The Hamiltonian Formulation

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

Abstract

In this chapter we return to the Hamiltonian formulation of the NS model in order to discuss the basic transformation of the inverse scattering method
from the Hamiltonian standpoint. We shall describe the Poisson structure on the scattering data of the auxiliary linear problem induced through f from the initial Poisson structure defined in Chapter I. Under the rapidly decreasing or finite density boundary conditions, the NS model proves to be a completely integrable system, with f defining a transformation to action-angle variables. In particular, we will show that the integrals of the motion introduced in Chapter I are in involution. In these terms scattering of solitons amounts to a simple canonical transformation.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A 1974]
    Arnold, V. I.: Mathematical Methods of Classical Mechanics. Moscow, Nauka 1974 [Russian]; English transl.: Graduate Texts in Mathematics 60, New York-Berlin-Heidelberg, Springer 1978Google Scholar
  2. [A 1981]
    Alber, S. I.: On stationary problems for equations of Korteweg-de Vries type. Comm. Pure Appl. Math. 34, 259–272 (1981)Google Scholar
  3. [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
  4. [B 1972]
    Baxter, R. J.: Partition function of the eight-vertex lattice model. Ann. Of Physics (N.Y.) 70, 193–228 (1972)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [B 1982]
    Baxter, R. J.: Exactly solved models in statistical mechanics. London, Academic Press 1982 Russian]; English transi. in Funct. Anal. Appl. 16, 159–180 (1982)Google Scholar
  6. [Be 1980]
    Russian]; English transi. in Funct. Anal. Appl. 14, 260–267 (1980)Google Scholar
  7. [BN 1976]
    Russian]; English transi. in Funct. Anal. Appl. 10, 8–11 (1976)Google Scholar
  8. [Bo 1976]
    Russian]; English transi. in Funct. Anal. Appl. 10, 92–95 (1976)Google Scholar
  9. [CD 1976]
    Calogero, F., Degasperis, A.: Nonlinear evolution equations solvable by the inverse spectral transform. I. Nuovo Cimento 32B, 201–242 (1976)ADSMathSciNetCrossRefGoogle Scholar
  10. [CD 1977]
    Calogero, F., Degasperis, A.: Nonlinear evolution equations solvable by the inverse spectral transform. II. Nuovo Cimento 39B, 1–54 (1977)MathSciNetCrossRefGoogle Scholar
  11. [D 1964]
    Dirac, P. A. M.: Lectures on quantum mechanics. Belfer Grad. School of Science, Yeshiva University, N.-Y. 1964Google Scholar
  12. [D 1981]
    Russian]; English transi. in Russian Math. Surveys 36 (2), 11–92 (1982)Google Scholar
  13. [DMN 1976]
    Russian]; English transi. in Russian Math. Surveys 31 (1), 59–146 (1976)Google Scholar
  14. [DN 1974]
    Russian]; English trans!. in Soy. Phys. JETP 40, 1058–1063Google Scholar
  15. [DN 1982]
    Russian]; English transi. in Sov. Math. Dokl. 26, 760–765 (1983)Google Scholar
  16. [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
  17. [F 1980]
    Faddeev, L. D.: Quantum completely integrable models in field theory. In: Mathematical Physics Review. Sect. C.: Math. Phys. Rev. 1, 107–155, Harwood Academic (1980)Google Scholar
  18. [FM 1976]
    Flaschka, H., McLaughlin, D.: Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions. Prog. of Theor. Phys. 55, 438–456 (1976)MathSciNetMATHGoogle Scholar
  19. [FT 1985]
    Faddeev, L. D., Takhtajan, L. A.: Poisson structure for the KdV equation. Lett. Math. Phys. 10, 183–188 (1985)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. [GD 1975]
    Russian]; English transi. in Russian Math. Surveys 30 (5), 77–113 (1975)Google Scholar
  21. [GD 1979]
