Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2006-08-31
PhysicaD235:109-125,2007
Nonlinear Sciences
Exactly Solvable and Integrable Systems
latex2e (a4paper, 12pt) using packages "amssymb,amsmath,amsthm", 44 pages, no figure; (v2) a few typos corrected, bibliographi
Scientific paper
10.1016/j.physd.2007.04.017
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.
Takasaki Kanehisa
Takebe Takashi
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