The dressing method and nonlocal Riemann-Hilbert problems

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.