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Basic Examples and Their General Properties

  • Ludwig D. Faddeev
  • Leon A. Takhtajan
Part of the Springer Series in Soviet Mathematics book series (CLASSICS)

Abstract

In this chapter we shall give a list of typical examples and establish their general properties: the zero curvature representation and the Hamiltonian formulation. Then, motivated by these examples, we shall outline a general scheme for constructing integrable equations and their solutions based on the matrix Riemann problem. A detailed study of the most important models and the Hamiltonian interpretation of the general scheme will be presented in the following chapters. The examples to be considered fall into two classes: dynamical systems generated by partial differential evolution equations (continuous models), and evolution systems of difference type (lattice models).

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References

  1. [A 1969]
    Adams, J. F.: Lectures on Lie groups. New York-Amsterdam, Benjamin 1969Google Scholar
  2. [A 1974]
    Arnold, V. I.: Mathematical Methods of Classical Mechanics. Moscow: Nauka 1974 [Russian]; English transi.: Graduate Texts in Mathematics 60, New York-Berlin-Heidelberg, Springer 1978Google Scholar
  3. [AKNS 1973]
    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)ADSMathSciNetCrossRefGoogle Scholar
  4. [AKNS 1974]
    Ablowitz, M. J., Kaup, D. J., Newell, A. C., Segur, H.: The inverse scattering transform - Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249 - 315 (1974)MathSciNetCrossRefMATHGoogle Scholar
  5. [AL 1976]
    Ablowitz, M. J., Ladik, J. F.: Nonlinear differential-difference equations and Fourier analysis. J. Math. Phys. 17, 1011 - 1018 (1976)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [AM 1980]
    Adler, M., van Moerbeke, P.: Completely integrable systems, Euclidean Lie algebras and curves. Adv. Math. 38, 267 - 317 (1980)MathSciNetCrossRefMATHGoogle Scholar
  7. [AYF 1983]
    Ablowitz, M. J., van Yaakov, D., Fokas, A. S.: On the inverse scattering transform for Kadomtsev-Petviashvili equation. Stud. Appl. Math. 69, 135 - 143 (1983)MathSciNetCrossRefMATHGoogle Scholar
  8. B 1984] Bogoyavlensky, O. I.: Integrable equations on Lie algebras arising in problems of mathematical physics. Izv. Akad. Nauk SSSR Ser. Mat. 48, 883938 (1984) [Russian]Google Scholar
  9. [BFT 1986]
    Buslaev, V. S., Faddeev, L. D., Takhtajan, L. A.: Scattering theory for the Korteweg-de Vries equation and its Hamiltonian interpretation. Physica D 18, 255 - 266 (1986)MATHGoogle Scholar
  10. [BR 1981]
    Russian]; English transl. in Theor. Math. Phys. 46, 242 - 248 (1981)Google Scholar
  11. Borovik, A. E., Robuk, V. N.: Linear pseudopotentials and conservation laws for the Landau-Lifshitz equation describing the nonlinear dynamics of a ferromagnet with uniaxial anisotropy. Teor. Mat. Fiz. 46, 371-381 (1981) BT 1977] Budagov, A. S., Takhtajan, L. A.: A nonlinear one-dimensional model of classical field theory with internal degrees of freedom. Dokl. Akad. Nauk SSSR 235,805-808 (1977) [Russian]Google Scholar
  12. Bu 1978] Budagov, A. S.: A completely integrable model of classical field theory with nontrivial particle interaction in two space-time dimensions. In: Problems in quantum field theory and statistical physics. I. Zap. Nauchn. Semin. LOMI 77, 24-56 (1978) [Russian]Google Scholar
  13. [C 1979 a]
    Cherednik, I. V.: Local conservation laws for principal chiral fields (d= 1). Teor. Mat. Fiz. 38, 179-185 (1979) Russian]; English trans]. in Theor. Math. Phys. 38, 120 - 124 (1979)Google Scholar
  14. [C 1979 b]
    Russian]; English transi. in Theor. Math. Phys. 41, 997 - 1002 (1980)Google Scholar
