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Article

Universal Whitham hierarchy, dispersionless Hirota equations and multicomponent KP hierarchy.

Physica D: Nonlinear Phenomena. 2007. Vol. 235. No. 1-2. P. 109-125.
Takebe T., Takasaki K.

The goal of this paper is to identify the universal Whitham hierarchy of genus zero with a dispersionless limit of the multi-component KP hierarchy. To this end, the multi-component 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 multi-component version of the so called differential Fay identity are derived from the Hirota equations of the tau-function of this ``charged'' multi-component KP hierarchy. These multi-component 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 S-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 multi-component KP hierarchy.