Universal Whitham hierarchy, dispersionless Hirota equations and multicomponent KP hierarchy

The goal of this paper is to identify the universal Whitham hierarchy of genus zero with a dispersionless limit of the multicomponent KP hierarchy. To this end, the multicomponent KP hierarchy is (re)formulated to depend on several discrete variables called “charges”. These discrete variables play the role of lattice coordinates in underlying Toda field equations. A multicomponent version of the so called differential Fay identity are derived from the Hirota equations of the ττ-function of this “charged” multicomponent KP hierarchy. These multicomponent differential Fay identities have a well-defined dispersionless limit (the dispersionless Hirota equations). The dispersionless Hirota equations turn out to be equivalent to the Hamilton–Jacobi equations for the SS-functions of the universal Whitham hierarchy. The differential Fay identities themselves are shown to be a generating functional expression of auxiliary linear equations for scalar-valued wave functions of the multicomponent KP hierarchy.