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.

💡 Research Summary

The paper investigates a Toda‑type extension of the DKP (also called coupled KP or Pfaff lattice) hierarchy, which the authors refer to as the Pfaff‑Toda hierarchy. The study is organized around three main achievements: (i) construction of an auxiliary linear problem (ALP) whose basic objects are difference operators rather than differential operators; (ii) derivation of a complete set of Fay‑type identities in difference form (difference Fay identities) that serve as generating functionals for the ALP; and (iii) analysis of the dispersionless limit of these identities, leading to dispersionless Hirota equations and the emergence of an elliptic spectral curve.

In the first part the authors introduce two families of infinite time variables ({t_n}{n\ge1}) and ({\bar t_n}{n\ge1}). Wave functions (\psi) and (\bar\psi) are defined by dressing operators (W) and (\bar W) acting on exponential phases (e^{\xi(t,z)}) and (e^{\xi(\bar t,z^{-1})}) respectively. Unlike the DKP case, (W) and (\bar W) belong to a non‑commutative algebra generated by forward and backward shift (difference) operators (\Delta) and (\bar\Delta). The auxiliary linear equations read (\Delta\psi = B\psi) and (\bar\Delta\bar\psi = \bar B\bar\psi), where the Lax operators (B) and (\bar B) are constructed from the positive parts of powers of the dressed shift operators. Evolution equations for the dressing operators are obtained in Lax form, \