  15. Cherednik, I. V.: Conservation laws and elements of scattering theory for principal chiral fields (d= 1). Teor. Mat. Fiz. 41, 236-244 (1979)C 1981] Cherednik, I. V.: Algebraic aspects of two-dimensional chiral fields. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 17, 175-218 (1981) [Russian]Google Scholar
  16. C 1983] Cherednik, I. V.: On the definition of a-function for generalized affine Lie algebras. Funk. Anal. Priloz. 17 (3), 93-95 (1983) [Russian]Google Scholar
  17. [D 1974]
    Druma, V. S.: Solution of the two-dimensional Korteweg-de Vries (KdV) equation. Pisma Zh. Exp. Teor. Fiz. 19, 753-755 (1974) Russian]; English transi. in Sov. Phys. JETP Letters 19, 387 - 388 (1974)Google Scholar
  18. [DB 1977]
    Dodd, R. K., Bullough, R. K.: Polynomial conserved densities for the Sine-Gordon equations. Proc. Roy. Soc. (London) A352, 481 - 503 (1977)ADSMathSciNetCrossRefGoogle Scholar
  19. [DJKM 1981]
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. [DKM 1981]
    Date, E., Kashiwara, M., Miwa, T.: Vertex operators and r-functions. Transformation groups for soliton equations. II. Proc. Japan Acad. 57A, 387 - 392 (1981)CrossRefMATHGoogle Scholar
  21. [DNF 1979]
    Dubrovin, B. A., Novikov, S. P., Fomenko, A. T.: Modern Geometry. Methods and Applications. Moscow: Nauka 1979 [Russian]; English translation of Part I: Graduate Texts in Mathematics 93; Part II: Graduate Texts in Mathematics 104, New York-Berlin-Heidelberg-Tokyo, Springer 1984, 1985Google Scholar
  22. [DS 1981]
    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 transi. in Sov. Math. Dokl. 23, 457 - 461 (1981)Google Scholar
  23. DS 1984] 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]Google Scholar
  24. [F 1974 a]
    Flaschka, H.: The Toda lattice. I. Existence of integrals. Phys. Rev. B9, 1924 - 1925 (1974)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. [F 1974b]
    Flaschka, H.: On the Toda lattice. II. Inverse scattering solution. Prog. Theor. Phys. 51, 703 - 716 (1974)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [FA 1983] Fokas, A. C., Ablowitz, M. J.: On the inverse scattering of the time dependent Schrödinger equation and the associated Kadomtsev-Petviashvili (I)
    equation. Stud. Appl. Math. 69, 211 - 228 (1983)Google Scholar
  27. [FK 1980]
    Frenkel, I. B., Kac, V. G.: Basic representation of affine Lie algebras and dual resonance models. Invent. Math. 62, 23 - 66 (1980)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. [FK 1983]
    Fordy, A. P., Kulish, P. P.: Nonlinear Schrödinger equations and simple Lie algebras. Comm. Math. Phys. 89, 427 - 443 (1983)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. [FR 1986] Faddeev, L. D., Rechetikhin, N. Yu.: Integrability of the principal chiral field model in 1 + 1 dimension. Ann. Phys. (N.Y.) 167, 227-256 (1986)
    FT 1985] Faddeev, L. D., Takhtajan, L. A.: Poisson structure for the KdV equation. Lett. Math. Phys. 10, 183 - 188 (1985)ADSMathSciNetCrossRefGoogle Scholar
  30. [G 1971]
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [GD 1975] 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)Google Scholar
  32. [GD 1977] Gelfand, I. M., Dikii, L. A.: The resolvent and Hamiltonian systems. Funk. Anal. Priloz. 11 (2), 11-17 (1977)
    Russian]; English transi, in Funct. Anal. Appl. 11, 93 - 104 (1977)Google Scholar
  33. [GGP 1966]
    Gelfand, I. M., Graev, M. I., Piatetski-Shapiro, I. I.: Representation theory and automorphic functions. Moscow, Nauka 1966 [Russian]; English transi.: Philadelphia, Saunders 1969Google Scholar
  34. [GIK 1984]
    Gerdjikov, V. S., Ivanov, M. I., Kulish, P. P.: Expansions over the “squared” solutions and difference evolution equations. J. Math-Phys. 25, 25 - 34 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  35. IK 1981] Izergin, A. G., Korepin, V. E.: A lattice model associated with the nonlinear Schrödinger equation. Dokl. Akad. Nauk SSSR 259,76-79 (1981) [Russian]Google Scholar
