Skip to main content
SpringerLink
Log in

Solitons in Mathematics

  • Conference paper
Structural Stability in Physics

Part of the book series: Springer Series in Synergetics ((SSSYN,volume 4))

  • 516 Accesses

Abstract

This second article presents a brief history of the inverse scattering method for solving nonlinear evolution equations and the Hamiltonian structure associated with it. It is not a comprehensive survey of the different mathematics now concerned with soliton theory. To attempt the latter would be impossible in the present compass. In any case, soliton theory already ramifies into areas of mathematics, algebraic geometry, theory of Jacobian varieties, on the edge of the mathematical range of one of us (RKB).

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

Access this chapter

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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. J.S. Russell, Report on Waves. British Association Reports (1844)

    Google Scholar 

  2. D.J. Korteweg and G. de Vries, Phil. Mag. 39, 422 (1895)

    Google Scholar 

  3. R.K. Bullough and P.J. Caudrey, ‘History of the soliton’ in “Solitons”, Springer Topics in Modern Physics Series. R.K. Bullough and P.J. Caudrey Eds. (Springer-Verlag, Heidelberg) To be published 1978.

    Google Scholar 

  4. G.B. Whitham, Linear and Non-linear Waves (John Wiley & Sons, New York, 1974) pp.580–585

    Google Scholar 

  5. J.D. Gibbon and J.C. Eilbeck, J.Phys. A: Gen. Phys. 5, L132 (1972)

    Article  ADS  Google Scholar 

  6. R.K. Dodd and R.K. Bullough, Proc. Roy. Soc. A 351, 499 (1976)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. R.K. Bullough, Solitons in Interaction of Radiation and Condensed Matter Vol.I, IAEA-SMR-20/51 (International Atomic Energy Agency, Vienna, 1977) pp.381–469

    Google Scholar 

  8. J.M. Burgers, Adv. Appl. Mech. 1, 171 (1948)

    Article  MathSciNet  Google Scholar 

  9. S. Coleman, Classical lumps and their quantum descendants Lectures at the 1975 international School of Subnuclear Physics “Ettore Majorana” (1975)

    Google Scholar 

  10. R. Jackiw, Rev. Mod. Phys. 49, 681 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  11. A.A. Belavin and V.E. Zakharov, Pis’ma Zh. Eksp. Teor. Fiz. 25, 603 (1977) (JETP Lett. 25, 567 (1978))

    Google Scholar 

  12. H. Haken, in Synergetics in Cooperative Phenomena Edited by H. Haken (North Holland, Amsterdam, 1974)

    Google Scholar 

  13. J.A. Krumhansl and J.R. Schrieffer, Phys. Rev. B 11, 3535 (1975)

    Article  ADS  Google Scholar 

  14. R.F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D 10, 4130 (1974)

    Article  ADS  Google Scholar 

  15. P.J. Caudrey, J.C. Eilbeck and J.D. Gibbon, Il Nuovo Cimento 25, 497 (1975)

    Article  Google Scholar 

  16. J.D. Gibbon and J.C. Eilbeck, J. Phys.A: Gen. Phys. 5, L122 (1972)

    Article  ADS  Google Scholar 

  17. P.J. Caudrey, J.C. Eilbeck and J.D. Gibbon, J. Inst. Math. Applics. 14, 375 (1975)

    Article  MathSciNet  Google Scholar 

  18. R. Hirota, Phys. Rev. Lett. 27, 1192 (1971)

    Article  ADS  MATH  Google Scholar 

  19. M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Phys. Rev. Lett. 31, 125 (1973)

    Article  MathSciNet  ADS  Google Scholar 

  20. M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Studies in Appl. Math. 53, 249 (1974)

    MathSciNet  Google Scholar 

  21. E. Fermi, J.R. Pasta and S.M. Ulam, Studies of nonlinear problems I, Los Alamos Rept. LA-1940 (May 1955) and Collected Works of E. Fermi Vol.II (Univ. of Chicago Press, 1965) pp.978–88

    Google Scholar 

  22. J. Gibbons, S.G. Thornhill, M.J. Wardrop, and D. ter Haar, On the theory of Langmuir Solitons, preprint Univ. of Oxford, Dept. of Theoretical Phys. Ref.36/76 (1976)

