Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 109, 34 pages      arXiv:0908.3569      http://dx.doi.org/10.3842/SIGMA.2009.109

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

Kanehisa Takasaki
Graduate School of Human and Environmental Studies, Kyoto University, Yoshida, Sakyo, Kyoto, 606-8501, Japan

Received August 27, 2009, in final form December 15, 2009; Published online December 19, 2009

Abstract
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.

Key words: integrable hierarchy; auxiliary linear problem; Fay-like identity; dispersionless limit; spectral curve; quasi-classical deformation.

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