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Recently proposed quasi-classical DBAR-dressing method provides a systematicapproach to study the weakly dispersive limit of integrable systems. We applythe quasi-classical DBAR-dressing method to describe dispersive corrections ofany order. We show how calculate the DBAR problems at any order for a rathergeneral class of integrable systems, presenting explicit results for the KPhierarchy case. We demonstrate the stability of the method at each order. Weconstruct an infinite set of commuting flows at first order which allow adescription analogous to the zero order (purely dispersionless) case,highlighting a Whitham type structure. Obstacles for the construction of thehigher order dispersive corrections are also discussed.Comment: 24 page

Quasi-classical -dressing approach to the weakly dispersive KP hierarchy