Differential Fay identities and auxiliary linear problem of integrable hiearchies

Details Differential Fay identities and auxiliary linear problem of integrable hiearchies Differential Fay identities and auxiliary linear problem of integrable hiearchies

2007-10-29

arxiv.org/abs/0710.5356v4

Advanced Studies in Pure Mathematics vol. 61 (2011), 387--441

Nonlinear Sciences

Exactly Solvable and Integrable Systems

latex2e, packages "amsmath,amssymb,amsthm", 50 pages, no figure, contribution to proceedings of conference "Exploration of new

Scientific paper

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.

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