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Integrable Systems.I

  • Chapter
Dynamical Systems IV

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 4))

Abstract

Integrable systems which do not have an “obvious“ group symmetry, beginning with the results of Poincaré and Bruns at the end of the last century, have been perceived as something exotic. The very insignificant list of such examples practically did not change until the 1960’s. Although a number of fundamental methods of mathematical physics were based essentially on the perturbation-theory analysis of the simplest integrable examples, ideas about the structure of nontrivial integrable systems did not exert any real influence on the development of physics.

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References

List of Additional References

  1. Aharonov A., Casher A.: Ground state of a spin-1/2 charged particle in a two-dimensional mag-netic field. Phys. Rev. A (3) 19, 2461–2462 (1979).

    Article  MathSciNet  Google Scholar 

  2. Ablowitz M., Benney D.: The evolution of multiple modes for nonlinear dispersive waves.Studies Appl. Math. 49, No 3, 225–238 (1970).

    MATH  Google Scholar 

  3. Avilov V.V., Novikov S.P.: Evolution of Whitham zone in the KdV theory. Dokl. AN SSSR (English translation: Sov. Phys.Dokl. 32,366–368 (1987)).

    Google Scholar 

  4. Avilov V.V., Krichever I.M., Novikov S.P.: Evolution of Whitham zone in the Korteweg-de Vries Theory. Dokl. AN SSSR 295, 345–349 (1987). English translation; Sov. Phys. Dokl. 31, 564–566, (1987)). Zb1. 655. 65132

    Google Scholar 

  5. Balinskij A.A., Novikov S.P., Poisson brackets of hydrodynamic type, Frobenious algebras and Lie algebras. Dokl. AN SSSR 283, 1036–1039 (1985)English translation: Sov.Math.Dokl. 32, 228–231 (1985)). Zb1. 606. 58018

    Google Scholar 

  6. Bobenko A.1.: Surfaces of constant mean curvature and integrable equations. Usp. Mat. Nauk 46,No. 4, 3–42 (1991) (English translation: Russian Math Surveys 46,No. 4, 1–45 (1991)).

    Google Scholar 

  7. Bogdanov L.V.: Novikov-Veselov equation as a natural twp-dimensional generalization of the Korteweg-de Vries equation. Teor.Mat.Fiz 70, No. 2, 309–314 (1987)English translation: Theor. Math. Phys. 70, 219–223 (1987)). Zb1. 639. 35072

    Google Scholar 

  8. Bogoyavlenskij 0.1.: On perturbations of periodic Toda lattice. Commentarii Math. Phys. 51, 201–209 (1976).

    Google Scholar 

  9. DeVega H.J., Shaposnik F.A.: Classical vortex solution of the Abelian Higgs model. Phys. Rev.D 14, 1100–1106 (1976).

    Google Scholar 

  10. Drinfeld V.G., Sokolov V.V.: Lie Algebras and equations of the Korteweg-de Vries type. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. (Contemporary problems in Mathematics)24. VINITI, Moscow 1984 English translation: J. Sov. Math. 30, No. 2, 1975–2036 (1985)).

    Google Scholar 

  11. Doyle P.W.: Differential geometric Poisson bivectors in one space variable. J. Math. Phys. 34, No. 4, 1314–1338 (1993). Zb1. 778. 58022

    Google Scholar 

  12. Dubrovin B.A.: Lectures on 2D Topological Field Theory, Lecture Notes in Math, 1620,120–348 (1996), Springer-Verlag.

    Google Scholar 

  13. Dubrovin B.A.: On the differentially geometric Poisson brackets on the lattice. Funkts. Anal. Prilozh 23, No. 2, 57–59 (1989).

    MathSciNet  Google Scholar 

  14. Dubrovin B.A., Novikov S.P.: Hydrodynamics of weakly deformed soliton lattices, Poisson brackets of Hydrodynamic type and Differential Geometry, lisp. Mat. Nauk 44, No. 6, 29–98 (1989)English translation: Russian Math Surveys 44, No. 6, 35–124 (1989)). Zb1. 712. 58032

    Google Scholar 

  15. DN2] Dubrovin B.A., Novikov S.P.: Ground states in a periodic field. Magnetoc Bloch functions and vector bundles. Dokl. AN SSSR 253,1293–1297 (1980) (English translation: Sov. Math. Dokl. 21,240–244 (1980)).

    Google Scholar 

  16. DN3] Dubrovin B.A., Novikov S.P.: Ground states of two-dimensional electron in a periodic magnetic field. Zh. Eksp. Teor. Fiz. 79,1006–1016 (1980) (English translation: Sov. Phys. JETP 52,511–516 (1980)).

    Google Scholar 

  17. Françoise J.-P., Novikov R.G.: Rational solutions of KdV type equations in dimension 2+1 and m-body problems on the line. C. R. Acad. Sci. Paris 314, Série I, 109–113, 1992.

    Google Scholar 

  18. Faddeev L.D., Takhtadzhyan A.L.: Hamiltonian Methods in the Theory of Solitons. Nauka

    Google Scholar 

  19. Moscow, 1986 (English translation: Springer-Verlag, Berlin, Heidelberg, New York, 1987). Fateev V.A., Lukyanov S.L.: Poisson-Lie groups and classical W-algebras. Internat. J. Modern Phys. A 7, No. 5, 853–876 (1992).

    Google Scholar 

  20. Feldman J., Knorrer H., Trubowitz E.: Riemann surfaces of the infinite genus, preprint ETH Zurich.

    Google Scholar 

  21. Ferapontov E.V.: Nonlocal hamiltonian operators of hydrodynamic type. Differential geometry and applications, Amer. Math. Soc. Translations, Ser. 2, 170,33–58 (1996). (ed by S.Novikov)

    Google Scholar 

  22. Flaschka H., Forrest M.G., and McLaughlin D.: Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation. Comm. Pure Appl. Math. 33, 739–784 (1980).

    MATH  Google Scholar 

  23. Gelfand I.M., Dikii L.A.: Fractional powers of operators and Hamiltonian systems. Funkts. Anal. Prilozh. 10,No. 4, 13–28 (1976) (English translation: Funct. Anal. Appl. 10,259–274 (1976)).

    Google Scholar 

  24. Gelfand I.M., Dorfman I.Ya.: Hamiltonian operators and infinite dimensional Lie algebras. Funkts. Anal. Prilozh. /5, No. 3, 23–40 (1981)English translation: Funct. Anal. Appl. 15, No. 3, 173–187 (1981).

    Google Scholar 

  25. Gelfand I.M., Fuks D.B.: Cohomology of Lie algebra of vector fields on the circle. Funkts. Anal. Prilozh 2, No. 4, 92–93 (1968).

    MathSciNet  Google Scholar 

  26. Grinevich P.G.: Rational solutions of the Veselov-Novikov equations are reflectinless two- dimensional potentials at fixed energy. Teor. Mat. Fiz. 69, No. 2, 307–310 (1986)English translation: Theor. Math. Phys. 69, 1170–1172 (1986)). Zb1. 617. 35121

    Google Scholar 

  27. English translation: Funct. Anal. Appl. 20, 94–103 (1986)). Zb1. 617. 35121

    Google Scholar 

  28. Grinevich P.G., Novikov R.G.: Analog of multisoliton potentials for the two-dimensional Schrödinger operator. Funkts. Anal. Prilozh. 19, No. 4, 32–42 (1985)English translation: Funct. Anal. Appl. 19, 276–285 (1985)). Zb1. 657. 35122

    Google Scholar 

  29. Grinevich P.G., Novikov S.P.: Spectral theory of commuting operators of rank two with periodic coefficients. Funkts. Anal. Prilozh. 16, No. 1, 25–26 (1982)English translation: Funct. Anal. Appl. 16, 19–20 (1982)). Zb1. 514. 47035

    Google Scholar 

  30. Grinevich P.G., Novikov S.P.: “Inverse scattering problem” for the two dimensional Schrödinger operator at a fixed negative enery and generalized analytic finctions. Funct. Anal. Appl. 22, No. 1, 19–27 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gurevich A. V., Pitaevskii L.P.: Nonstationary structure of the collisionless shock wave, Zh.Ehksp. Teor. Fiz. 65, No. 2, 590–604 (1973).

    Google Scholar 

  32. Krichever I.M.: The potentials with zero reflection coefficient on the background of the finite-gap potentials. Funkts. Anal. Prilozh. 9, No. 2, 77–78 (1975)English translation: Funct. Anal. Appl. 9, 161–163 (1975)). Zb1. 333. 34022

    Google Scholar 

  33. Krichever I.M.: The periodic nonabelian Toda lattice and two-dimensional generalization. Usp. Mat. Nauk 36,No. 2, 72–77 (1981) (English translation: Russian Math. Surveys 36,No. 2, ?? (1981))

    Google Scholar 

  34. Krichever I.M.: Spectral theory of two-dimensional periodic operators and its applications. Usp. Mat. Nauk 44, No. 2, 121–184 (1989).English translation: Russ. Math. Surv. 44, No. 2, 145–25 (1989)). Zb1. 699. 35188

    Google Scholar 

  35. Krichever I.M.: Method of averaging for two-dimensional integrable equations. Funkts. Anal. Prilozh. 22, No. 3, 37–52 (1988)English translation: Funct. Anal. Appl. 22, No. 3, 200–213 (1988)). Zb1. 688. 35088

    Google Scholar 

  36. Krichever I.M.: The dispersionless Lax equations and topological minimal models. Commun. Math. Phys 143, 415–429 (1992). Zb1. 753. 35095

    Google Scholar 

  37. Krichever I.M.: The Tau-function of the universal Whitham hierarchy, matrix modles and topological field theories. Commun. Pure Appl. Math. 47, 437–475 (1994). Zb1. 811. 58064

    Google Scholar 

  38. Krichever l.M.: Two-dimensional periodic difference operators and algebraic geometry, Dokl. Akad. Nauk SSSR 285, No. I, 31–36 (1985).

    Google Scholar 

  39. Krichever I., Babelon 0., Billey E., Talon M.: Spin generalization of the Calogero-Moser system and the Matrix KP equation, Amer. Math. Soc. Transi. (Ser. 2) 170, 83–119 (1995). Zb1. 843. 58069

    Google Scholar 

  40. Krichever I., Lipan O., Wiegmann P., Zabrodin A.: Quantum integrable systems and discrete classical Hirota equations. Commun. Math. Phys. 188, 267–304 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  41. Krichever I.M., Novikov S.P.: Virasoro-type algebras, Riemann surfaces and soliton theory. Funkts. Anal. Prilozh. 21, No. 2, 46–63 (1987).

    MathSciNet  Google Scholar 

  42. English translation: Funct. Anal. Appl. 21, No. 4, 294–307 (1987)). Zb1. 659. 17012

    Google Scholar 

  43. Krichever I.M., Phong D.H.: On the integrable geometry of soliton equations and N = 2 su-persymmetric gauge theories. J. Diff. Geom. 45, No. 2, 349–389 (1997). Zb1.970. 4860 I

    Google Scholar 

  44. English translation: Russ. Math. Surv. 50, No. 6, 1101–1150 (1995)). Zb1. 960. 43357

    Google Scholar 

  45. Knörrer H., Trubowitz E.: A directional compactification of the complex Bloch variety. Com- ment. Math. Helvetici 65, 114–149 (1990).

    Article  MATH  Google Scholar 

  46. Laplace P.: Recherches sur le calcul integral aux differences partielles. Histore de l’Academie de Sciences (1977).

    Google Scholar 

  47. Leznov A., Saveliev M.: Representation of zero curvature for the system of nonlinear partial differential equation Xa.:z = exp 1K X)„. Lett. Math. Phys. 3, No. 6, 489–494 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  48. Lax P., Levermore C.: The small dispersion limit of the Korteweg-de-Vries equation, I, II, III. Commun. Pure Appl. Math. 36, 253–290, 571–593, 809–830 (1983)

    Article  MathSciNet  Google Scholar 

  49. Marchenko V.A., Ostrovski 1.: see in the book Marchenko V.A.: Sturm-Liouville operator and applications. Kiev, Naukova Dumka (1977) (in Russian)

    Google Scholar 

  50. Manakov S.V.: The method of inverse scattering problem and two-dimensional evolution equa-tions. Usp. Mat. Nauk 31, No. 5, 245–246 (1976).

    MathSciNet  MATH  Google Scholar 

  51. Maslov V.P.: Limit for h. 0 of Heisenberg equation. Teor. Mat. Fiz. 1, No. 3, 378–383 (1969).

    Article  Google Scholar 

  52. Mikhailov A.V.: On the integrability of the two-dimensional generalization of Toda lattices.Zh. Ehksp. Teor. Fiz. 30, 443–448 (1979) ( English translation: Sov. Phys. JETP )

    Google Scholar 

  53. Mokhov O.I.: On the Poisson brackets of the Dubrovin-Novikov type (DN-brackets), Funkts. Anal. Prilozh. 12,No. 4, 92–93 (1988) (English translation: Funct. Anal. Appl. 22,336–337 (1988))

    Google Scholar 

  54. Mokhov 0.1., Ferapontov E.V.: Nonlocal hamiltonian operators of hydrodynamic type asso-ciated with constant curvature metrics. Usp. Mat. Nauk 45, No. 3, 191–192 (1990) (English translation: Russian Math Surveys 45,No. 3, 218–219 (1990))

    Google Scholar 

  55. Maltsev A.: Averaged Poisson brackets, preprint Landau Intitute, Chemogolovka, 1997.

    Google Scholar 

  56. Natanzon S.M.: Differential equation for Prym fuction. A criterion for two-dimensional finite-gap purely potential Schrödinger operator to be real. Funct. Anal. Appl. 26, 13–20 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  57. Natanzon S.M.: Real nonsingular finite-zone solutions of soliton Equations, AMS Transi. (ser.2) 170,153–184 (1996) (ed. S. Novikov).

    Google Scholar 

  58. Novikov S.P.: Periodic and conditionaly periodic analogues of the many-soliton solutions of the Korteweg-de Vries equation. Zh. Ekspr. Teor. Fiz. 67,2131–2144 (1974) (English translation: Sov, Phys. JETP 40,1055–1063 (1974)).

    Google Scholar 

  59. Novikov S.P.: Topology. Itogi Nauki Tekh., Socr. Probl. Mat., Fund. Napr. 12, VINITI, Mos-cow 1986 (English translation: Encyclopaedia Math. Sciences, Vol. 12, Springer-Verlag, 1996 )

    Google Scholar 

  60. English translation: Russ. Math. Surv. 40, No. 4, 85–98 (1985)) Zb1. 654. 76004

    Google Scholar 

  61. Novikov S.P.: Magnetic Bloch functions and vector bundles. Typical dispertion relations and their quantum numbers. Sov. Math. Dokl. 23, No. 2, 298–303 (1981).