    Google Scholar 

  23. V.E. Zakharov, Zh. Eksp. Teor. Fiz. 62, 1745 (1972) (Soviet Phys. J.E.T.P. 35, 908 (1975)

    Google Scholar 

  24. G.L. Lamb, Rev. Mod. Phys. 43, 99 (1971)

    Article  MathSciNet  ADS  Google Scholar 

  25. J.C. Eilbeck, P.J. Caudrey, J.D. Gibbon and R.K. Bullough, J.Phys.A: Math. Nucl. & Gen. 6, 1337 (1973)

    Article  ADS  Google Scholar 

  26. R.K. Bullough, P.M. Jack and P.W. Kitchenside, Physica Scripta (1978)

    Google Scholar 

  27. R.F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D 11, 3424 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  28. B.B. Kadomtsev and V.I. Petviashvili, Dokl. Akad. Nauk. SSR, 192, 753 (1970)

    Google Scholar 

  29. V.E. Zakharov and A.B. Shabat, Funkt. Anal, i Ego Prilozh. 8, 43 (1974)

    Google Scholar 

  30. S.V. Manakov, V.E. Zakharov, L.A. Bordag, A.R. Its and V.B. Matveev, Phys. Lett. 63A, 205 (1977)

    ADS  Google Scholar 

  31. H.C. Yuen and B.M. Lake, Phys. Fluids 18, 956 (1975)

    Article  ADS  MATH  Google Scholar 

  32. V.E. Zakharov and A.B. Shabat, Zh. Eksp. Teor. Fiz. 64, 1627 (1973) (Sov. Phys. JETP 37, 823 (1973))

    Google Scholar 

  33. R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics Vol.III (Addison-Wesley, Reading, Mass. 1965) pp.21.14–21.18

    MATH  Google Scholar 

  34. H.M. Gibbs and R.E. Slusher, Phys. Rev. A 6, 2326 (1972)

    Article  ADS  Google Scholar 

  35. T.A. Fulton and R.C. Dynes, Solid State Comm. 12, 57 (1973)

    Article  ADS  Google Scholar 

  36. S.P. Novikov, Funkt Anal, i Ego Prilozh. 8, 54 (1974)

    Article  Google Scholar 

  37. P.D. Lax, Comm. Pure and Appl. Maths. 28, 141 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  38. R.K. Bullough, P.J. Caudrey, J.D. Gibbon, S. Duckworth, H.M. Gibbs, B. Bölger and L. Baede, Optics Communications 18, 200 (1976)

    Article  ADS  Google Scholar 

  39. R.K. Bullough and P.J. Caudrey, Bumping spin waves in the B-phase of liquid 3 He. Preprint (April, 1977)

    Google Scholar 

  40. R.K. Bullough and R.K. Dodd, Solitons I Basic Concepts. II Mathematical Structures in Synergetics. A Workshop ed. by H. Haken (Springer-Verlag, Heidelberg, 1977) pp.92–103 and 104–119

    Chapter  Google Scholar 

  41. N. Zabusky and M.D. Kruskal, Phys. Rev. Lett. 15, 240 (1965)

    Article  ADS  MATH  Google Scholar 

  42. C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Phys. Rev. Lett. 19, 1095 (1967)

    Article  ADS  MATH  Google Scholar 

  43. J. Liouville, J. Mathematiques Pures et Appliquées (Paris) 18, (1) 71–72 (1853)

    Google Scholar 

  44. R.K. Dodd and R.K. Bullough, Proc. Roy. Soc. (London) A 351, 499 (1976)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. R.K. Dodd and R.K. Bullough, “Integrability of nonlinear evolution equations: prolongations and solitons” To be published

    Google Scholar 

  46. Details communicated by W. Shadwick, Warsaw Meeting, September (1977)

    Google Scholar 

  47. F. Lund, Phys. Rev. Lett. 38, 1175 (1977); Proc. of NATO Advanced Study Institute on Nonlinear Problems in Physics and Mathematics (Istanbul, August 1977), ed. by A.O. Barut (D. Reidel Publishing Co., Dordrecht, Holland 1978)

    Article  MathSciNet  ADS  Google Scholar 

  48. V.E. Zakharov and A.B. Shabat, Zh. Eksp. Teor. Fis. (Soviet) 61, 118 (1971); JETP (Soviet) 34, 62 (1972)

    Google Scholar 

  49. K. Pohlmeyer, Comm. Math. Phys. 46, 207 (1976); New Developments in Quantum Field Theory and Statistical Mechanics, ed. by M. Levy and P. Nitter (Plenum Press, New York 1977) p.339

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16, 1 (1975); 17, 1293 (1976)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. V.L. Ginzburg and L.P. Pitaevskii, JETP 34, 1240 (1958)

