A. G., Semenov-Tian-Shansky, M. A.: Current algebras and nonlinear partial differential equations. Dokl. Akad. Nauk SSSR 251, 1310–1314 (1980) [Russian]; English transi. in Sov. Math. Dokl. 21, 630–634 (1980)
Reyman, A. G., Semenov-Tian-Shansky, M. A.: A family of Hamiltonian structures, a hierarchy of Hamiltonians and reduction for first order matrix differential operators. Funk. Anal. Priloz. 14 (2), 77–78 (1980) [Russian]; English transi. in Funct. Anal. Appl. 14, 146–148 (1980)
Reyman, A. G., Semenov-Tian-Shansky, M. A.: Reduction of Hamiltonian systems, affine Lie algebras and Lax equations. II. Invent. Math. 63, 423–432 (1981)
Reyman, A. G., Semenov-Tian-Shansky, M. A.: Hamiltonian structure of equations of the Kadomtsev-Petviashvili type. In: Differential geometry, Lie groups and mechanics. VI. Zap. Nauchn. Semin. LOMI 133, 212–227 (1984) [Russian]; English transi. in J. Sov. Math. 31, 3399–3410 (1985)
Reyman, A. G., Semenov-Tian-Shansky, M. A.: Lie algebras and Lax equations with spectral parameter on an elliptic curve. In: Problems of quantum field theory and statistical physics 6. Zap. Nauchn. Semin. LOMI 150,104–118 (1986) [Russian]
Reyman, A. G., Semenov-Tian-Shansky, M. A., Frenkel, I. B.: Graded Lie algebras and completely integrable systems. Dokl. Akad. Nauk SSSR 247, 802–805 (1979) [Russian]; English transi. in Soy. Math. Dokl. 20, 811–814 (1979)
Souriau, J.-M.: Structure des systèmes dynamiques. Paris, Dunod 1970
Sklyanin, E. K.: On complete integrability of the Landau-Lifshitz equation. Preprint LOMI E-3–79, Leningrad 1979
Symes, W.: Systems of Toda type, inverse spectral problems and representation theory. Invent. math. 59, 13–53 (1980)
Semenov-Tian-Shansky, M. A.: What is a classical r-matrix. Funk. Anal. Priloz. 17 (4), 17–33 (1983) [Russian]; English transl. in Funct. Anal. Appl. 17, 259–272 (1983)
Semenov-Tian-Shansky, M. A.: Group-theoretic methods in the theory of integrable systems. Doctoral thesis, Leningrad 1985 [Russian]
Semenov-Tian-Shansky, M. A.: Dressing transformations and Poisson group actions. Publ. RIMS 21, 1237–1260 (1985)
Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. IHES 61, 5–65 (1985)
Trofimov, V. V.: Completely integrable geodesic flows of left-invariant metries on Lie groups connected with commutative graded algebras with Poincare duality. Dokl. Akad. Nauk SSSR 263, 812–816 (1982) [Russian]; English transi. in Sov. Math. Dokl. 25, 449–453 (1982)
Takhtajan, L. A.: Solutions of the triangle equations with 7ZNX7ZN symmetry as matrix analogues of the Weierstrass zeta and sigma functions. In: Differential geometry, Lie groups and mechanics. VI. Zap. Nauchn. Semin. LOMI 133,258–276 (1984) [Russian]
Weinstein, A.: The local structure of Poisson manifolds. J. Diff. Geometry 18, 523–557 (1983)
Zakharov, V. E., Manakov, S. V., Novikov, S. P., Pitaievski, L. P.: Theory of Solitons. The Inverse Problem Method. Moscow, Nauka 1980 [Russian]; English transl.: New York, Plenum 1984
Zakharov, V. E., Shabat, A. B.: Integration of the nonlinear equations of mathematical physics by the method of the inverse scattering problem. II. Funk. Anal. Priloz. 13 (3), 13–22 (1979) [Russian]; English transl. in Funct. Anal. Appl. 13, 166–174 (1979)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Faddeev, L.D., Takhtajan, L.A. (2007). Lie-Algebraic Approach to the Classification and Analysis of Integrable Models. In: Hamiltonian Methods in the Theory of Solitons. Springer Series in Soviet Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69969-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-69969-9_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69843-2
Online ISBN: 978-3-540-69969-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)