Dispersionless scalar integrable hierarchies, Whitham hierarchy, and the quasiclassical ∂̄-dressing method

The quasiclassical limit of the scalar nonlocal ∂-problem is derived and a quasiclassical version of the ∂-dressing method is presented. Dispersionless Kadomtsev–Petviashvili (KP), modified KP, and dispersionless two-dimensional Toda lattice (2DTL) hierarchies are discussed as illustrative examples. It is shown that the universal Whitham hierarchy is nothing but the ring of symmetries for the quasiclassical ∂-problem. The reduction problem is discussed and, in particular, the d2DTL equation of B type is derived.