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Geometry of the akns — ZS inverse scattering scheme

  • R. Sasaki
  • R. K. Bullough
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 120)

Abstract

We review the geometrical theory of nonlinear evolution equations (NEEs) solvable by the AKNS1-generalised Zakharov-Shabat2 scattering problem introduced by one of us previously3,4. We show how the theory contains within it the canonical structure known to be associated with integrable NEEs. We exploit the “gaugerd transformations of the geometric theory to derive an infinite set of non-local Hamiltonian densities for the sine-Gordon equation. We show that it is from these that the hierarchy of Lax-type sine-Gordon equations can be derived. We summarise the relation between the geometric theory and the theory of prolungation structures due to Wahlquist and Estabrook.

Keywords

Riccati Equation Scattering Problem Canonical Structure Soliton Equation Linearise Dispersion Relation 
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References

  1. 1.
    M.J.Ablowitz, D.J.Kaup, A.C.Newell and H.Segur, Phys.Rev.Lett. 31 (1973) 125.Google Scholar
  2. 2.
    V.E.Zakharov and A.B.Shabat, JETP (Sov.Phys.) 34 (1972) 62.Google Scholar
  3. 3.
    R.Sasaki, Phys.Lett. 71A (1979) 390.Google Scholar
  4. 4.
    R.Sasaki, Nucl.Phys. B154 (1979) 343.Google Scholar
  5. 5.
    R.K.Dodd and R.K.Bullough, Phys.Scripta, 1979, in press.Google Scholar
  6. 6.
    R.Sasaki, Niels Bohr Inst. prep.,NBI-HE-79-31,Sept.1979,to be publ.in Proc.Roy.Soc.Google Scholar
  7. 7.
    R.Hermann, Phys.Rev.Lett. 36 (1976) 835, M.Crampin, Phys.Lett. 66A (1978) 170.Google Scholar
  8. 8.
    F.Lund, Phys.Rev.Lett. 38 (1977) 1175.Google Scholar
  9. 9.
    G.L.Lamb, Jr., Phys.Rev.Lett. 37 (1976) 235.Google Scholar
  10. 10.
    M.Lakshmanan, Phys.Lett. 64A (1978) 354.Google Scholar
  11. 11.
    A.Sym and J.Corones, Phys.Rev.Lett. 42 (1979) 1099.Google Scholar
  12. 12.
    R.Sasaki, Phys.Lett. 73A (1979) 77.Google Scholar
  13. 13.
    H.D.Wahlquist and F.B.Estabrook, J.Math.Phys. 16 (1975) 1.Google Scholar
  14. 14.
    M.J.Ablowitz, D.J.Kaup, A.C.Newell and H.Segur, Phys.Rev.Lett. 30 (1973) 1262.Google Scholar
  15. 15.
    P.D.Lax, Comm.Pure and Appl.Math. 21 (1968) 467.Google Scholar
  16. 16.
    M.J.Ablowitz, D.J.Kaup, A.C.Newell and H.Segur, Stud.in Math.Phys. 53 (1974) 249.Google Scholar
  17. 17.
    R.Sasaki,R.K.Bullough,Niels Bohr Ins t. prep. NBI-HE-79-32,to be publ.in Proc.Roy.Soc.Google Scholar
  18. 18.
    H.Flanders, “Differential Forms” (Academic Press, New York, 1963).Google Scholar
  19. 19.
    R.K.Bullough,P.W.Kitchenside,P.M.Jack and R.Saunders, Phys.Scripta,1979 in press.Google Scholar
  20. 20.
    L.P.Eisenhart, “A treatise on the differential geometry of curves and surfaces” (Dover Publ., New York, 1960).Google Scholar
  21. 21.
    “Solitons”, Springer Topics in Modern Physics Series, R.K.Bullough and P.J.Caudrey eds. (Springer-Verlag, Heidelberg) In press. To appear early 1980.Google Scholar
  22. 22.
    A.V. Bäcklund, Lund Univ. Arsskrift, 110 (1875) and 119 (1883).Google Scholar
  23. 23.
    K.Konno, H.Sanuki and Y.H.Ichikawa, Prog.Theor.Phys. 52 (1974) 886. M.Wadati, H.Sanuki and K.Konno, Prog.Theor.Phys. 53 (1975) 419.Google Scholar
  24. 24.
    R.M.Miura, C.S.Gardner and M.D.Kruskal, J.Math.Phys. 9 (1968) 1204.Google Scholar
  25. 25.
    R.K.Dodd and R.K.Bullough, Proc.Roy.Soc. A352 (1977) 481.Google Scholar
  26. 26.
    P.J. Caudrey, J.D. Gibbon, J.C. Eilbeck, R.K. Bullough, Phys.Rev.Lett. 30 (1973).Google Scholar
  27. 27.
    L.A.Takhtadzhyan and L.D.Faddev, Theor.and Math.Phys. 21 (1974) 1041.Google Scholar
  28. 28.
    H.Flaschka and A.C.Newell, in “Dynamical Systems Theory and Applications”, J.Moser ed. (Springer, Heidelberg 1975).Google Scholar
  29. 29.
    R.K.Bullough and R.K.Dodd, in “Synergetics. A Workshop”, H.Haken ed. (Springer, Heidelberg 1977).Google Scholar
  30. 30.
    R.K.Bullough and P.J.Caudrey, in “Nonlinear evolution equationssolvable by the spectral transform”, F.Calogero ed. (Pitman, London 1978). R.K.Bullough, in “Nonlinear equations in physics and mathematics”, A.O.Barut ed. (Reidel.Publ.Dordrecht, Holland 1978).Google Scholar
  31. 31.
    H. H. Chen, Phys.Rev.Lett. 33 (1974) 925. R.Konno and M.Wadati, Prog.Theor.Phys. 53 (1975) 1652.Google Scholar
  32. 32.
    J.Corones, J.Math.Phys. 18 (1977) 163, H.C.Morris, J.Math.Phys. 18 (1977) 533, R.K.Dodd and J.D.Gibbon, Proc.Roy.Soc. A359 (1978) 411.Google Scholar
  33. 33.
    D.Maisson, Phys.Rev.Lett. 41 (1978) 521.Google Scholar
  34. 34.
    B.K.Harrison, Phys.Rev.Lett. 41 (1978) 1197.Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • R. Sasaki
    • 1
  • R. K. Bullough
    • 2
  1. 1.Niels Bohr InstituteCopenhagen 0Denmark
  2. 2.NorditaCopenhagenDenmark

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