Abstract:
Along with the well-known hierarchy of integrable Kadomtsev-Petviashvili (KP)
type equations, there is a more general (and much less studied) Pfaffian
version of it, so named because some of its explicit solutions are expressed
not in terms of determinants, but in terms of Pfaffians. This hierarchy, first
briefly mentioned in the work of Jimbo and Miwa in 1983, was rediscovered
several times later and appears in the literature under various names (DKP,
coupled KP, etc.). The talk will be mostly devoted to the dispersionless limit
of this hierarchy, which is interesting because the most important object in
it is a certain elliptic curve built in the structure of the hierarchy, which
allows (as a result of the uniformization of the curve by elliptic functions)
to give its compact and nice formulation. A similar formulation exists for the
usual dispersionless KP hierarchy, but there the elliptic curve degenerates to
a rational one, which is uniformized by trigonometric functions. Recently, all
this has been generalized into a multicomponent Pfaff-KP hierarchy, which will
also be discussed in the talk, if time permits.
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