    MathSciNet  Google Scholar 

  52. L.P. Pitaevskii, JETP 35, 408 (1968)

    Google Scholar 

  53. R.Y. Chiao, E. Garmire and C.H. Townes, Phys. Rev. Lett. 13, 479 (1964)

    Article  ADS  Google Scholar 

  54. P.L. Kelley, Phys. Rev. Lett. 15, 1005 (1965)

    Article  ADS  Google Scholar 

  55. T.B. Benjamin and J.E. Feir, J. Fluid. Mech. 27, 417 (1966)

    Article  ADS  Google Scholar 

  56. D.J. Benney and A.C. Newell, J. Math. Phys. 46, 133 (1967)

    MathSciNet  MATH  Google Scholar 

  57. V.I. Bespalov, A.G. Litvak and V.I. Tulanov, Nauka, 2nd All-Union Symposium on Nonlinear Optics, Collection of Papers, (Russian)(Moscow, 1968)

    Google Scholar 

  58. M.D. Kruskal, in Proceedings of Symposium on Nonlinear Evolution Equations Solvable by the Inverse Spectral Transform, Accademia del Lincei, Home, June 1977, ed. by F. Calogero (Pitman V London 1978); Proceedings of NATO Advanced Study Institute on Nonlinear Problems in Physics and Mathematics (Istanbul, August 1977) ed. by A.O. Barut (D. Reidel Publishing Co., Dordrecht, Holland 1978)

    Google Scholar 

  59. P.D. Lax, Comm. Pure Appl. Maths. 21, 467 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  60. R.K. Bullough, in Proceedings of NATO Advanced Study Institute on Nonlinear Problems in Physics and Mathematics (Istanbul, August 1977), ed. by A.O. Barut (D. Reidel Publishing Co., Dordrecht, Holland 1978)

    Google Scholar 

  61. R.M. Miura, C.S. Gardner and M.D. Kruskal, J. Math. Phys. 9, 1204 (1968)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  62. R.M. Miura, J. Math. Phys. 9, 1202 (1968)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  63. F. Calogero, Nonlinear evolution equations solvable by the inverse spectral transform, invited lecture presented at International Conference on Mathematical Problems in Theoretical Physics, Rome University (June 6–15, 1977)

    Google Scholar 

  64. A.C. Newell, The Inverse Scattering Transform, in Solitons, Springer Topics in Modern Physics Series, ed. by R.K. Bullough and P.J. Caudrey (Springer-Verlag, Berlin, Heidelberg, New York, 1978)

    Google Scholar 

  65. R. Hirota, in, for example, Direct Methods in Soliton Theory, in Solitons Springer Topics in Modern Physics Series, ed. R.K. Bullough and P.J. Caudrey (Springer-Verlag, Berlin, Heidelberg, New York, 1978)

    Google Scholar 

  66. P.J. Caudrey, Proceedings of NATO Advanced Study Institute on Nonlinear Problems in Physics and Mathematics (Istanbul, August 1977), ed. by A.O. Barut (D. Reidel Publishing Co., Dordrecht, Holland, 1978)

    Google Scholar 

  67. R. Hirota, Phys. Rev. Lett. 27, 1192 (1971)

    Article  ADS  MATH  Google Scholar 

  68. P.J. Caudrey, J.C. Eilbeck and J.D. Gibbon, J. Inst. Maths. Applics. 14, 375 (1974)

    Article  MathSciNet  Google Scholar 

  69. I.M. Gel’fand and L. Dikii, Uspeki mat. nauk. 30, 67 (1975); Russian Maths. Surveys 30, 77 (1975), Funkt. Anal, i Ego Prilog. 10, 18 (1976)

    MathSciNet  Google Scholar 

  70. Yu. I. Manin, Itogi Nauki i Tekniki 11, 5 (1978)

    MathSciNet  Google Scholar 

  71. S.P. Novikov, Funkt. Anal, i Ego Prilozh. 8, (3)54 (1974)

    Google Scholar 

  72. B.A. Dubrovin, I.M. Krichever and S.P. Novikov, Dokl. AN SSSR(1976)

    Google Scholar 

  73. C.S. Gardner, J. Math. Phys. 12, 1548 (1971)

    Article  ADS  MATH  Google Scholar 

  74. H. Flanders: Differential Forms with Applications to the Physical Sciences (Academic Press, New York, 1963)

    MATH  Google Scholar 

  75. This argument and some earlier remarks were partly stimulated by access to notes by Prof. David Simms on Lectures by H.P. McKean at Calgary, 1978. We have not been able to check original sources and rely on our recollections

    Google Scholar 

  76. V.E. Zakharov and L.D. Fadeev, Funkt. Anal, i Ego Prilozh. 5, 18 (1971)

    Google Scholar 

  77. M. Toda, in Studies on a Nonlinear Lattice, Ark. for Der Fysiske Sem., Trondheim 2 (1974); On a Nonlinear Lattice - the Toda Lattice in Solitons, Springer Topics in Modern Physics Series, ed. by R.K. Bullough and P.J. Caudrey (Springer-Verlag, Berlin, Heidelberg, New York, 1978)