    MATH  Google Scholar 

  62. Novikov S.P.: Solitons and Geometry, Academia dei Lincei, Scuola Normale Superiore, Lezzio-ni Fermiane, Pisa 1992 (published in 1994 )

    Google Scholar 

  63. Novikov S.P., Maltsev A.: Liouville form of the averaged Poisson brackets. Usp. Mat. Nauk 48, No. 1, 155–156 (1993).

    MathSciNet  MATH  Google Scholar 

  64. Novikov S.P., Veselov A.P.: Laplace transformations and exactly solvable two-dimensional Schrödinger operators. Usp. Mat. Nauk 50, No. 6, 71–72 (1995) ( English translation: Russian Math Surveys - short notes )

    Google Scholar 

  65. Potemin G.: On Poisson brackets of differential geometric type. Dokl. AN SSSR 286, No. I, 39–42 (1986)English translation: Sov. Math. Dokl. 33, 30–33 (1986)). Zb1. 606. 53010

    Google Scholar 

  66. Potemin G.: Algebro-geometrical construction of self-similar solution to Whitham equation, Usp. Mat. Nauk 43, No. 5, 211–212 (1988) (English translation: Russian Math. Surv. 43, No. 5, 252–253 (1988))

    Google Scholar 

  67. Shiota T.: Characterization of Jacobian varieties in terms of soliton equations, Invent. Math. 83, 333–382 (1986). Zb1. 621. 35097

    Google Scholar 

  68. Shabat A.B., Sokolov V.V.: (L, A)pairs and Ricatti type transformations. Funkts. Anal. Pri-lozh 14, No. 2, 79–80 (1980).

    Google Scholar 

  69. Shabat A.B., Veselov A.P.: A dressing chain and the spectral theory of the Schrödinger opera-tor. Funkts. Anal. Prilozh. 27, No. 2, 1–21 (1993).

    MathSciNet  Google Scholar 

  70. Seiberg N, Witten E.: Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nuclear Phys. B 426, 19–52 (1994).

    Google Scholar 

  71. Seiberg N, Witten E.: Monopoles, duality, and chiral symmetry breaking in N = 2 supersym-metric QCD. Nuclear Phys. B 431, 484–550 (1994).

    Google Scholar 

  72. Tian F.R.: The initial value problem for the Whitham averaged system. Commun. Math. Phys. 79–115 (1994).

    Google Scholar 

  73. Taimanov I.A.: Prym theta functions and hierarhies of nonlinear equations, Mat. Zametki, 50(1991)English translation: Math. Notes 50, No. I, 723–730 (1991).

    Google Scholar 

  74. Taimanov 1.A.: Modified Novikov-Veselov equation and differential geometry of surfaces, Amer. Math. Soc. translations, Ser. 2, 179, 133–151 (1997).

    Google Scholar 

  75. Tsarev S.P.: Poisson brackets and one-dimensional systems of hydrodynamic type. Sov. Math.Dokl. 34, 534–537 (1987).

    Google Scholar 

  76. Tsarev S.P.: Geometry of the Hydrodynamic type systems. Generalized Hodpgraph method. Izvest. AN SSSR (ser. mat.) 54, No. 5, 1048–1068 (1990).

    Google Scholar 

  77. Tsiseika G.: Geometrie Differentielle projective des reseans. Paris-Budapest, 1924.

    Google Scholar 

  78. Weiss J.: Periodic fixed points of Backlund transformations and the KdV equation. J. Math.Phys. 27, 2647–2676 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  79. Whitham G.B.: Non-linear dispersive waves. Proc Royal Society A 283,238–261 (1965). [Wh2] Whitham G.B.: Linear and nonliear waves, John Wiley Interscience Publ., New York, 1977 [YaR] Yanenko N., Rozhdestvenskij B.: Systems of quasilinear equations and their applications in gas dynamics. Moscow, Nauka, 1986 (second edition)

    Google Scholar 

  80. Zehnanov E.I.: On a class of local translation invariant Lie algebras. Dokl. AN SSSR 292, 1294-1297 (1987) English translation: Sov. Math. Doklady 35, 216–218 (1987)). Zb1.629.17002 Integrable Systems I 323

    Google Scholar 

References

  1. Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Method for solving the sine-Gordon equation. Phys. Rev. Lett. 30, 1262–1264 (1973).

    Article  MathSciNet  Google Scholar 

  2. Adler, M.: On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg—Devries type equations. Invent. Math. 50, 219–248 (1979). Zbl. 393. 35058

    Google Scholar 

  3. Adler, M., van Moerbeke, P.: Completely integrable systems, Euclidean Lie algebras and curves. Adv. Math. 38, 267–317 (1980). Zbl. 455. 58017

    Google Scholar 

  4. Airault, H., McKean, H.P., Moser, J.: Rational and elliptic solutions of the Korteweg—de Vries equation and a related many-body problem. Commun. Pure Appl. Math. 30, 95–148 (1977). Zbl. 338. 35024

    Google Scholar 

  5. Akhiezer, N.I.: A continuous analogue of orthogonal polynomials on a system of intervals. Dokl. Akad. Nauk SSSR 141, 263–266 (1961) (English translation: Sov. Math., Dokl. 2, 1409–1412 (1961)). Zbl. 109. 296

    Google Scholar 

  6. Al’ber, S.I.: On stationary problems for equations of Korteweg—de Vries type. Commun. Pure Appl. Math. 34, 259–272 (1981). Zbl. 456. 35075

    Google Scholar 

  7. Arnol’d, V.I.: Mathematical Methods of Classical Mechanics. Nauka, Moscow 1974, 431 p. (English translation: Graduate Texts in Mathematics 60. Springer-Verlag, New York—Heidelberg—Berlin 1978, 462 p.). Zbl. 386. 70001

    Google Scholar 

  8. Baker, H.F.: Abelian Functions (the full title is: Abel’s Theorem and the Allied Theory Including the Theory of the Theta Functions). Cambridge University Press, Cambridge 1897, 684 p. FdM 28, 331

    Google Scholar 

  9. Baker, H.F.: Note on the foregoing paper “Commutative ordinary differential operators,” by J.L. Burchnall and J.W. Chaundy. Proc. R. Soc. Lond., Ser. A 118, 584–593 (1928). FdM. 54, 439

    Google Scholar 

  10. Bar’yakhtar, V.G., Belokolos, E.D., Golod, P.I.: One-dimensional magnetic structures and the higher-order Landau—Lifshitz equations (in Russian). Prepr. ITF-84–128 R, Kiev 1984, 28 p.