  36. [K 1980] Krichever, I. M.: Analogue of the D’Alembert formula for the equations of principal chiral field and the sine-Gordon equation. Dokl. Akad. Nauk SSR 253, 288-292 (1980)
    Russian]; English transi. in Sov. Math. Dokl. 22, 79 - 84 (1981)Google Scholar
  37. K 1983] Krichever, I. M.: Nonlinear equations and elliptic curves. Itogi Nauki Teich., Ser. Sovrem. Probl. Math. 23,79-136, Moscow, VINITI 1983 [Russian]Google Scholar
  38. [KM 1979]
    Kako, F., Mugibayashi, N.: Complete integrability of general nonlinear differential-difference equations solvable by the inverse method. II. Prog. Theor. Phys. 61, 776 - 790 (1979)ADSMathSciNetCrossRefMATHGoogle Scholar
  39. [KN 1978] Krichever, I. M., Novikov, S. P.: Holomorphic bundles over Riemann surfaces and the Kadomtsev-Petviashvili equation. Funk. Anal. Priloz. 12 (4), 41-52 (1978)
    Russian]; English transi. in Funct. Anal. Appl. 12, 276 - 286 (1978)Google Scholar
  40. KR 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]Google Scholar
  41. L 1968] Lax, P. D.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure and Appl. Math. 21,467-490 (1968)Google Scholar
  42. L 1977] Lakshmanan, M.: Continuum spin system as an exactly solvable dynamical system. Phys. Lett. 61A, 53-54 (1977)Google Scholar
  43. [LS 1979]
    Leznov, A. N., Saveliev, M. A.: Representation of zero curvature for the system of nonlinear partial differential equations xrsz=(expkx),. and its integrability. Lett. Math. Phys. 3, 489 - 494 (1979)ADSMathSciNetCrossRefMATHGoogle Scholar
  44. M 1973] Manakov, S. V.: On the theory of two-dimensional stationary self-focusing of electro-magnetic waves. Zh. Exp. Teor. Fiz. 65,505-516 (1973) [Russian]; English transi. in Sov. Phys. JETP 38, 248-253 (1974)Google Scholar
  45. [M 1974] Manakov, S. V.: Complete integrability and stochastization of discrete dynamical systems. Zh. Exp. Teor. Fiz. 67, 543-555 (1974)
    Russian]; English transi. in Soy. Phys. JETP 40, 269 - 274 (1975)Google Scholar
  46. [M 1976] Manakov, S. V.: Note on the integration of Euler’s equations of the dynamics of an n-dimensional rigid body. Funk. Anal. Priloi. 10 (4), 93-94 (1976)
    Russian]; English trans[. in Funct. Anal. Appl. 10, 328 - 329 (1976)Google Scholar
  47. [M 1981]
    Manakov, S. V.: The inverse scattering transform for the time-dependent Schrödinger equation and Kadomtsev-Petviashvili equation. Physica D3, 420 - 427 (1981)MathSciNetMATHGoogle Scholar
  48. [MF 1978] Miscenko, A. S., Fomenko, A. T.: Euler equations on finite-dimensional Lie groups. Izv. Akad. Nauk SSSR Ser. Mat. 42, 396-415 (1978)
    Russian]; English transi. in Math. USSR, Izv. 12, 371 - 389 (1978)Google Scholar
  49. [Mi 1979] Mikhailov, A. V.: Integrability of the two-dimensional generalization of the Toda chain. Pisma Zh. Exp. Teor. Fiz. 30, 443-448 (1979)
    Russian]; English trans]. in Sov. Phys. JETP Letters 30, 414 - 418 (1979)Google Scholar
  50. Mi 1981] Mikhailov, A. V.: The reduction problem and the inverse scattering method. Physica D3, 73-107 (1981)Google Scholar
  51. [MOP 1981]
    Mikhailov, A. V., Olshanetsky, M. A., Perelomov, A. M.: Two-dimensional generalized Toda lattice. Comm. Math. Phys. 79, 473 - 488 (1981)ADSMathSciNetCrossRefGoogle Scholar
  52. [N 1974] 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)Google Scholar
  53. [N 1982 a] Novikov, S. P.: Hamiltonian formalism and a multi-valued analogue of Morse theory. Usp. Mat. Nauk 37 (5), 3-49 (1982)
    Russian]; English transi. in Russian Math. Surveys 37 (5), 1 - 56 (1982)Google Scholar
  54. [N 1982 b]
    Novikov, S. P.: Hamiltonian formalism and variational-topological methods for finding periodic trajectories of conservative dynamical systems. Soviet Scientific Reviews, Sect. C, Math. Physics Reviews 3, 3 - 51 (1982)Google Scholar
  55. [Ne 1979]
    Newell, A. C.: The general structure of integrable evolution equations. Proc. Royal Soc. (London) A365, 283 - 311 (1979)MATHGoogle Scholar
