Journal of Nonlinear Science

, Volume 2, Issue 1, pp 109–134 | Cite as

The dressing method and nonlocal Riemann-Hilbert problems

  • A. S. Fokas
  • V. E. Zakharov


We consider equations in 2+1 solvable in terms of a nonlocal Riemann-Hilbert problem and show that for such an equation there exists a unified dressing method which yields: (i) a Lax pair suitable for obtaining solutions that are perturbations of an arbitrary exact solution of the given equation; (ii) certain integrable generalizations of the given equation. Using this generalized dressing method large classes of solutions of these equations, including dromions and line dromions, can be obtained. The method is illustrated by using theN-wave interactions, the Davey-Stewartson I, and the Kadomtsev-Petviashvili I equations. We also show that a careful application of the usual dressing method yields a certain generalization of theN-wave interactions.

Key words

solitons in two spatial dimensions dressing method 

AMS/MOS classification numbers

58F07 35R58 35Q51 


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  1. [1]
    M. J. Ablowitz and H. Segur,Solitons and the Inverse Scattering Transform, SIAM Stud. Appl. Math., Philadelphia (1981).Google Scholar
  2. [2]
    V. G. Makhankov,Soliton Phenomenology, Kluwer Academic Publishers, Mathematics and Its Applications, Holland (1990).Google Scholar
  3. [3]
    A. S. Davydov,Solitons in Molecular Systems, D. Reidel Publishing Co., Mathematics and Its Applications, Holland (1985).Google Scholar
  4. [4]
    D. J. Kaup, Physica6D, 143–54 (1983).Google Scholar
  5. [5]
    O. Legrand and C. Montes, J.E. Phys.50, 3–147 (1989).Google Scholar
  6. [6]
    F. Calogero, inWhat is Integrability?, ed. by V. E. Zakharov, Springer-Verlag, New York (1990).Google Scholar
  7. [7]
    M. Boiti, J. Leon, L. Martina, and F. Pempinelli, Phys. Lett. A,132, 432 (1988).MathSciNetCrossRefGoogle Scholar
  8. [8]
    A. S. Fokas and P. M. Santini, Phys Rev. Lett.63, 1329 (1989);Dromions and a Boundary Value Problem for the Davey-Stewartson I Equation, Physica D44, 99 (1990).MathSciNetCrossRefGoogle Scholar
  9. [9]
    S. V. Manakov, Physica D3, 420 (1981).MathSciNetCrossRefGoogle Scholar
  10. [10]
    A. S. Fokas and M. J. Ablowitz, Stud. Appl. Math.69, 211 (1983). M. J. Ablowitz, D. BarYaacov, and A. S. Fokas, Stud. Appl. Math.69, 135 (1983). A. S. Fokas, Phys. Rev. Lett.51, 3 (1983). A. S. Fokas and M. J. Ablowitz, J. Math. Phys.25, No. 8, 2494–2505 (1984).MathSciNetGoogle Scholar
  11. [11]
    R. Beals and R. R. Coifman, Proc. Symp. Pure Math.43, Amer. Math. Soc. Providence, 45 (1985);The Spectral Problem for the Davey-Stewartson and Ishimori Hierarchies, Proc. Conf. on Nonlinear Evolution Equations: Integrability and Spectral Methods, Como, University of Manchester (1988).Google Scholar
  12. [12]
    J. Bona, Private communication.Google Scholar
  13. [13]
    V. E. Zakharov and P. B. Shabat, Funct. Anal. Appl.8, 226 (1974).CrossRefGoogle Scholar
  14. [14]
    V. E. Zakharov and P. B. Shabat, Funct. Anal. Appl.13, 166 (1979).MathSciNetGoogle Scholar
  15. [15]
    V. E. Zakharov and S. V. Manakov, Funct. Anal. App.19, No. 2, 11 (1985).MathSciNetGoogle Scholar
  16. [16]
    P. M. Santini, M. J. Ablowitz, and A. S. Fokas, J. Math. Phys.25, 2619 (1984).Google Scholar
  17. [17]
    E. A. Kuznetsov and A. V. Mikhailov, Zh. Eksp. Teor. Fiz.67, 1718 (1974).Google Scholar
  18. [18]
    V. E. Zakharov,On the Dressing Method, to appear (1990).Google Scholar
  19. [19]
    L.-Y. Sung and A. S. Fokas, Inverse Problems ofN × N Hyperbolic Systems on the Plane and the N wave Interactions, Comm. Pure and Appl. Math.XLIV, 535–571 (1991).Google Scholar
  20. [20]
    X. Zhou, Inverse Scattering Transform for the Time Dependent Schrödinger Equation and KPI, to appear in Comm. in Math. Phys.Google Scholar
  21. [21]
    M. Boiti, F. Pempinelli, A. K. Pogrebkov, and M. C. Polivanov, Inverse Problems7, 43 (1991).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc 1992

Authors and Affiliations

  • A. S. Fokas
    • 1
  • V. E. Zakharov
    • 2
  1. 1.Department of Mathematics and Computer ScienceClarkson UniversityPotsdamUSA
  2. 2.Landau Institute of Theoretical PhysicsAcademy of SciencesMoscowRussia

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