  Recently proposed quasi-classical DBAR-dressing method provides a systematic
approach to study the weakly dispersive limit of integrable systems. We apply
the quasi-classical DBAR-dressing method to describe dispersive corrections of
any order. We show how calculate the DBAR problems at any order for a rather
general class of integrable systems, presenting explicit results for the KP
hierarchy case. We demonstrate the stability of the method at each order. We
construct an infinite set of commuting flows at first order which allow a
description analogous to the zero order (purely dispersionless) case,
highlighting a Whitham type structure. Obstacles for the construction of the
higher order dispersive corrections are also discussed.