  56. [P 1976]
    Pohlmeyer, K.: Integrable Hamiltonian systems and interaction through quadratic constraints. Comm. Math. Phys. 46, 207 - 223 (1976)ADSMathSciNetCrossRefMATHGoogle Scholar
  57. [P 1981] Perelomov, A. M.: Several remarks on the integrability of the equations of motion of a rigid body in ideal fluid. Funk. Anal. Priloi. 15 (2), 83-85 (1981)
    Russian]; English transi. in Funct. Anal. Appl. 15, 144 - 146 (1981)Google Scholar
  58. [R 1980]
    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. Soy. Math. 19, 1507 - 1545 (1982)Google Scholar
  59. R 1983] 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]Google Scholar
  60. [R 1984]
    Ramadas, T. R: The Wess-Zumino term and fermionic solutions. Comm. Math. Phys. 93, 355 - 365 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  61. [RS 1979]
    Reyman, A. G., Semenov-Tian-Shansky, M. A.: Reduction of Hamiltonian systems, affine Lie algebras and Lax equations. I. Inventiones math. 54, 81 - 100 (1979)ADSMathSciNetCrossRefMATHGoogle Scholar
  62. [RS 1980] Reyman, A. G., Semenov-Tian-Shansky, M. A.: Current algebras and nonlinear partial differential equations. Dokl. Akad. Nauk SSSR 251, 13101314 (1980)
    Russian]; English trans!. in Soy. Math. Dokl. 21, 630 - 634 (1980)Google Scholar
  63. [RSF 1979] Reyman, A. G., Semenov-Tian-Shansky, M. A., Frenkel, I. B.: Graded Lie algebras and completely integrable dynamical systems. Dokl. Akad. Nauk SSSR 247, 802-805 (1979)
    Russian]; English transl. in Soy. Math. Dokl. 20, 811 - 814 (1979)Google Scholar
  64. S 1975] Shabat, A. B.: The inverse scattering problem for a system of differential equations. Funk. Anal. Priloz. 9 (3), 75-78 (1975) [Russian]Google Scholar
  65. [S 1979] Shabat, A. B.: An inverse scattering problem. Differencialnye Uravneniya 15, 1824-1834 (1979)
    Russian]; English transl. in Diff. Equations 15, 1299 - 1307 (1980)Google Scholar
  66. [Se 1981]
    Segal, G.: Unitary representations of some infinite dimensional groups. Comm. Math. Phys. 80, 301 - 342 (1981)ADSMathSciNetCrossRefMATHGoogle Scholar
  67. [Sk 1979]
    Sklyanin, E. K.: On complete integrability of the Landau-Lifshitz equation. Preprint LOMI E-3-79, Leningrad 1979Google Scholar
  68. [Sk 1982] Sklyanin, E. K.: Algebraic structures connected with the Yang-Baxter equation. Funk. Anal. Priloz. 16 (4), 27-34 (1982)
    Russian]; English trans!. in Funct. Anal. Appl. 16, 263 - 270 (1982)Google Scholar
  69. [T 1970]
    Toda, M.: Waves in nonlinear lattice. Prog. Theor. Phys. Suppl. 45, 174 - 200 (1970)ADSCrossRefGoogle Scholar
  70. T 1974] Takhtajan, L. A.: Exact theory of propagation of ultra-short optical pulses in two-level media. Zh. Exp. Teor. Fiz. 66, 476-489 (1974) [Russian]Google Scholar
  71. [T 1977]
    Takhtajan, L. A.: Integration of the continuous Heisenberg spin chain through the inverse scattering method. Phys. Lett. 64A, 235 - 237 (1977)MathSciNetCrossRefGoogle Scholar
  72. [TF 1974] Takhtajan, L. A., Faddeev, L. D.: Essentially nonlinear one-dimensional model of classical field theory. Teor. Mat. Fiz. 21 (2), 160-174 (1974)
    Russian]; English transl. in Theor. Math. Phys. 21, 1046 - 1057 (1974)Google Scholar
  73. TF 1982] Takhtajan, L. A., Faddeev, L. D.: A simple connection between geometrical and Hamiltonian representations for the integrable nonlinear equations. In: Boundary-value problems of mathematical physics and related questions in function theory. 14. Zapiski Nauchn. Semin. LOMI 115, 264273 (1982) [Russian]Google Scholar
  74. [TTF 1983] Tarasov, V. O., Takhtajan, L. A., Faddeev, L. D.: Local Hamiltonians for integrable quantum models on the lattice. Teor. Mat. Fiz. 57, 163-181 (1983)
    Russian]; English transi. in Theor. Math. Phys. 57, 1059 - 1073 (1983)Google Scholar
  75. [V 1931]