    Russian]; English transi. in Funct. Anal. Appl. 13, 6–15 (1979)Google Scholar
  22. [GD 1980]
    Russian]; English transi. in Funct. Anal. Appl. 14, 223–226 (1980)Google Scholar
  23. [GD 1981]
    Russian]; English transi. in Funct. Anal. Appl. 15, 173–187 (1981)Google Scholar
  24. [GGKM 1974]
    Gardner, C. S., Greene, J. M., Kruskal, M. D., Miura, R. M.: Korteweg-de Vries equation and generalizations. VI. Methods for exact solution. Comm. Pure Appl. Math. 27, 97–133 (1974)MATHGoogle Scholar
  25. [GK 1980 a]
    Gerdjikov, V. S., Khristov, E. Kh.: On the evolution equations solvable through the inverse scattering method. I: Spectral Theory. Bulg. J. Phys. 7, 28–41 (1980) [Bulgarian]Google Scholar
  26. [GK 1980 b]
    Gerdjikov, V. S., Khristov, E. Kh.: On the evolution equations solvable through the inverse scattering method. II: Hamiltonian structure and Bäcklund transformations. Bulg. J. Phys. 7,119–133 (1980) [Bulgarian]Google Scholar
  27. [H 1912]
    Hermite, Ch.: Sur l’équation de Lamé - Cours d’analyse de l’École poly- techn. Paris, 1872–1873, 32-e leçon; Oeuvres, T. III, 118–122, Paris 1912Google Scholar
  28. [I 1976]
    Its, A. R.: Inversion of hyperelliptic integrals and integration of nonlinear differential equations. Vestnik Leningrad Univ., ser. mat.-mech.-astr. 7 (2), 39–46 (1976) [Russian]Google Scholar
  29. [IK 1976]
    Its, A. R., Kotlyarov, V. P.: Explicit formulas for solutions of the nonlinear Schrödinger equation. Dokl. Akad. Nauk Ukr. SSR, ser. A., 11,965–968 (1976) [Russian]Google Scholar
  30. [IK 1981]
    Izergin, A. G., Korepin, V. E.: The inverse scattering method approach to the quantum Shabat-Mikhailov model. Comm. Math. Phys. 79, 303–316 (1981)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [IK 1982]
    Russian]; English transi. in Soviet J. Particles and Nuclei 13 (3), 207–223 (1982)Google Scholar
  32. [IM 1975]
    Russian]; English transi. in Theor. Math. Phys. 23, 343–355 (1976)Google Scholar
  33. [K 1976]
    Kaup, D. J.: Closure of the squared Zakharov-Shabat eigenstates. J. Math. Anal. Appl. 54, 849–864 (1976)MathSciNetCrossRefMATHGoogle Scholar
  34. [K 1977]
    Russian]; English transi. in Russian Math. Surveys 32 (6), 185–213 (1977)Google Scholar
  35. [K 1980]
    Kulish, P. P.: Generating operators for integrable nonlinear evolution equations. In: Boundary-value problems of mathematical physics and related questions in function theory. 12. Zapiski Nauchn. Semin. LOMI 96, 105–112 (1980) [Russian]; English transi. in J. Sov. Math. 21, 717–723 (1983)Google Scholar
  36. [KMF 1976]
    Russian]; English transi. in Theor. Math. Phys. 28, 615–620 (1977)Google Scholar
  37. [KO 1976]
    Kotlyarov, V. P.: Periodic problem for the nonlinear Schrödinger equation. In: Problems of mathematical physics and functional analysis (seminar notes) 1, 121–131, Kiev, Naukova Dumka, 1976 [Russian]Google Scholar
  38. [KR 1978]
    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. Sov. Math. 22, 1627–1637 (1983)Google Scholar
  39. [KS 1980]
    Kulish, P. P., Sklyanin, E. K.: Solutions of the Yang-Baxter equation. In: Differential geometry, Lie groups and mechanics. III. Zapiski Nauchn. Semin. LOMI 95, 129–160 (1980) [Russian]; English transi. in J. Soy. Math. 19 (5), 1596–1620 (1982)Google Scholar
  40. [KS 1982]
    Kulish, P. P., Sklyanin, E. K.: Quantum spectral transform method. Recent developments. Lecture Notes in Physics, vol. 151, 61–119, Berlin-Heidelberg-New York, Springer 1982Google Scholar
  41. [L 1975]
    Lax, P. D.: Periodic solutions of the KdV equation. Comm. Pure Appl. Math. 28, 141–188 (1975)Google Scholar
  42. [L 1981]
