Skip to main content

Introduction

  • 446 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1232)

Keywords

  • Solitary Wave
  • Inverse Scattering
  • Nonlinear Evolution Equation
  • Nonlinear Partial Differential Equation
  • Nonlinear Wave Equation

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   29.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   39.95
Price excludes VAT (Canada)
  • 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.

References

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

    CrossRef  ADS  MathSciNet  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.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, Philadelphia, SIAM, 1981.

    CrossRef  MATH  Google Scholar 

  4. F. van der Blij, Some details of the history of the Korteweg-de Vries equation, Nieuw Archief voor Wiskunde 26 (1978), 54–64.

    MATH  Google Scholar 

  5. R.K. Bullough and P.J. Caudrey, The soliton and its history, in: Solitons, Topics in Current Physics 17, Springer, New York, 1980 (edited by the same).

    Google Scholar 

  6. F. Calogero and A. Degasperis, Spectral Transform and Solitons, Amsterdam, North-Holland, 1982.

    MATH  Google Scholar 

  7. R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, 1982.

    Google Scholar 

  8. W. Eckhaus and A. van Harten, The Inverse Scattering Transformation and the Theory of Solitons, North-Holland Mathematics Studies 50, 1981 (2nd ed. 1983).

    Google Scholar 

  9. 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.

    CrossRef  ADS  MATH  Google Scholar 

  10. C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Korteweg-de Vries equation and generalizations VI, Comm. Pure Appl. Math. 27 (1974), 97–133.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. I.M. Gel'fand and B.M. Levitan, On the determination of a differential equation from its spectral function, Izvest. Akad. Nauk 15 (1951), 309–360, AMST 1 (1955), 253–309.

    MathSciNet  Google Scholar 

  12. D.J. Kaup and A.C. Newell, The Goursat and Cauchy problems for the sine-Gordon equation, SIAM J. Appl. Math. 34 (1978), 37–54.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. 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, Phil. Mag. 39 (1895), 422–443.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. G.L. Lamb, Jr., Elements of Soliton Theory, Wiley-Interscience, 1980.

    Google Scholar 

  15. G.L. Lamb, Jr. and D.W. McLaughlin, Aspects of soliton physics, in: Solitons (Ed. R.K. Bullough and P.J. Caudrey) Topics in Current Physics 17, Springer, New York, 1980.

    CrossRef  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. R.M. Miura, The Korteweg-de Vries equation: A survey of results, SIAM Review 18 (1976), 412–459.

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  18. A.C. Scott, F.Y.F. Chu and D. McLaughlin, The soliton: a new concept in applied science, Proc. IEEE 61 (1973), 1443–1483.

    CrossRef  ADS  MathSciNet  Google Scholar 

  19. J. Scott Russell, Report on waves in: Report of the fourteenth meeting of the British association for the advancemen of science, John Murray, London, 1844, 311–390.

    Google Scholar 

  20. S. Tanaka, Modified Korteweg-de Vries equation and scattering theory, Proc. Japan Acad. 48 (1972), 466–489.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. S. Tanaka, On the N-tuple wave solutions of the Korteweg-de Vries equation, Publ. R.I.M.S. Kyoto Univ. 8 (1972), 419–427.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. S. Tanaka, Non-linear Schrödinger equation and modified Korteweg-de Vries equation; construction of solutions in terms of scattering data, Publ. R.I.M.S. Kyoto Univ. 10 (1975), 329–357.

    CrossRef  MATH  Google Scholar 

  23. M. Wadati, The exact solution of the modified Korteweg-de Vries equation, J. Phys. Soc. Japan 32 (1972) 1681.

    CrossRef  ADS  MathSciNet  Google Scholar 

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

    CrossRef  ADS  MATH  Google Scholar 

  25. V.E. Zakharov, S.V. Manakov, S.P. Novikov and L.P. Pitaievski, Theory of Solitons. The Inverse Problem Method, Nauka, Moscow, 1980 (in Russian).

    Google Scholar 

  26. V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media, Soviet Phys. JETP (1972), 62–69.

    Google Scholar 

Download references

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 1986 Springer-Verlag

About this chapter

Cite this chapter

Schuur, P.C. (1986). Introduction. In: Asymptotic Analysis of Soliton Problems. Lecture Notes in Mathematics, vol 1232. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073055

Download citation

  • DOI: https://doi.org/10.1007/BFb0073055

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17203-1

  • Online ISBN: 978-3-540-47387-9

  • eBook Packages: Springer Book Archive