    Volterra, V.: Leçons sur la théorie Mathématique de la Lutte pour la Vie. Paris, Gauthier-Villars 1931MATHGoogle Scholar
  76. [V 1983] Veselov, A. P.: The Landau-Lifshitz equation and integrable systems of classical mechanics. Dokl. Akad. Nauk SSSR 270, 1094-1097 (1983)
    Russian]; English transi. in Sov. Phys. Dokl. 28, 458 - 459 (1983)Google Scholar
  77. [VGG 1973] Vershik, A. M., Gelfand, I. M., Graev, M. I.: Representations of the group SL(2, R), where R is a ring of functions. Usp. Mat. Nauk 28 (5), 83-128 (1973)
    Russian]; English transi. in Russian Math. Surveys 28 (5), 87 - 132 (1973)Google Scholar
  78. VT 1984] Veselov, A. P., Takhtajan, L. A.: The integrability of Novikov’s equations for the principal chiral fields with a multi-valued Lagrangian. Dokl. Akad. Nauk SSSR 279, 1097-1110 (1984) [Russian]Google Scholar
  79. VV 1985] Vladimirov, V. S., Volovich, I. V.: Local and nonlocal currents for nonlinear equations. Teor. Mat. Fiz. 62, 3-29 (1985) [Russian]Google Scholar
  80. [W 1979]
    Wilson, G.: Commuting flows and conservation laws for Lax equations. Math. Proc. Cambridge Phil. Soc. 86, 131 - 143 (1979)MathSciNetCrossRefMATHGoogle Scholar
  81. [W 1981]
    Wilson, G.: On two constructions of conservation laws for Lax equations. Quart. J. Math. Oxford 32, 491 - 512 (1981)MathSciNetCrossRefMATHGoogle Scholar
  82. [W 1983]
    Witten, E.: Global aspects of current algebra. Nucl. Phys. B223, 422 - 432 (1983)ADSMathSciNetCrossRefGoogle Scholar
  83. [W 1984]
    Witten, E.: Nonabelian bosonization in two dimensions. Comm. Math. Phys. 92, 455 - 472 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  84. [WZ 1971]
    Wess, J., Zumino, B.: Consequences of anomalous Ward identities. Phys. Lett. 37B, 95 - 97 (1971)Google Scholar
  85. [ZF 1971] Zakharov, V. E., Faddeev, L. D.: Korteweg-de Vries equation, a completely integrable Hamiltonian system. Funk. Anal. Priloz. 5 (4) 18-27 (1971)
    Russian]; English transl. in Funct. Anal. Appl. 5, 280 - 287 (1971)Google Scholar
  86. [ZM 1975] Zakharov, V. E., Manakov, S. V.: The theory of resonant interaction of wave packets in nonlinear media. Zh. Eksp. Teor. Fiz. 69, 1654-1673 (1975)
    Russian]; English transi. in Soy. Phys. JETP 42, 842 - 850 (1976)Google Scholar
  87. [ZM 1978] Zakharov, V. E., Mikhailov, A. V.: Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method. Zh. Exp. Teor. Fiz. 74, 1953-1973 (1978)
    Russian]; English transl. in Sov. Phys. JETP 47, 1017 - 1027 (1978)Google Scholar
  88. [ZMNP 1980]
    Zakharov, V. E., Manakov, S. V., Novikov, S. P., Pitaievski, L. P.: Theory of Solitons. The Inverse Problem Method. Moscow, Nauka 1980 [Russian]; English transi.: New York, Plenum 1984Google Scholar
  89. ZS 1971] Zhiber, A. V., Shabat, A. B.:Klein-Gordon equations with a nontrivial group. Dokl. Akad. Nauk SSSR 247, 1103-1106 (1971) [Russian]Google Scholar
  90. [ZS 1974] Zakharov, V. E., Shabat, A. B.: Integration of the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I. Funk. Anal. Priloz. 8 (3), 43-53 (1974)
    Russian]; English transi. in Funct. Anal. Appl. 8, 226 - 235 (1974)Google Scholar
  91. [ZS 1979] 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)Google Scholar
  92. [ZT 1979] Zakharov, V. E., Takhtajan, L. A.: Equivalence of the nonlinear Schrödinger equation and the Heisenberg ferromagnet equation. Teor. Mat. Fiz. 38 (1), 26-35 (1979)
    Russian]; English transi. in Theor. Math. Phys. 38, 17 - 23 (1979)Google Scholar
  93. [ZTF 1974] Zakharov, V. E., Takhtajan, L. A., Faddeev, L. D.: A complete description of solution of the “sine-Gordon” equation. Dokl. Akad. Nauk SSSR 219, 1334-1337 (1974)
    Russian]; English transl. in Soy. Phys. Dokl. 19, 824 - 826 (1975)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Ludwig D. Faddeev
    • 1
  • Leon A. Takhtajan
    • 1
  1. 1.Steklov Mathematical InstituteLeningradUSSR

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