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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
Article

Summary

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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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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