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

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.