    Russian]; English transi. in Math. USSR–Izv. 18, 249–273 (1982)Google Scholar
  43. [L 1984]
    Levitan, B. M.: Inverse Sturm-Liouville Problems. Moscow, Nauka 1984 [Russian]Google Scholar
  44. [LL 1965]
    Landau, L. D., Lifshitz, E. M.: Mechanics. Course of Theoretical Physics, vol. I. Moscow, Nauka 1965 [Russian]; English transl. Oxford-LondonNew York-Paris, Pergamon Press 1960Google Scholar
  45. [M 1974]
    Marchenko, V. A.: The periodic Korteweg-de Vries problem. Mat. Sb. 95, 331–356 (1974) [Russian]Google Scholar
  46. [M 1977]
    Marchenko, V. A.: Sturm-Liouville Operators and their Applications. Kiev, Naukova Dumka 1977 [Russian]Google Scholar
  47. [M 1978]
    Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19, 1156–1162 (1978)ADSMathSciNetCrossRefMATHGoogle Scholar
  48. [MKM 1975]
    McKean, H. P., van Moerbeke, P.: The spectrum of Hill’s equation. Invent. Math. 30, 217–274 (1975)ADSMathSciNetCrossRefMATHGoogle Scholar
  49. [MKT 1976]
    McKean, H. P., Trubowitz, E.: Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points. Comm. Pure Appl. Math. 29, 143–226 (1976)MATHGoogle Scholar
  50. [N 1974]
    Russian]; English transl. in Func. Anal. Appl. 8, 236–246 (1974)Google Scholar
  51. [N 1984]
    Novikov, S. P.: Algebraic-topological approach to reality problems. Realaction variables in the theory of finite-gap solutions of the sine-Gordon equation. In: Differential geometry, Lie groups and mechanics. VI. Zapiski Nauchn. Semin. LOMI 133,177–196 (1984) [Russian]Google Scholar
  52. [Ne 1978]
    Newell, A. C.: Near-integrable systems, nonlinear tunneling and solutions in slowly changing media. In: Calogero, F. (ed.). Nonlinear evolution equations solvable by the spectral transform. Research Notes in Mathematics 26, 127–179, London, Pitman 1978Google Scholar
  53. [S 1979 a]
    Russian]; English transi. in Soy. Phys. Dokl. 24, 107–110 (1979)Google Scholar
  54. [S 1979b]
    Sklyanin, E. K.: On complete integrability of the Landau-Lifshitz equation. Preprint LOMI, E-3–79, Leningrad 1979Google Scholar
  55. [S 1980]
    Sklyanin, E. K.: Quantum version of the inverse scattering problem method. In: Differential geometry, Lie groups and mechanics. III. Zapiski Nauchn. Semin. LOMI 95, 55–128 (1980) [Russian]; English transi. in J. Sov. Math. 19, 1546–1595 (1982)Google Scholar
  56. [SF 1978]
    Russian]; English transi. in Sov. Phys. Dokl. 23, 902–906 (1978)Google Scholar
  57. [STF 1979]
    -220 (1979) [Russian]; English transi. in Theor. Math. Phys. 40 (2) 688–706 (1980)Google Scholar
  58. [T 1973]
    Takhtajan, L. A.: Hamiltonian systems connected with the Dirac equation, in: Differential geometry, Lie groups and mechanics. I. Zapiski Nauchn. Semin. LOMI 37, 66–76 (1973) [Russian]; English transl. in J. Sov. Math. 8, 219–228 (1977)Google Scholar
  59. [TF 1979]
    Russian]; English trans!. in Russian Math. Surveys 34 (5), 11–68 (1979)Google Scholar
  60. [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
  61. [VN 1982]
    Russian]; English transl. in Sov. Math. Dokl. 26, 357–362 (1983)Google Scholar
  62. [VN 1984]
    Veselov, A. P., Novikov, S. P.: Poisson brackets and complex tori. Trudy Mat. Inst. Steklov 165,49–61 (1984) [Russian]Google Scholar
  63. [ZF 1971]
    Russian]; English transi. in Func. Anal. Appl. 5, 280–287 (1972)Google Scholar
  64. [ZM 1974]
    Russian]; English transl. in Theor. Math. Phys. 19, 551–560 (1975)Google Scholar
  65. [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 transl.: New York, Plenum 1984Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

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

Personalised recommendations