    Google Scholar 

  78. F. Calogero, A. Degasperis, in Solitons, Springer Topics in Modern Physics Series, ed. by R.K. Bullough and P.J. Caudrey (Springer-Verlag, Berlin, Heidelberg, New York, 1978); Proc. NATO Advanced Study Institute on Nonlinear Problems in Physics and Mathematics (Istanbul, August 1977), ed. by A.O. Barut (D. Reidel Publishing Co., Dordrecht, Holland 1978)

    Google Scholar 

  79. R.K. Dodd and R.K. Bullough, The Generalized Marchenko Equation and the Canonical Structure of the AKNS-ZS Inverse Method (to be published 1978)

    Google Scholar 

  80. R.K. Dodd and R.K. Bullough, Phys. Lett. 62A, 70 (1977)

    MathSciNet  ADS  Google Scholar 

  81. A.C. Newell, “The general structure of integrable evolution equations” (Preprint 1977)

    Google Scholar 

  82. J.D. Gibbon, P.J. Caudrey, R.K. Bullough and J.C. Eilbeck, Lett, al Nuovo Cimento 8, 775 (1973)

    Article  Google Scholar 

  83. M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Phys. Rev. Lett. 30, 1262 (1973)

    Article  MathSciNet  ADS  Google Scholar 

  84. L.A. Taktadjan and L.D. Fadeev, Teor. Mat. Fis. 21, 160 (1974)

    Google Scholar 

  85. R.K. Dodd and R.K. Bullough, Proc. Roy. Soc. (London) A 352, 481 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  86. R.F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D 11, 3424 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  87. V.E. Korepin and L.D. Fadeev, Teor. Mat. Fis. 25, 47 (1975)

    Google Scholar 

  88. A Luther, Phys. Rev. B. 14, 2153 (1976)

    Article  ADS  Google Scholar 

  89. A.R. Bishop, in Solitons and Condensed Matter Physics, ed. by A.R. Bishop and T. Schneider (Springer-Verlag, Heidelberg, 1978)

    Google Scholar 

  90. A.C. Newell, in Solitons and Condensed Matter Physics, ed. by A.R. Bishop and T. Schneider (Springer-Verlag, Heidelberg, 1978)

    Google Scholar 

  91. P.W. Kitchenside, A.L. Mason, R.K. Bullough and P.J. Caudrey, in Solitons and Condensed Matter Physics, ed. by A.R. Bishop and T. Schneider (Springer-Verlag, Heidelberg, 1978)

    Google Scholar 

  92. R.K. Dodd, Proc. NATO Advanced Study Institute on Nonlinear Problems in Physics and Mathematics (Istanbul, August 1977), ed. by A.O. Barut (D. Reidel Publishing Co., Dordrecht, Holland, 1978)

    Google Scholar 

  93. V.E. Zakharov and A.B. Shabat, Funkt. Analiz. i Ego Prilozh. 8, 43 (1974)

    Google Scholar 

  94. V.E. Zakharov, The inverse scattering method in Solitons, Springer Topics in Modern Physics Series, ed. by R.K. Bullough and P.J. Caudrey (Springer-Verlag, Berlin, Heidelberg, New York, 1978)

    Google Scholar 

  95. R.K. Bullough and R.K. Dodd, Solitons in Mathematics: Brief History in Solitons and Condensed Matter Physics, ed. by A.R. Bishop and T. Schneider (Springer-Verlag, Heidelberg, 1978).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Institute of Science and Technology, University of Manchester, P.O. Box 88, Manchester, M60 1QD, UK

    R. K. Bullough

  2. School of Mathematics, Trinity College, Dublin 2, Ireland

    R. K. Dodd

Authors
  1. R. K. Bullough
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. K. Dodd
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute for Information Sciences, University of Tübingen, D-7400, Tübingen, Fed. Rep. of Germany

    Werner Güttinger  & Horst Eikemeier  & 

  2. Department of Physics and Astronomy, University of Wyoming, Laramie, WY, 82071, USA

    Werner Güttinger

Rights and permissions

Reprints and permissions

Copyright information

© 1979 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bullough, R.K., Dodd, R.K. (1979). Solitons in Mathematics. In: Güttinger, W., Eikemeier, H. (eds) Structural Stability in Physics. Springer Series in Synergetics, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-67363-4_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-67363-4_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-67365-8

  • Online ISBN: 978-3-642-67363-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation