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Universal Whitham hierarchy, dispersionless Hirota equations and multi-component KP hierarchy

Kanehisa Takasaki·Takashi Takebe
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Abstract

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 $τ$-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.

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Kanehisa Takasaki, Takashi Takebe. 2007-08-31. Universal Whitham hierarchy, dispersionless Hirota equations and multi-component KP hierarchy. https://doi.org/10.1016/j.physd.2007.04.017

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