Skip to main content
SpringerLink
Log in

The Hamiltonian Formulation

  • Chapter
Hamiltonian Methods in the Theory of Solitons

Part of the book series: Springer Series in Soviet Mathematics ((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

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQ % dadaqadaqaaiabeI8a5naabmaabaGaamiEaaGaayjkaiaawMcaaiaa % cYcacuaHipqEgaqeamaabmaabaGaamiEaaGaayjkaiaawMcaaaGaay % jkaiaawMcaaiabgkziUoaabmaabaGaamOyamaabmaabaGaeq4UdWga % caGLOaGaayzkaaGaaiilaiqadkgagaqeamaabmaabaGaeq4UdWgaca % GLOaGaayzkaaGaai4oaiaaykW7cqaH7oaBdaWgaaWcbaGaamOAaaqa % baGccaGGSaGaeq4SdC2aaSbaaSqaaiaadQgaaeqaaaGccaGLOaGaay % zkaaaaaa!57BB!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$f:\left( {\psi \left( x \right),\bar \psi \left( x \right)} \right) \to \left( {b\left( \lambda \right),\bar b\left( \lambda \right);\,{\lambda _j},{\gamma _j}} \right)$$

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arnold, V. I.: Mathematical Methods of Classical Mechanics. Moscow, Nauka 1974 [Russian]; English transl.: Graduate Texts in Mathematics 60, New York-Berlin-Heidelberg, Springer 1978

    Google Scholar 

  2. Alber, S. I.: On stationary problems for equations of Korteweg-de Vries type. Comm. Pure Appl. Math. 34, 259–272 (1981)

    Google Scholar 

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baxter, R. J.: Partition function of the eight-vertex lattice model. Ann. Of Physics (N.Y.) 70, 193–228 (1972)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. 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. Russian]; English transi. in Funct. Anal. Appl. 14, 260–267 (1980)

    Google Scholar 

  7. Russian]; English transi. in Funct. Anal. Appl. 10, 8–11 (1976)

    Google Scholar 

  8. Russian]; English transi. in Funct. Anal. Appl. 10, 92–95 (1976)

    Google Scholar 

  9. Calogero, F., Degasperis, A.: Nonlinear evolution equations solvable by the inverse spectral transform. I. Nuovo Cimento 32B, 201–242 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  10. Calogero, F., Degasperis, A.: Nonlinear evolution equations solvable by the inverse spectral transform. II. Nuovo Cimento 39B, 1–54 (1977)

    Article  MathSciNet  Google Scholar 

  11. Dirac, P. A. M.: Lectures on quantum mechanics. Belfer Grad. School of Science, Yeshiva University, N.-Y. 1964

    Google Scholar 

  12. Russian]; English transi. in Russian Math. Surveys 36 (2), 11–92 (1982)

    Google Scholar 

  13. Russian]; English transi. in Russian Math. Surveys 31 (1), 59–146 (1976)

    Google Scholar 

  14. Russian]; English trans!. in Soy. Phys. JETP 40, 1058–1063

    Google Scholar 

  15. Russian]; English transi. in Sov. Math. Dokl. 26, 760–765 (1983)

    Google Scholar 

  16. 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, 1985

    Google Scholar 

  17. 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. 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)

    MathSciNet  MATH  Google Scholar 

  19. Faddeev, L. D., Takhtajan, L. A.: Poisson structure for the KdV equation. Lett. Math. Phys. 10, 183–188 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Russian]; English transi. in Russian Math. Surveys 30 (5), 77–113 (1975)

    Google Scholar 

  21. Russian]; English transi. in Funct. Anal. Appl. 13, 6–15 (1979)

    Google Scholar 

  22. Russian]; English transi. in Funct. Anal. Appl. 14, 223–226 (1980)

    Google Scholar 

  23. Russian]; English transi. in Funct. Anal. Appl. 15, 173–187 (1981)

    Google Scholar 

  24. 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)

    MATH  Google Scholar 

  25. 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. 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. 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 1912

    Google Scholar 

  28. 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. 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. Izergin, A. G., Korepin, V. E.: The inverse scattering method approach to the quantum Shabat-Mikhailov model. Comm. Math. Phys. 79, 303–316 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Russian]; English transi. in Soviet J. Particles and Nuclei 13 (3), 207–223 (1982)

    Google Scholar 

  32. Russian]; English transi. in Theor. Math. Phys. 23, 343–355 (1976)

    Google Scholar 

  33. Kaup, D. J.: Closure of the squared Zakharov-Shabat eigenstates. J. Math. Anal. Appl. 54, 849–864 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  34. Russian]; English transi. in Russian Math. Surveys 32 (6), 185–213 (1977)

    Google Scholar 

  35. 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. Russian]; English transi. in Theor. Math. Phys. 28, 615–620 (1977)

    Google Scholar 

  37. 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. 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. 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. 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 1982

    Google Scholar 

  41. Lax, P. D.: Periodic solutions of the KdV equation. Comm. Pure Appl. Math. 28, 141–188 (1975)

    Google Scholar 

  42. Russian]; English transi. in Math. USSR–Izv. 18, 249–273 (1982)

    Google Scholar 

  43. Levitan, B. M.: Inverse Sturm-Liouville Problems. Moscow, Nauka 1984 [Russian]

    Google Scholar 

  44. 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 1960

    Google Scholar 

  45. Marchenko, V. A.: The periodic Korteweg-de Vries problem. Mat. Sb. 95, 331–356 (1974) [Russian]

    Google Scholar 

  46. Marchenko, V. A.: Sturm-Liouville Operators and their Applications. Kiev, Naukova Dumka 1977 [Russian]

    Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. 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)

    MATH  Google Scholar 

  50. Russian]; English transl. in Func. Anal. Appl. 8, 236–246 (1974)

    Google Scholar 

  51. 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. 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 1978

    Google Scholar 

  53. Russian]; English transi. in Soy. Phys. Dokl. 24, 107–110 (1979)

    Google Scholar 

  54. Sklyanin, E. K.: On complete integrability of the Landau-Lifshitz equation. Preprint LOMI, E-3–79, Leningrad 1979

    Google Scholar 

  55. 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. Russian]; English transi. in Sov. Phys. Dokl. 23, 902–906 (1978)

    Google Scholar 

  57. -220 (1979) [Russian]; English transi. in Theor. Math. Phys. 40 (2) 688–706 (1980)

    Google Scholar 

  58. 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. Russian]; English trans!. in Russian Math. Surveys 34 (5), 11–68 (1979)

    Google Scholar 

  60. 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. Russian]; English transl. in Sov. Math. Dokl. 26, 357–362 (1983)

    Google Scholar 

  62. Veselov, A. P., Novikov, S. P.: Poisson brackets and complex tori. Trudy Mat. Inst. Steklov 165,49–61 (1984) [Russian]

    Google Scholar 

  63. Russian]; English transi. in Func. Anal. Appl. 5, 280–287 (1972)

    Google Scholar 

  64. Russian]; English transl. in Theor. Math. Phys. 19, 551–560 (1975)

    Google Scholar 

  65. 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

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, Fontanka 27, 191011, Leningrad, USSR

    Ludwig D. Faddeev & Leon A. Takhtajan

Authors
  1. Ludwig D. Faddeev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leon A. Takhtajan
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Faddeev, L.D., Takhtajan, L.A. (2007). The Hamiltonian Formulation. 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_4

Download citation

Publish with us

Policies and ethics

Search

Navigation