Skip to main content
SpringerLink
Log in
Book cover

Harmonic Maps and Integrable Systems pp 7–28Cite as

A Historical Introduction to Solitons and Bäcklund Transformations

  • Chapter

Part of the book series: Aspects of Mathematics ((ASMA,volume E 23))

Abstract

Soliton theory developed after the discovery by Gardner, Greene, Kruskal and Miura (GGKM)[11] of the Inverse Scattering Transform for the Korteweg de Vries (KdV) equation (see 1.1 below). They had been led to this by the earlier discovery of solitons by Kruskal and Zabusky [35], who were studying the Fermi-Pasta-Ulam problem of 1—dimensional lattices. Thus started the modern development of soliton theory.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   49.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   64.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. 149, CUP, Cambridge, 1991.

    Book  MATH  Google Scholar 

  2. M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H. Segur, The inverse scattering transform–Fourier analysis for nonlinear problems, Stud.Appl.Math. 53 (1974), 249–315.

    MathSciNet  Google Scholar 

  3. M. Antonowicz and A.P. Fordy. Hamiltonian structures of nonlinear evolution equations, in: Fordy [7], 273–312.

    Google Scholar 

  4. A.I. Bobenko. Surfaces in terms of 2 by 2 matrices. Old and new integrable cases,this volume.

    Google Scholar 

  5. J. Bolton and L.M. Woodward. The affine Toda equations and minimal surfaces, this volume.

    Google Scholar 

  6. A.P. Fordy. Equations associated with simple Lie algebras and symmetric spaces,in: Fordy [7], 315–337.

    Google Scholar 

  7. A.P. Fordy, ed. Soliton Theory: A Survey of Results, MUP, Manchester, 1990.

    Google Scholar 

  8. A.P. Fordy and J. Gibbons, Factorization of operators I: Miura transformations, J. Math. Phys 21 (1980), 2508–2510.

    Article  MathSciNet  MATH  Google Scholar 

  9. A.P. Fordy and J. Gibbons. Integrable nonlinear Klein-Gordon equations and Toda lattices, Commun. Math. Phys. 77 (1980), 21–30.

    Article  MathSciNet  MATH  Google Scholar 

  10. C.S. Gardner, The Korteweg-de Vries equation and generalizations IV. The Kortewegde Vries equation as a Hamiltonian system, J. Math.Phys. 12 (1971), 1548–1551.

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  12. C.S. Gardner, J.M. Greene, M.D. Kruskal, and R.M. Miura, The Korteweg-de Vries equation and generalizations VI. Methods for exact solution, Commun. Pure Appl. Math. 27 (1974), 97–133.

    Article  MathSciNet  MATH  Google Scholar 

  13. I.M. Gel’fand and L. Dikii, Fractional powers of operators and Hamiltonian systems, Funkts. Anal. Prilozh. 10 (1976), 13–29

    MathSciNet  MATH  Google Scholar 

  14. I.M. Gel’fand and L. Dikii, Fractional powers of operators and Hamiltonian systems, Funct. Anal. Appl. 10 (1976), 259–273.

    Article  Google Scholar 

  15. I.M. Gel’fand and B.M. Levitan. On the determination of a differential equation from its spectral function, Izv. Akad. Nauk. SSR Ser.Math 15 (1951), 309–360

    MathSciNet  Google Scholar 

  16. I.M. Gel’fand and B.M. Levitan. On the determination of a differential equation from its spectral function, Amer. Math. Soc. Trans!. Ser. 2, 1 (1955), 259–309.

    Google Scholar 

  17. R. Hirota, Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Letts. 27 (1972), 1192–1194.

    Article  Google Scholar 

  18. D.J. Kaup and A.C. Newell, An exact solution for a derivative nonlinear Schrödinger equation, J. Math.Phys. 19 (1978), 798–801.

    Article  MathSciNet  MATH  Google Scholar 

  19. B.G. Konopelchenko, Nonlinear Integrable Equations, Springer, Berlin, 1987.

    Book  MATH  Google Scholar 

  20. D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. Ser. 5, 39 (1895), 422–443.

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. I. McIntosh. Infinite dimensional Lie groups and the two-dimensional Toda lattice,this volume.

