The dressing method and nonlocal Riemann-Hilbert problems

A. S. Fokas, Vladimir E Zakharov

Research output: Contribution to journalArticle

48 Citations (Scopus)

Abstract

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 the N-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 the N-wave interactions.

Original languageEnglish (US)
Pages (from-to)109-134
Number of pages26
JournalJournal of Nonlinear Science
Volume2
Issue number1
DOIs
StatePublished - Mar 1992
Externally publishedYes

Fingerprint

Nonlocal Problems
Riemann-Hilbert Problem
Wave Interaction
wave interaction
Lax Pair
Exact Solution
Perturbation
perturbation
Line
Arbitrary

Keywords

  • AMS/MOS classification numbers: 58F07, 35R58, 35Q51
  • dressing method
  • solitons in two spatial dimensions

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Applied Mathematics
  • Mathematics(all)
  • Mechanics of Materials
  • Computational Mechanics

Cite this

The dressing method and nonlocal Riemann-Hilbert problems. / Fokas, A. S.; Zakharov, Vladimir E.

In: Journal of Nonlinear Science, Vol. 2, No. 1, 03.1992, p. 109-134.

Research output: Contribution to journalArticle

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