## Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy

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Takasaki, Kanehisa

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Recently the study of Fay-type identities revealed some new features of the
DKP hierarchy (also known as "the coupled KP hierarchy" and "the Pfaff
lattice"). Those results are now extended to a Toda version of the DKP
hierarchy (tentatively called "the Pfaff-Toda hierarchy"). Firstly, an
auxiliary linear problem of this hierarchy is constructed. Unlike the case of
the DKP hierarchy, building blocks of the auxiliary linear problem are
difference operators. A set of evolution equations for dressing operators of
the wave functions are also obtained. Secondly, a system of Fay-like identities
(difference Fay identities) are derived. They give a generating functional
expression of auxiliary linear equations. Thirdly, these difference Fay
identities have well defined dispersionless limit (dispersionless Hirota
equations). As in the case of the DKP hierarchy, an elliptic curve is hidden in
these dispersionless Hirota equations. This curve is a kind of spectral curve,
whose defining equation is identified with the characteristic equation of a
subset of all auxiliary linear equations. The other auxiliary linear equations
are related to quasi-classical deformations of this elliptic spectral curve.

Comment: 49 pages, no figure, usepackage amsmath,amssymb,amsthm; (v2) several typos are corrected; (v3) published version

Comment: 49 pages, no figure, usepackage amsmath,amssymb,amsthm; (v2) several typos are corrected; (v3) published version

##### Keywords

Nonlinear Sciences - Exactly Solvable and Integrable Systems, High Energy Physics - Theory, Mathematical Physics