    Google Scholar 

  23. M. Manas, The principal chiral model as an integrable system,this volume.

    Google Scholar 

  24. M. Melko and I. Sterling, Integrable systems, harmonic maps and the classical theory of surfaces, this volume.

    Google Scholar 

  25. R.M. Miura, Korteweg–de Vries equation and generalizations I. A remarkable explicit nonlinear transformation, J. Math. Phys. 9 (1968), 1202–1204.

    Article  MathSciNet  MATH  Google Scholar 

  26. R.M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Review, 18 (1976), 412–559.

    Article  MathSciNet  MATH  Google Scholar 

  27. T. Miwa, Infinite-dimensional Lie algebras of hidden symmetries of soliton equations, in: Fordy [7], 338–353.

    Google Scholar 

  28. A.C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, 1985.

    Google Scholar 

  29. J.J.C. Nimmo Hirota’s method, in Fordy [7], 75–96.

    Google Scholar 

  30. S.P. Novikov, S.V. Manakov, L.P. Pitaevskii and V.E. Zakharov, Theory of Solitons, Plenum, NY, 1984.

    MATH  Google Scholar 

  31. C. Rogers, Bäcklund transformations in soliton theory, in: Fordy [7], 97–130.

    Google Scholar 

  32. J. Scott Russell. Report on waves, Fourteenth meeting of the British Association for the Advancement of Science, 1844.

    Google Scholar 

  33. E. Schrödinger, A method of determining quantum mechanical eigenvalues and eigen-functions, Proc. R. Ir. Acad. 46A (1940), 9–16.

    MATH  Google Scholar 

  34. N.J. Vilenkin. Special Functions and the Theory of Group Representations, Amer. Math. Soc., Providence, R. I. 1968.

    Google Scholar 

  35. R.S. Ward. Sigma models in 2 + 1 dimensions, this volume.

    Google Scholar 

  36. J.C. Wood, Harmonic maps into symmetric spaces and integrable systems,this volume.

    Google Scholar 

  37. N.J. Zabusky and M.D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965), 240–243.

    Article  MATH  Google Scholar 

  38. V.E. Zakharov and A.V. Mikhailov. Relativistically invariant two-dimensional model of field theory which is integrable by means of the inverse scattering problem method, Zh. Eksp. Teor.Fiz. 74 (1978), 1953–1973

    MathSciNet  Google Scholar 

  39. V.E. Zakharov and A.V. Mikhailov. Relativistically invariant two-dimensional model of field theory which is integrable by means of the inverse scattering problem method, Soy. Phys. JETP 47 (1978) 1017–1027.

    Google Scholar 

  40. V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self focusing and one-dimensional self modulation of waves in nonlinear media,Zh. Eksp. Teor. Fiz. 61 (1971)118–134

    Google Scholar 

  41. V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self focusing and one-dimensional self modulation of waves in nonlinear media, Sov. Phys. JETP 34 (1972), 62–69.

    MathSciNet  Google Scholar 

  42. V.E. Zakharov and A.B. Shabat., A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem I, Funkts. Anal. Prilozh. 8 (1974), 43–53

    Google Scholar 

  43. V.E. Zakharov and A.B. Shabat., A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem I, Func. Anal. Appl. 8 (1974), 226–235.

    Article  MATH  Google Scholar 

Download references

Authors
  1. A. P. Fordy
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Mathematics, Univ. of Leeds, LS2 9JT, Leeds, UK

    A. P. Fordy  & J. C. Wood  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer Fachmedien Wiesbaden

About this chapter

Cite this chapter

Fordy, A.P. (1994). A Historical Introduction to Solitons and Bäcklund Transformations. In: Fordy, A.P., Wood, J.C. (eds) Harmonic Maps and Integrable Systems. Aspects of Mathematics, vol E 23. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-14092-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-663-14092-4_2

  • Publisher Name: Vieweg+Teubner Verlag, Wiesbaden

  • Print ISBN: 978-3-528-06554-6

  • Online ISBN: 978-3-663-14092-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation