A Unified Approach to Recursion Operators

  • A. S. Fokas
  • P. M. Santini
Conference paper
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 25)


The recent theory of recursion operators in multidimensions is reviewed. Furthermore, a simple, unifying, algorithmic way of constructing recursion operators is given. In particular it is shown that recursion operators in 1 + 1 and 2 + 1 are concrete realizations of more abstract structures.


Eigenvalue Problem Bilinear Form Poisson Bracket Hamiltonian Operator Linear Eigenvalue Problem 
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  1. [1]
    M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segue, Stud. Appl. Math., 53 (1974), p. 249Google Scholar
  2. M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segue, Phys. Rev. Lett., 30 (1983), p. 1262;ADSCrossRefGoogle Scholar
  3. M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segue, Phys. Rev. Lett., 31 (1973), p. 125.MathSciNetADSCrossRefGoogle Scholar
  4. [2]
    P.D. Lax, Comm. Pure Appl. Math., 21 (1968), p. 467.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [3]
    W. Symes, J. Math. Phys., 20 (1979), p. 721.MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. [4]
    A.C. Newell, Proc. Roy. Soc. London Ser., A 365 (1979), p. 283.MathSciNetADSCrossRefGoogle Scholar
  7. [5]
    H. Flaschka and A.C. Newell, Lecture Notes in Phys., 38 (1975), p. 355.MathSciNetADSCrossRefGoogle Scholar
  8. [6]
    V.S. Gerdjikov, Lett. Math. Phys., 6 (1982), p. 315.MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. [7]
    M. Born, F. Pempinelli, and G.Z. Tu, Nuovo Cimento, B 79 (1984), p. 231.ADSGoogle Scholar
  10. [8]
    B.G. Konopelchenko, Nonlinear Integrable Equations, Springer-Verlag, 270 (1988).Google Scholar
  11. [9]
    D.J. Kaup, Siam J. Appl. Math., 31 (1976), pp. 121–133.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [10]
    D.J. Kaup and A.C. Newell, Proc. R. Soc. Lond., A, 361 (1976), pp. 113–446.Google Scholar
  13. [11]
    P.J. Olver, J. Math. Phys., 18 (1977), p. 1212.MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. [12]
    A.S. Fokas, J. Math. Phys., 21 (1980), pp. 1318–1325.MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. [13]
    B. Fuchssteiner, Nonlinear Anal, 3 (1979), p. 849.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [14]
    A.S. Fokas and R.L. Anderson, J. Math. Phys., 23 (1982), p. 1066.MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. [15]
    F. Magri, J. Math. Phys., 19 (1979), p. 1156;ADSCrossRefGoogle Scholar
  18. F. Magri, in Nonlinear Evolution Equations and Dynamical Systems (M. Boiti, F. Pempinelli, and G. Soliani, Eds.), Lecture Notes in Phys., Vol 120, Springer, New York, (1980), p. 233.CrossRefGoogle Scholar
  19. [16]
    A.S. Fokas and B. Fuchssteiner, Lett. Nuovo Cimento, 28 (1980), p. 299;MathSciNetCrossRefGoogle Scholar
  20. B. Fuchssteiner and A.S. Fokas, Phys. D, 4 (1981), p. 47.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [17]
    I.M. Gel’fand and I. YA. Dorfman, Functional Anal. Appl., 13 (1979), p. 13;MathSciNetzbMATHGoogle Scholar
  22. I.M. Gel’fand and I. YA. Dorfman, Functional Anal. Appl., 14 (1980), p. 71.MathSciNetzbMATHGoogle Scholar
  23. [18]
    F. Calogero and A. Degasperis, Nuovo Cimento B, 39 (1977), p. 1.MathSciNetADSCrossRefGoogle Scholar
  24. [19]
    P.M. Santini and A.S. Fokas, Comm. Math. Phys., 115 (1988), pp. 375–419.MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. [20]
    A.S. Fokas and P.M. Santini, Comm. Math. Phys., 116 (1988), pp. 449–474.MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. [21]
    A.S. Fokas and P.M. Santini, Stud. Appl. Math., 75 (1986), p. 179.MathSciNetzbMATHGoogle Scholar
  27. [22]
    A.S. Fokas and P.M. Santini, J. Math. Phys., 29, (3) (1988), pp. 604–617.MathSciNetADSzbMATHCrossRefGoogle Scholar
  28. [23]
    P.M. Santini, Bi-Hamiltonian formulation of the intermediate long wave equation, preprint INS# 80.Google Scholar
  29. [24]
    M. Boiti, J.J.P. Leon, and F. Pempinelli, Canonical and non-Canonical recursion operators in multidimensions, preprint, Montepellier (1987).Google Scholar
  30. [25]
    F. Magri and C. Morosi, An algebraic approval to KP, in “Topics in Soliton Theory”, Ed. by M.J. Ablowitz, B. Fuchssteiner, and M. Kruscal, World Scientific (1987).Google Scholar
  31. [26]
    P.M. Santini and A.S. Fokas, The Bi-Hamiltonian formulation of Integrable Evolution Equations in Multidimensions, Balaruc Les Bains, France (1987).Google Scholar
  32. [27]
    P.M. Santini, Dimensional Deformation of Integrable Systems: An Approach to Integrability in Multidimensions, preprint 1988.Google Scholar
  33. [28]
    P.M. Santini, Algebraic Structures Underlying Integrability and Solvable Algebraic Systems, (preprint 1988 ).Google Scholar
  34. [29]
    P. Olver, Applications of Lie Groups to Differential Equations, 107 Springer-Verlag (1986).Google Scholar
  35. [30]
    A.S. Fokas and P.M. Santini, Conservation Laws for Integrable Systems, Bogota, Columbia, Feb. 22–27, 1988, ed. by G. Violini, D. Levi and P. Winternitz.Google Scholar
  36. [31]
    A.S. Fokas and B. Fuchssteiner, Phys. Lett. A, 86 (1981), p. 341.MathSciNetADSCrossRefGoogle Scholar
  37. [32]
    A.S. Fokas, Phys. Rev. Lett., 57 (1986), p. 159.MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • A. S. Fokas
    • 1
  • P. M. Santini
    • 2
  1. 1.Department of Mathematics and Computer ScienceClarkson UniversityPotsdamUSA
  2. 2.Istituto di Fisica “Guglielmo Marconi”University Degli Studi - RomaRomaItaly

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