2007/10/31 by Kanehisa Takasaki · 2 citations
Physics and Astronomy · Mathematics · #nlin.SI #hep-th #math-ph #math.MP
published as Advanced Studies in Pure Mathematics vol. 61 (2011), 387--441 · latex2e, packages "amsmath,amssymb,amsthm", 50 pages, no figure, contribution to proceedings of conference "Exploration of new structures and natural constructions in mathematical physics" (Nagoya University, March, 2007); (v2) a few references added; (v3) final version for publication; (v4) typos in)(5.1), (5.2), (5.3) and first equation in 5.2 corrected
arxiv created 2008/10/06 · arxiv updated 2011/11/08
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.