Differential Fay identities and auxiliary linear problem of integrable hiearchies
arXiv:0710.5356
Abstract
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.
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
References in corpus (4)
Cited by in corpus (10)
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- The Multicomponent KP Hierarchy: Differential Fay Identities and Lax Equations
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- Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy
- On Addition Formulae of KP, mKP and BKP Hierarchies
- Integrable structure of melting crystal model with two q-parameters
- Multi-variable reductions of the dispersionless DKP hierarchy
- Loewner equations and reductions of dispersionless hierarchies
- Infinite-Dimensional Frobenius Manifolds for 2+1 Integrable Systems