On the Linear Equations of Soliton Theory

  • G. L. LambJr.
Conference paper


During the past few years there has been an increasing interest in solitons. The soliton and its remarkable properties are well known to those already familiar with the history of coherent optics since one of the earliest examples of a soliton was the 2π pulse of self-induced transparency [1]. In that instance a satisfactory mathematical setting for the soliton was completely provided by the coupled Maxwell and Schroedinger-Bloch equations. The situation is not as satisfactory in regard to the equations that have more recently been shown to exhibit soliton behavior, however. To solve these equations, certain linear equations analogous to the equations for the two-level atom have been introduced [2] in a somewhat ad hoc fashion.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S.L. McCall and E.L. Hahn, Phys. Rev. 183, 457 (1969).ADSCrossRefGoogle Scholar
  2. 2.
    M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Studies in Appl. Math. 53, 249 (1974).zbMATHMathSciNetGoogle Scholar
  3. 3.
    For other geometric approaches to solitons see: H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16, 1 (1975);ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. H.D. Wahlquist and F.B. Estabrook, 17, 1293 (1976);zbMATHMathSciNetGoogle Scholar
  5. F. Lund and T. Regge, Phys. Rev. D14, 1524 (1976);ADSzbMATHMathSciNetGoogle Scholar
  6. R. Hermann, Phys. Rev. Letters 37, 235 (1976);CrossRefGoogle Scholar
  7. M. Lakshmanan, Phys. Lett. 64A, 354 (1978).ADSCrossRefGoogle Scholar
  8. 4.
    L.P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces (Dover, New York, 1960), Sect. 13.Google Scholar
  9. 5.
    G.L. Lamb, Jr., Phys. Rev. A9, 422 (1974).ADSCrossRefGoogle Scholar
  10. 6.
    H. Hasimoto, J. Fluid Mech. 51, 477 (1972).ADSCrossRefzbMATHGoogle Scholar
  11. 7.
    The author is indebted to P.G. Saffman for bringing this work to his attention.Google Scholar
  12. 8.
    G.L. Lamb. Jr., Phys. Rev. Letters 37, 235 (1976)ADSCrossRefMathSciNetGoogle Scholar
  13. 9.
    R. Hirota, J. Math. Phys. 14, 805 (1974).ADSCrossRefMathSciNetGoogle Scholar
  14. 10.
    G.L. Lamb, Jr., J. Math. Phys, 18, 1654 (1977).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 11.
    W.E. Lamb, Jr., Phys. Rev. 134, A1429 (1964).ADSCrossRefGoogle Scholar
  16. 12.
    A similar transformation is considered by D.W. McLaughlin and J. Corones, Phys. Rev. A10, 2051 (1974).ADSCrossRefGoogle Scholar
  17. 13.
    M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, J. Math. Phys. 15, 1852 (1974).ADSCrossRefGoogle Scholar
  18. 14.
    A.C. Scott, F.Y.F. Chu and D.W. McLaughlin, Proc, IEEE 61, 1449 (1973).ADSCrossRefMathSciNetGoogle Scholar
  19. 15.
    F.T. Arecchi and R. Bonifacio, IEEE J. Quantum Electron. 1, 169 (1965).ADSCrossRefGoogle Scholar
  20. 16.
    G.L. Lamb., Jr., Rev. Mod. Phys. 43, 99 (1971).ADSCrossRefMathSciNetGoogle Scholar
  21. 17.
    R. Hoppe, J. f. Math. 60, 182 (1862).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • G. L. LambJr.
    • 1
  1. 1.University of ArizonaTucsonUSA

Personalised recommendations