    Google Scholar 

  11. Bateman, H. (Erdélyi, A., editor): Higher transcendental functions (in three volumes). McGraw-Hill Book Company, New York—Toronto—London 1953/1953/1955, 302/396/292 p. Zbl. 51,303 Zbl. 52,295 Zbl. 64, 63

    Google Scholar 

  12. Belokolos, E.D.: Peierls—Fröhlich problem and potentials with finite number of gaps. I, II. I: Teor. Mat. Fiz. 45,No. 2, 268–275 (1980); II: Teor. Mat. Fiz. 48,No. 1, 60–69 (1981) (English translation: I: Theor. Math. Phys. 45,1022–1026 (1980); II: Theor. Math. Phys. 48,604–6610 (1981))

    Google Scholar 

  13. Belokolos, E.D., Ehnol’skij, V.Z.: Generalized Lamb ansatz. Teor. Mat. Fiz. 53, No. 2, 271–282 (1982) (English translation: Theor. Math. Phys. 53, 1120–1127 (1982)). Zbl. 507. 35060

    Google Scholar 

  14. Berezin, F.A.: Some remarks on the universal enveloping algebra of a Lie algebra. Funkts. Anal. Prilozh. /, No. 2, 1–14 (1967) (English translation under the title: Some remarks about the associated envelope of a Lie algebra. Funct. Anal. Appl. 1, 91–102 (1967)). Zbl. 227. 22020

    Google Scholar 

  15. Bobenko, A.I.: Euler equations in the algebras e(3) and so(4). Isomorphisms of integrable cases. Funkts. Anal. Prilozh. 20, No. I, 64–66 (1986) (English translation: Funct. Anal. Appl. 20, 53–56 (1986)). Zbl. 622. 58010

    Google Scholar 

  16. Bobenko, A.I., Bikbaev, R.F.: On finite gap integration of the Landau—Lifshitz equation. The XYZ case (in Russian). Prepr. LOMI, E-8–83, Leningrad 1983, 27 p.

    Google Scholar 

  17. Bogolyubov (Bogoliubov), N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields, Third Edition. Nauka, Moscow 1976, 479 p. (English translation: Wiley—Interscience, John Wiley & Sons, New York—Chichester—Brisbane—Toronto 1980, 620 p.)

    Google Scholar 

  18. Bogoyavlenskij, O.I.: Integrals of higher-order stationary KdV equations and eigenvalues of the Hill operator. Funkts. Anal. Prilozh. 10, No. 2, 9–12 (1976) (English translation: Funct. Anal. Appl. 10, 92–95 (1976)). Zbl. 359. 34023

    Google Scholar 

  19. Bogoyavlenskij, O.I.: Integrable Euler equations on Lie algebras arising in problems of mathematical physics. Izv. Akad. Nauk SSSR, Ser. Mat. 48, 883–938 (1984) (English translation: Math. USSR, Izv. 25, 207–257 (1985)). Zbl. 583. 58012

    Google Scholar 

  20. Bogoyavlenskij, O.I., Novikov, S.P.: The relationship between Hamiltonian formalisms of stationary and non-stationary problems. Funkts. Anal. Prilozh. 10, No. 1, 9–13 (1976) (English translation: Funct. Anal. Appl. 10, 8–11 (1976)). Zbl. 345. 70011

    Google Scholar 

  21. Bogoyavlensky, O.I.: On perturbations of the periodic Toda lattice. Commun. Math. Phys. 51, 201–209 (1976)

    Article  MathSciNet  Google Scholar 

  22. Borovik, A.E.: N-soliton solutions of the nonlinear Landau-Lifshitz equation. Pis’ma Zh. Ehksp. Teor. Fiz. 28,629–632 (1978) (English translation: JETP Lett. 28,581–584 (1978))

    Google Scholar 

  23. Brazovskij, S.A., Dzyaloshinskij, I.E., Krichever, I.M.: Discrete Peierls models with exact solutions. Zh. Ehksp. Teor. Fiz. 83,389–415 (1982) (English translation: Sov. Phys.-JETP 56,212–225 (1982))

    Google Scholar 

  24. Brazovskij, S.A., Gordyunin, S.A., Kirova, N.N.: An exact solution of the Peierls model with an arbitrary number of electrons in the unit cell. Pis’ma Zh. Ehksp. Teor. Fiz. 31,486–491 (1980) (English translation: JETP Lett. 31,456–461 (1980))

    Google Scholar 

  25. Bullough, R.K., Caudrey, P.J. (editors): Solitons. Topics in Current Physics 17. Springer-Verlag, Berlin-Heidelberg-New York 1980, 389 p. Zbl. 428. 00010

    Google Scholar 

  26. Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators. I, II. I: Proc. Lond. Math. Soc., II. Ser. 21,420–440 (1923); II: Proc. R. Soc. Lond., Ser. A 118, 557–583 (1928). FdM 49,311 FdM 54, 439

    Google Scholar 

  27. Burgatti, P.: Determinazione dell’equazioni di Hamilton-Jacobi integrabili mediante la separazione delle variabili. Rend. R. Accad. Lincei 20, 108–111 (1° semestre 1911 )

    Google Scholar 

  28. Cherednik, I.V.: Reality conditions in “finite-zone integration”. Dokl. Akad. Nauk SSSR 252, 1104–1108 (1980) (English translation: Sov. Phys., Dokl. 25, 450–452 (1980)). Zbl. 491. 35044

    Google Scholar 

  29. Cherednik, I.V.: Integrability of the equation of a two-dimensional asymmetric chiral O(3)-field and of its quantum analog. Yad. Fiz. 33,278–282 (1981) (English translation: Sov. J. Nucl. Phys. 33,144–145 (1981))

    Google Scholar 

  30. Cherednik, I.V., Gel’fand, I.M.: The abstract Hamiltonian formalism for the classical Yang-Baxter families. Usp. Mat. Nauk 38, No. 3, 3–21 (1983) (English translation: Russ. Math. Surv. 38, No. 3, 1–22 (1983) (The title of the English translation speaks of Yang-Baxter bundles, translating the corresponding Russian word literally but inappropriately; what is being referred to is in fact a parametrized family of functions. Trans!. note.)). Zbl. 536. 58006

    Google Scholar 

  31. Choodnovsky, D.V., Choodnovsky, G.V.: Pole expansions of nonlinear partial differential equations. Nuovo Cimento B 40, 339–353 (1977)

    Article  MathSciNet  Google Scholar 

  32. Dall’Acqua, F.A.: Sulla integrazione delle equazioni di Hamilton-Jacobi per separazione di variabili. Math. Ann. 66, 398–415 (1909)

    Article  MATH  Google Scholar 

  33. Date, E., Tanaka, S.: Analogue of inverse scattering theory for the discrete Hill’s equation and exact solutions for the periodic Toda lattice. Prog. Theor. Phys. 55, 457–465 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  34. Dikij, L.A., Gel’fand, I.M.: Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations. Usp. Mat. Nauk 30, No. 5, 67–100 (1975) (English translation: Russ. Math. Surv. 30, No. 5, 77–113 (1975)). Zbl. 334. 58007

    Google Scholar 

  35. Drinfel’d, V.G.: On commutative subrings of certain noncommutative rings. Funkts. Anal. Prilozh. 11, No. 1, 11–14 (1977) (English translation: Funct. Anal. Appl. 11, 9–12 (1977)). Zbl. 359. 14011

    Google Scholar 

  36. Drinfel’d, V.G.: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations. Dokl. Akad. Nauk SSSR 268, 285–287 (1983) (English translation: Soy. Math., Dokl. 27, 68–71 (1983)). Zbl. 526. 58017

    Google Scholar 

  37. Dryuma, V.S.: Analytic solution of the two-dimensional Korteweg-de Vries (KdV) equation. Zh. Ehksp. Teor. Fiz. Pis’ma 19,753–755 (1974) (English translation: JETP Lett. 19,387–388 (1974))

    Google Scholar 

  38. Dubrovin, B.A.: Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials. Funkts. Anal. Prilozh. 9, No. 3, 41–51 (1975) (English translation: Funct. Anal. Appl. 9, 215–223 (1975)). Zbl. 338. 35022

    Google Scholar 

  39. Dubrovin, B.A.: Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties. Funkts. Anal. Prilozh. 11, No. 4, 28–41(1977) (English translation: Funct. Anal. Appt. 11, 265–277 (1977)). Zbl. 377. 58008

    Google Scholar 

  40. Dubrovin, B.A.: Theta functions and non-linear equations. Usp. Mat. Nauk 36, No. 2, 11–80 (1981) (English translation: Russ. Math. Surv. 36, No. 2, 11–92 (1981)). Zbl. 478. 58038

    Google Scholar 

  41. Dubrovin, B.A.: Matrix finite-zone operators, in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. (Contemporary Problems of Mathematics) 23. VINITI, Moscow 1983, 33–78 (English translation: J. Soy. Math. 28, 20–50 (1985)). Zbl. 561. 58043

    Google Scholar 

  42. Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry—Methods and Applications. Nauka, Moscow 1979, 759 p. (English translation in two parts: Part I: The geometry of surfaces, transformation groups, and fields. Graduate Texts in Mathematics 93, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo 1984, 464 p.; Part II. The geometry and topology of manifolds. Graduate Texts in Mathematics 104, Springer-Verlag, New York-BerlinHeidelberg-Tokyo 1985, 430 p.). Zbl. 529.53002, Zbl. 565. 57001

    Google Scholar 

  43. Dubrovin, B.A., Krichever, I.M., Novikov, S.P.: The Schrödinger equation in a periodic field and Riemann surfaces. Dokl. Akad. Nauk SSSR 229, 15–18 (1976) (English translation: Sov. Math., Dokl. 17, 947–951 (1976)). Zbl. 441. 35021

    Google Scholar 

  44. Dubrovin, B.A., Krichever, I.M., Novikov, S.P.: Topological and algebraic geometry methods in contemporary mathematical physics. II. Sov. Sci. Rev., Sect. C, Math. Phys. Rev. 3, 1–150 (1982). Zbl. 534. 58002

    Google Scholar 

  45. Dubrovin, B.A., Matveev, V.B., Novikov, S.P.: Non-linear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties. Usp. Mat. Nauk 31, No. 1, 55–136 (1976) (English translation: Russ. Math. Surv. 31, No. 1, 59–146 (1976)). Zbl. 326. 35011

    Google Scholar 

  46. Dubrovin, B.A., Natanzon, S.M.: Real two-zone solutions of the sine-Gordon equation. Funkts. Anal. Prilozh. /6, No. I, 27–43 (1982) (English translation: Funct. Anal. Appt. 16, 21–33 (1982)). Zbl. 554. 35100

    Google Scholar 

  47. Dubrovin, B.A., Novikov, S.P.: Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation. Zh. Ehksp. Teor. Fiz. 67,2131–2144 (1974) (English translation: Sov. Phys.-JETP 40,1058–1063 (1974))

    Google Scholar 

  48. Dubrovin, B.A., Novikov, S.P.: Hamiltonian formalism of one-dimensional systems of hydrodynamic type, and the Bogolyubov-Whitham averaging method. Dokl. Akad. Nauk SSSR 270, 781–785 (1983) (English translation: Sov. Math., Dokl. 27, 665–669 (1983) (in the translation, the name Whitham is given incorrectly as “Whitman”)). Zbl. 553. 35011

    Google Scholar 

  49. Dubrovin, B.A., Novikov, S.P.: On Poisson brackets of hydrodynamic type. Dokl. Akad. Nauk SSSR 279, 294–297 (1984) (English translation: Soy. Math., Dokl. 30, 651–654 (1984)). Zbl. 591. 58012

    Google Scholar 

  50. Dzyaloshinskij, I.E., Krichever, I.M.: Commensurability effects in the discrete Peierls model. Zh. Ehksp. Teor. Fiz. 83,1576–1586 (1982) (English translation: Sov. Phys.-JETP 56,908–913 (1982))

    Google Scholar 

  51. Dzyaloshinskij, I.E., Krichever, I.M.: Sound and charge-density wave in the discrete Peierls model. Zh. Ehksp. Teor. Fiz. 85,1771–1789 (1983) (English translation: Soy. Phys.-JETP 58,1031–1040 (1983))

    Google Scholar 

  52. Faddeev, L.D.: Integrable models in 1+1 dimensional quantum theory. Preprint S. Ph. T. 82/76. CEN Saclay, Saclay 1982, 97 p.

    Google Scholar 

  53. Faddeev, L.D., Slavnov, A.A.: Introduction to the Quantum Theory of Gauge Fields. Nauka, Moscow 1978, 239 p. (English translation: Gauge Fields: Introduction to Quantum Theory. Frontiers in.Physics 50, Benjamin-Cummings, Reading, Mass. 1980, 232 p.). Zbl. 486. 53052

    Google Scholar 

  54. Faddeev, L.D., Zakharov, V.E.: Korteweg-de Vries equation: a completely integrable Hamiltonian system. Funkts. Anal. Prilozh. 5, No. 4, 18–27 (1971) (English translation: Funct. Anal. Appt. 5, 280–287 (1971)). Zbl. 257. 35074

    Google Scholar 

  55. Flaschka, H.: The Toda lattice. II. Existence of integrals. Phys. Rev., B 9, 1924–1925 (1974), and On the Toda Lattice. II — inverse-scattering solution. Prog. Theor. Phys. 51, 703–716 (1974)

    MathSciNet  Google Scholar 

  56. Flaschka, H., McLaughlin, D.W.: Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions. Prog. Theor. Phys. 55, 438–456 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  57. Fomenko, A.T., Mishchenko, A.S.: Euler equations on finite-dimensional Lie groups. lzv. Akad. Nauk SSSR, Ser. Mat. 42, 396–415 (1978) (English translation: Math. USSR, Izv. /2, 371–389 (1978)). Zbl. 383. 58006

    Google Scholar 

  58. Fomenko, A.T., Trofimov, V.V.: Liouville integrability of Hamiltonian systems on Lie algebras. Usp. Mat. Nauk 39, No. 2, 3–56 (1984) (English translation: Russ. Math. Surv. 39, No. 2, 1–67 (1984)). Zbl. 549. 58024

    Google Scholar 

  59. Gardner, C.S.: Korteweg-de Vries equation and generalizations. IV. The Korteweg—de Vries equation as a Hamiltonian system. J. Math. Phys. 12, 1548–1551 (1971). Zbl. 283. 35021

    Google Scholar 

  60. Gardner, C.S., Green, J., Kruskal, M., Miura, R.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967)

    Article  MATH  Google Scholar 

  61. Gamier, R.: Sur une classe de systèmes différentiels abéliens déduits de la théorie des équations lineaires. Rend. Circ. Mat. Palermo 43, 155–191 (1919). FdM 47, 404

    Google Scholar 

  62. Glaser, V., Grosse, H., Martin, A.: Bounds on the number of eigenvalues of the Schrödinger operator. Commun. Math. Phys. 59, 197–212 (1978). Zbl. 373. 35050

    Google Scholar 

  63. Golubev, V.V.: Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point. Gostekhizdat, Moscow 1953, 287 p. (English translation: Coronet Books, Philadelphia 1953, 240 p.). Zbl. 51, 151

    Google Scholar 

  64. Its, A.R., Matveev, V.B.: A class of solutions of the Korteweg-de Vries equation (in Russian), in: Probl. Mat. Fiz. (Problems in Mathematical Physics) 8. Leningrad State University, Leningrad 1976, 70–92.

    Google Scholar 

  65. Kirchhoff, G.: Vorlesungen über mathematische Physik. Mechanik. B.G. Teubner, Leipzig 1876, 466 p. FdM 8, 542

    Google Scholar 

  66. Kirillov, A.A.: Elements of the Theory of Representations. Nauka, Moscow 1972, 336 p. (English translation: Grundlehren der mathematischen Wissenschaften 220, Springer-Verlag, Berlin-Heidelberg-New York 1976, 315 p.). Zbl. 264. 22011

    Google Scholar 

  67. Kostant, B.: Quantization and unitary representations. Part I: Prequantization, in: Lecture Notes in Modern Analysis and Applications III. Lect. Notes Math. 170. Springer-Verlag, Berlin-Heidelberg-New York 1970, 87–208. Zbl. 223. 53028

    Google Scholar 

  68. Kotlyarov, V.P., Kozel, V.A.: Almost periodic solutions of the equation u„ - uxx+sin u=0 (in Russian). Dokl. Akad. Nauk Ukr. SSR, Ser. A no. /0, 878–881 (1976). Zbl. 337. 35003

    Google Scholar 

  69. Kötter, F.: Über die Bewegung eines festen Körpers in einer Flüssigkeit. J. Reine Angew. Math. /09, 51–81, 89–111 (1892). FdM 23, 977

    Google Scholar 

  70. Kötter, F.: Die von Steklow und Liapunow entdeckten integrablen Fälle der Bewegung eines starren Körpers in einer Flüssigkeit. Sitzungsber., K. Preuss. Akad. Wiss. Berlin 1900, 79–87.

    Google Scholar 

  71. Kozlov, V.V.: Methods of Qualitative Analysis in the Dynamics of a Rigid Body (in Russian). Moscow State University, Moscow 1980, 230 p. Zbl. 557. 70009

    Google Scholar 

  72. Kozlov, V.V.: Integrability and non-integrability in Hamiltonian mechanics. Usp. Mat. Nauk 38, No. 1, 3–67 (1983) (English translation: Russ. Math. Surv. 38, No. 1, 1–76 (1983)). Zbl. 525. 70023

    Google Scholar 

  73. Kozlov, V.V., Onishchenko, D.A.: Nonintegrability of Kirchhoff’s equations. Dokl. Akad. Nauk SSSR 266, 1298–1300 (1982) (English translation: Soy. Math., Dokl. 26, 495–498 (1982)), Zbl. 541. 70009

    Google Scholar 

  74. Krichever, I.M.: Algebraic-geometric construction of the Zakharov-Shabat equations and their periodic solutions. Dokl. Akad. Nauk SSSR 227, 291–294 (1976) (English translation: Soy. Math., Dokl. 17, 394–397 (1976)). Zbl. 361. 35007

    Google Scholar 

  75. Krichever, I.M.: Integration of nonlinear equations by the methods of algebraic geometry. Funkts. Anal. Prilozh. 11, No. 1, 15–31 (1977) (English translation: Funct. Anal. Appl. 11, 12–26 (1977)). Zbl. 346. 35028

    Google Scholar 

  76. Krichever, I.M.: Methods of algebraic geometry in the theory of non-linear equations. Usp. Mat. Nauk 32, No. 6, 183–208 (1977) (English translation: Russ. Math. Surv. 32, No. 6, 185–213 (1977)). Zbl. 372. 35002

    Google Scholar 

  77. Krichever, I.M.: Rational solutions of the Kadomtsev-Petviashvili equation and integrable systems of N particles on a line. Funkts. Anal. Prilozh. 12, No. 1, 76–78 (1978) (English translation: Funct. Anal. Appl. 12, 59–61 (1978)). Zbl. 374. 70008

    Google Scholar 

  78. Krichever, I.M.: Commutative rings of ordinary linear differential operators. Funkts. Anal. Prilozh. 12, No. 3, 20–31(1978) (English translation: Funct. Anal. Appl. 12, 175–185 (1978)). Zbl. 408. 34008

    Google Scholar 

  79. Krichever, I.M.: Algebraic curves and non-linear difference equations. Usp. Mat. Nauk 33, No. 4, 215–216 (1978) (English translation: Russ. Math. Surv. 33, No. 4, 255–256 (1978)). Zbl. 382. 39003

    Google Scholar 

  80. Krichever, I.M.: Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles. Funkts. Anal. Prilozh. 14, No. 4, 45–54 (1980) (English translation: Funct. Anal. Appl. /4, 282–290 (1980)). Zbl. 462. 35080

    Google Scholar 

  81. Krichever, I.M.: The Peierls model. Funkts. Anal. Prilozh. 16,No. 4, 10–26 (1982) (English translation: Funct. Anal. Appl. 16, 248–263 (1982))

    Google Scholar 

  82. Krichever, I.M.: Algebro-geometric spectral theory of the Schrödinger difference operator and the Peierls model. Dokl. Akad. Nauk SSSR 265, 1054–1058 (1982) (English translation: Sov. Math., Dokl. 26, 194–198 (1982)). Zbl. 527. 39001

    Google Scholar 

  83. Krichever, I.M.: Nonlinear equations and elliptic curves, in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. (Contemporary Problems of Mathematics) 23. VINITI, Moscow 1983, 79–136 (English translation: J. Soy. Math. 28, 51–90 (1985)). Zbl. 595. 35087

    Google Scholar 

  84. Krichever, I.M., Novikov, S.P.: Holomorphic bundles over Riemann surfaces and the Kadomtsev-Petviashvili (KP) equation. I. Funkts. Anal. Prilozh. 12, No. 4, 41–52 (1978) (English translation: Funct. Anal. Appl. 12, 276–286 (1978)). Zbl. 393. 35061

    Google Scholar 

  85. Krichever, I.M., Novikov, S.P.: Holomorphic fiberings and nonlinear equations. Finite zone solutions of rank 2. Dokl. Akad. Nauk SSSR 247, 33–37 (1979) (English translation: Soy. Math., Dokl. 20, 650–654 (1979)). Zbl. 434. 35078

    Google Scholar 

  86. Krichever, I.M., Novikov, S.P.: Holomorphic bundles over algebraic curves and non-linear equations. Usp. Mat. Nauk 35, No. 6, 47–68 (1980) (English translation: Russ. Math. Surv. 35, No. 6, 53–79 (1980)). Zbl. 501. 35071

    Google Scholar 

  87. Kruskal, M.D.: The Korteweg-de Vries equation and related evolution equations, in: Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem. Clarkson Coll. of Technology, Potsdam, N.Y. 1972). Lect. Appl. Math. 15. American Mathematical Society, Providence 1974, pp. 61–83. Zbl. 292. 35017

    Google Scholar 

  88. Lamb, G.L: Analytical descriptions of ultrashort optical pulse propagation in a resonant medium. Rev. Mod. Phys. 43, 99–124 (1971)

    Article  MathSciNet  Google Scholar 

  89. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968). Zbl. 162, 411

    Google Scholar 

  90. Lax, P.D.: Periodic solutions of the KdV equation. Commun. Pure Appl. Math. 28, 141–188 (1975). Zbl. 295. 35004

    Google Scholar 

  91. Levi-Civita, T.: Sulla integrazione della equazione di Hamilton-Jacobi per separazione di variabili. Math. Ann. 59, 383–397 (1904). FdM 35, 362

    Google Scholar 

  92. Levi-Civita, T., Amaldi, U.: Lezioni di Meccanica Razionale. Nuova edizione riveduta e corretta. I, II Parte 1, II Parte 2. N. Zanichelli, Bologna 1950/1951/1952, 816/510/673 p. Zbl. 45,123, Zbl. 47, 173

    Google Scholar 

  93. Lie, S. (unter Mitwirkung von F. Engel): Theorie der Transformationsgruppen. I.-III. Abschnitt. B.G. Teubner, Leipzig 1888/1890/1893, 632/554/830 p. (reprint: B.G. Teubner, Leipzig-Berlin 1930). I: FdM 20, 368

    Google Scholar 

  94. Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19, 1156–1162 (1978). Zbl. 383. 35065

    Google Scholar 

  95. Maki, K., Ebisawa, H.: Exact magnetic ringing solutions in superfluid’He-B. Phys. Rev., B 13, 2924–2930 (1976)

    Article  Google Scholar 

  96. Manakov, S.V.: Complete integrability and stochastization of discrete dynamical systems. Zh. Ehksp. Teor. Fiz. 67,543–555 (1974) (English translation: Soy. Phys: JETP 40,269–274 (1974))

    Google Scholar 

  97. Manakov, S.V.: Note on the integration of Euler’s equations of the dynamics of an n-dimensional rigid body. Funkts. Anal. Prilozh. 10, No. 4, 93–94 (1976) (English translation: Funct. Anal. Appl. 10, 328–329 (1976)). Zbl. 343. 70003

    Google Scholar 

  98. Manakov, S.V.: The method of the inverse scattering problem and two-dimensional evolution equations (in Russian). Usp. Mat. Nauk 31, No. 5, 245–246 (1976). Zbl. 345. 35055

    Google Scholar 

  99. Manakov, S.V., Zakharov, V.E., Bordag, L.A., Its, A.R., Matveev, V.B.: Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction. Phys. Lett., A 63, 205–206 (1977)

    Article  Google Scholar 

  100. Marchenko, V.A.: Sturm-Liouville Operators and Applications. Naukova Dumka, Kiev 1977, 331 p. (English translation: Operator Theory: Advances and Applications, Vol. 22. Birkhäuser, Basel-Boston-Stuttgart 1986, 367 p.). Zbl. 399. 34022

    Google Scholar 

  101. Marsden, J. Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5,121–130 (1974). Zbl. 327.58005

    Google Scholar 

  102. McKean, H.P. van Moerbeke, P.: The spectrum of Hill’s equation. Invent. Math. 30,217–274 (1975). Zbl. 319.34024

    Google Scholar 

  103. McKean, H.P., Trubowitz, E.: Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points. Commun. Pure Appl. Math. 29, 143–226 (1976). Zbl. 339. 34024

    Google Scholar 

  104. Meiman, N.N.: The theory of one-dimensional Schrödinger operators with a periodic potential. J. Math. Phys. 18, 834–848 (1977)

    Article  Google Scholar 

  105. Meshcheryakov, M.V.: The integration of the equations for geodesics of left-invariant metrics on simple Lie groups using special functions. Mat. Sb., Nov. Ser. 117 (159), 481–493 (1982) (English translation: Math. USSR, Sb. 45, 473–485 (1983)). Zbl. 506. 58028

    Google Scholar 

  106. Mikhajlov, A.V., Zakharov, V.E.: Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method. Zh. Ehksp. Teor. Fiz. 74,.1953–1973 (1978) (English translation: Soy. Phys.-JETP 47, 1017–1027 (1978))

    Google Scholar 

  107. Milne-Thompson, L.M.: Theoretical Hydrodynamics (Fourth Edition). Macmillan, London/St. Martin’s Press, New York 1960, 660 p. Zbl. 89, 426

    Google Scholar 

  108. Mishchenko, A.S.: Integrals of geodesic flows on Lie groups. Funkts. Anal. Prilozh. 4, No. 3, 73–77 (1970) (English translation under the incorrect title: Integral geodesics of a flow on Lie groups. Funct. Anal. Appl. 4, 232–235 (1970)). Zbl. 241. 22022

    Google Scholar 

  109. Moser, J.: Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16,197–220 (1975). Zbl. 303.34019

    Google Scholar 

  110. Moser, J.: Various aspects of integrable Hamiltonian systems, in: Dynamical Systems, C.I.M.E. Lectures, Bressanone, Italy, June 1978. Prog. Math., Boston 8. Birkhäuser, Boston-BaselStuttgart 1980, pp. 233–289. Zbl. 468. 58011

    Google Scholar 

  111. Novikov, S.P.: The periodic problem for the Korteweg-de Vries equation. Funkts. Anal. Prilozh. 8, No. 3, 54–66 (1974) (English translation: Funct. Anal. Appl. 8, 236–246 (1974)). Zbl. 299. 35017

    Google Scholar 

  112. Novikov, S.P.: The Hamiltonian formalism and a many-valued analogue of Morse theory. Usp. Mat. Nauk 37, No. 5, 3–49 (1982) (English translation: Russ. Math. Surv. 37, No. 5, 1–56 (1982)). Zbl. 571. 58011

    Google Scholar 

  113. Novikov, S.P.: Two-dimensional Schrödinger operators in periodic fields, in: Itogi Nauki Tekh., Ser. Sovrem. Probt. Mat. (Contemporary Problems of Mathematics) 23. VINITI, Moscow 1983, 3–32 (English translation: J. Sov. Math. 28, 1–20 (1985)). Zbl. 564. 35083

    Google Scholar 

  114. Novikov, S.P., Grinevich, P.G.: Spectral theory of commuting operators of rank two with periodic coefficients. Funkts. Anal. Prilozh. 16, No. 1, 25–26 (1982) (English translation: Funct. Anal. Appl. 16, 19–20 (1982)). Zbl. 514. 47035

    Google Scholar 

  115. Novikov, S.P., Manakov, S.V., Pitaevskij (Pitaevskii), L.P., Zakharov, V.E.: Theory of Solitons. The Inverse Scattering Method. Nauka, Moscow 1980, 319 p. (English translation: Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York-London 1984, 276 p.). Zbl. 598. 35003

    Google Scholar 

  116. Novikov, S.P., Veselov, A.P.: On Poisson brackets compatible with algebraic geometry and Korteweg-de Vries dynamics on the set of finite-zone potentials. Dokl. Akad. Nauk SSSR 266, 533–537 (1982) (English translation: Soy. Math., Dokl. 26, 357–362 (1982)). Zbl. 555. 35106

    Google Scholar 

  117. Novikov, S.P., Veselov, A.P.: Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formulas and evolution equations. Dokl. Akad. Nauk SSSR 279, 20–24 (1984) (English translation: Sov. Math., Dokl. 30, 588–591 (1984)). Zbl. 613. 35020

    Google Scholar 

  118. Novikov, S.P., Veselov, A.P.: Poisson brackets and complex tori. Tr. Mat. Inst. Steklova 165, 49–61 (1984) (English translation: Proc. Steklov Inst. Math. /65, 53–65 (1985:3)). Zbl. 565. 58022

    Google Scholar 

  119. Perelomov, A.M.: Some remarks on the integrability of the equations of motion of a rigid body in an ideal fluid. Funkts. Anal. Prilozh. /5, No. 2, 83–85 (1981) (English translation: Funct. Anal. Appl. 15, 144–146 (1981)). Zbl. 495. 70016

    Google Scholar 

  120. Perelomov, A.M.: Integrable systems of classical mechanics and Lie algebras. Systems with constraints (in Russian). Prepr., Inst. Teor. Ehksp. Fiz., Moskva, 1983, No. 116, Moscow 1983, 63 p.

    Google Scholar 

  121. Perelomov, A.M.: Integrable systems of classical mechanics and Lie algebras. The motion of a rigid body about a fixed point (in Russian). Prepr., Inst. Teor. Ehksp. Fiz., Moskva, 1983, No. 147, Moscow 1983, 64 p.

    Google Scholar 

  122. Pohlmeyer, K.: Integrable Hamiltonian systems and interactions through quadratic constraints. Commun. Math. Phys. 46, 207–221 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  123. Rejman, A.G.: Integrable Hamiltonian systems connected with graded Lie algebras. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 95, 3–54 (1980) (English translation: J. Sov. Math. 19, 1507–1545 (1982)). Zbl. 488. 70013

    Google Scholar 

  124. Semenov-Tyan-Shanskij, M.A.: What is a classical r-matrix? Funkts. Anal. Prilozh. 17, No. 4, 17–33 (1983) (English translation: Funct. Anal. Appl. 17, 259–272 (1983)). Zbl. 535. 58031

    Google Scholar 

  125. Shabat, A.B., Zakharov, V.E.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Zh. Ehksp. Teor. Fiz. 61,118–134 (1971) (English translation: Sov. Phys.-JETP 34,62–69 (1972))

    Google Scholar 

  126. Shabat, A.B., Zakharov, V.E.: A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I. Funkts. Anal. Prilozh. 8, No. 3, 43–53 (1974) (English translation: Funct. Anal. Appl. 8, 226–235 (1974)). Zbl. 303. 35024

    Google Scholar 

  127. Shabat, A.B., Zakharov, V.E.: Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II. Funkts. Anal. Prilozh. 13, No. 3, 13–22 (1979) (English translation: Funct. Anal. Appl. /3, 166–174 (1979)). Zbl. 448. 35090

    Google Scholar 

  128. Sklyanin, E.K.: On complete integrability of the Landau-Lifshitz equation (in Russian). Prepr. LOMI, E-3–79, Leningrad 1979, 32 p. Zbl. 449. 35089

    Google Scholar 

  129. Sklyanin, E.K.: Some algebraic structures connected with the Yang-Baxter equation. Funkts. Anal. Prilozh. /6, No. 4, 27–34 (1982) (English translation: Funct. Anal. Appl. 16, 263–270 (1982)). Zbl. 513. 58028

    Google Scholar 

  130. Sokolov, V.V., Svinolupov, S.I.: Evolution equations with nontrivial conservative laws. Funkts. Anal. Prilozh. 16, No. 4, 86–87 (1982) (English translation: Funct. Anal. Appl. 16, 317–319 (1982)). Zbl. 508. 35072

    Google Scholar 

  131. Springer, G.: Introduction to Riemann Surfaces. Addison-Wesley, Reading, Mass. 1957, 307 p. Zbl. 78, 66

    Google Scholar 

  132. Steklov, V.A.: On the motion of a rigid body in a fluid (in Russian). Khar’kov 1893, 98 p.

    Google Scholar 

  133. Steklov, V.A. (Stekloff, W.): Sur le mouvement d’un corps solide ayant une cavité de forme ellipsoïdale remplie par un liquide incompressible et sur les variations des latitudes. Ann. Fac. Sci. Toulouse, III. Ser., Math.-Phys. 1, 145–256 (1909)

    Google Scholar 

  134. Takhtadzhyan, L.A.: Exact theory of propagation of ultrashort optical pulses in two-level media. Zh. Ehksp. Teor. Fiz. 66476–489 (1974) (English translation: Sov. Phys.-JETP 39228–233 (1974))

    Google Scholar 

  135. Takhtadzhyan, L.A., Veselov, A.P.: Integrability of the Novikov equations for principal chiral fields with a multivalued Lagrangian. Dokl. Akad. Nauk SSSR 279,1097–1100 (1984) (English translation: Sov. Phys., Dokl. 29,994–996 (1984))

    Google Scholar 

  136. Veselov, A.P.: Finite-zone potentials and integrable systems on a sphere with quadratic potential. Funkts. Anal. Prilozh. 14, No. 1, 48–50 (1980) (English translation: Funct. Anal. Appl. 14, 37–39 (1980)). Zbl. 574. 34001

    Google Scholar 

  137. Veselov, A.P.: The Landau—Lifshits equation and integrable systems of classical mechanics. Dokl. Akad. Nauk SSSR 2701094–1097 (1983) (English translation: Sov. Phys., Dokl. 28458–459 (1983))

    Google Scholar 

  138. Veselov, A.P.: On integrability conditions for the Euler equations on so(4). Dokl. Akad. Nauk SSSR 270, 1298–1300 (1983) (English translation: On integrability conditions for the Euler equations on SO(4). Sov. Math., Dokl. 27, 740–742 (1983)). Zbl. 539. 58013

    Google Scholar 

  139. Veselov, A.P.: Cnoidal solutions of the Landau—Lifshitz equation for two-sublattice magnetic materials. Dokl. Akad. Nauk SSSR 276590–593 (1984) (English translation: Sov. Phys., Dokl. 29393–395 (1984))

    Google Scholar 

  140. Weber, H.: Anwendung der Thetafunctionen zweier Veränderlicher auf die Theorie der Bewegung eines festen Körpers in einer Flüssigkeit. Math. Ann. 14, 173–206 (1879)

    Article  MATH  Google Scholar 

  141. Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Fourth Edition). Cambridge University Press, Cambridge 1937, 456 p. FdM 63, 1286

    Google Scholar 

  142. Zakharov, V.E.: On stochastization of one-dimensional chains of nonlinear oscillators. Zh. Ehksp. Teor. Fiz. 65,219–225 (1973) (English translation: Sov. Phys.-JETP 38,108–110 (1974))

    Google Scholar 

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Authors
  1. B. A. Dubrovin
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  2. I. M. Krichever
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  3. S. P. Novikov
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Editors and Affiliations

  1. Steklov Mathematical Institute, ul. Gubkina 8, 117966, Moscow, Russia

    V. I. Arnold

  2. CEREMADE, Université Paris 9 — Dauphine, Place du Marechal de Lattre de Tassigny, 75775, Paris Cedex 16-e, France

    V. I. Arnold

  3. Institute of Physical Sciences and Technology, University of Maryland, 20742-2431, College Park, MD, USA

    S. P. Novikov

  4. L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 2 Kosygina str., 117940, Moscow, Russsia

    S. P. Novikov

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Dubrovin, B.A., Krichever, I.M., Novikov, S.P. (2001). Integrable Systems.I. In: Arnold, V.I., Novikov, S.P. (eds) Dynamical Systems IV. Encyclopaedia of Mathematical Sciences, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06791-8_3

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  • DOI: https://doi.org/10.1007/978-3-662-06791-8_3

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