Differential Fay identities and auxiliary linear problem of integrable hierarchies

We review the notion of differential Fay identities and demonstrate, through case studies, its new role in integrable hierarchies of the KP type. These identities are known to be a convenient tool for deriving dispersionless Hirota equations. We show that differential (or, in the case of the Toda hierarchy, difference) Fay identities play a more fundamental role. Namely, they are nothing but a generating functional expression of the full set of auxiliary linear equations, hence substantially equivalent to the integrable hierarchies themselves. These results are illustrated for the KP, Toda, BKP and DKP hierarchies. As a byproduct, we point out some new features of the DKP hierarchy and its dispersionless limit. The author is grateful to Takashi Takebe for collaboration. This research was partially supported by Grant-in-Aid for Scientific Research No. 16340040, No. 18340061 and No. 19540179 from the Japan Society for the Promotion of Science.