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

The recently proposed quasi-classical ∂̄-dressing method provides a systematic approach to studying the weakly dispersive limit of integrable systems. We apply the quasi-classical ∂̄-dressing method to describe dispersive corrections of any order. We show how to calculate the ∂̄ 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 allows 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.