Skip to main content
SpringerLink
Log in

A Unified Approach to Recursion Operators

  • Conference paper
Solitons in Physics, Mathematics, and Nonlinear Optics

Part of the book series: The IMA Volumes in Mathematics and Its Applications ((IMA,volume 25))

  • 391 Accesses

Abstract

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.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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. M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segue, Stud. Appl. Math., 53 (1974), p. 249

    Google Scholar 

  2. M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segue, Phys. Rev. Lett., 30 (1983), p. 1262;

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  4. P.D. Lax, Comm. Pure Appl. Math., 21 (1968), p. 467.

    Article  MathSciNet  MATH  Google Scholar 

  5. W. Symes, J. Math. Phys., 20 (1979), p. 721.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. A.C. Newell, Proc. Roy. Soc. London Ser., A 365 (1979), p. 283.

    Article  MathSciNet  ADS  Google Scholar 

  7. H. Flaschka and A.C. Newell, Lecture Notes in Phys., 38 (1975), p. 355.

    Article  MathSciNet  ADS  Google Scholar 

  8. V.S. Gerdjikov, Lett. Math. Phys., 6 (1982), p. 315.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. M. Born, F. Pempinelli, and G.Z. Tu, Nuovo Cimento, B 79 (1984), p. 231.

    ADS  Google Scholar 

  10. B.G. Konopelchenko, Nonlinear Integrable Equations, Springer-Verlag, 270 (1988).

    Google Scholar 

  11. D.J. Kaup, Siam J. Appl. Math., 31 (1976), pp. 121–133.

    Article  MathSciNet  MATH  Google Scholar 

  12. D.J. Kaup and A.C. Newell, Proc. R. Soc. Lond., A, 361 (1976), pp. 113–446.

    Google Scholar 

  13. P.J. Olver, J. Math. Phys., 18 (1977), p. 1212.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. A.S. Fokas, J. Math. Phys., 21 (1980), pp. 1318–1325.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. B. Fuchssteiner, Nonlinear Anal, 3 (1979), p. 849.

    Article  MathSciNet  MATH  Google Scholar 

  16. A.S. Fokas and R.L. Anderson, J. Math. Phys., 23 (1982), p. 1066.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. F. Magri, J. Math. Phys., 19 (1979), p. 1156;

    Article  ADS  Google 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.

    Chapter  Google Scholar 

  19. A.S. Fokas and B. Fuchssteiner, Lett. Nuovo Cimento, 28 (1980), p. 299;

    Article  MathSciNet  Google Scholar 

  20. B. Fuchssteiner and A.S. Fokas, Phys. D, 4 (1981), p. 47.

    Article  MathSciNet  MATH  Google Scholar 

  21. I.M. Gel’fand and I. YA. Dorfman, Functional Anal. Appl., 13 (1979), p. 13;

    MathSciNet  MATH  Google Scholar 

  22. I.M. Gel’fand and I. YA. Dorfman, Functional Anal. Appl., 14 (1980), p. 71.

    MathSciNet  MATH  Google Scholar 

  23. F. Calogero and A. Degasperis, Nuovo Cimento B, 39 (1977), p. 1.

    Article  MathSciNet  ADS  Google Scholar 

  24. P.M. Santini and A.S. Fokas, Comm. Math. Phys., 115 (1988), pp. 375–419.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. A.S. Fokas and P.M. Santini, Comm. Math. Phys., 116 (1988), pp. 449–474.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. A.S. Fokas and P.M. Santini, Stud. Appl. Math., 75 (1986), p. 179.

    MathSciNet  MATH  Google Scholar 

  27. A.S. Fokas and P.M. Santini, J. Math. Phys., 29, (3) (1988), pp. 604–617.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. P.M. Santini, Bi-Hamiltonian formulation of the intermediate long wave equation, preprint INS# 80.

    Google Scholar 

  29. M. Boiti, J.J.P. Leon, and F. Pempinelli, Canonical and non-Canonical recursion operators in multidimensions, preprint, Montepellier (1987).

    Google Scholar 

  30. 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. 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. P.M. Santini, Dimensional Deformation of Integrable Systems: An Approach to Integrability in Multidimensions, preprint 1988.

    Google Scholar 

  33. P.M. Santini, Algebraic Structures Underlying Integrability and Solvable Algebraic Systems, (preprint 1988 ).

    Google Scholar 

  34. P. Olver, Applications of Lie Groups to Differential Equations, 107 Springer-Verlag (1986).

    Google Scholar 

  35. 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. A.S. Fokas and B. Fuchssteiner, Phys. Lett. A, 86 (1981), p. 341.

    Article  MathSciNet  ADS  Google Scholar 

  37. A.S. Fokas, Phys. Rev. Lett., 57 (1986), p. 159.

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Clarkson University, Potsdam, New York, 13676, USA

    A. S. Fokas

  2. Istituto di Fisica “Guglielmo Marconi”, University Degli Studi - Roma, Piazzale dell Scienze, 5, 1-00185, Roma, Italy

    P. M. Santini

Authors
  1. A. S. Fokas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. M. Santini
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Mathematics, University of Minnesota, 55455, Minneapolis, MN, USA

    Peter J. Olver  & David H. Sattinger  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Springer-Verlag New York Inc.

About this paper

Cite this paper

Fokas, A.S., Santini, P.M. (1990). A Unified Approach to Recursion Operators. In: Olver, P.J., Sattinger, D.H. (eds) Solitons in Physics, Mathematics, and Nonlinear Optics. The IMA Volumes in Mathematics and Its Applications, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9033-6_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-9033-6_5

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9035-0

  • Online ISBN: 978-1-4613-9033-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation