Constrained KP Hierarchy and Bi-Hamiltonian Structures

摘要

The Kadomtsev-Petviashvili (KP) hierarchy is considered together with theevolutions of eigenfunctions and adjoint eigenfunctions. By constraining the KP flows in terms of squared eigenfunctions, 1 + 1 dimensional integrable equations with scattering problems given by pseudodifferential Lax operators are obtained. The bi-Hamiltonian nature of these systems is shown by a systematic construction of two general Poisson brackets in the algebra of associated Lax operators. Gauge transformations provide Miura links to modified equations. These systems are constrained flows of the modified KP hierarchy, for which again a general description of their bi-Hamiltonian nature is given. The gauge transformations are shown to be Poisson maps relating the bi-Hamiltonian structures of the constrained KP hierarchy and the modified KP hierarchy. The simplest realization of this scheme yields the AKNS hierarchy and its Miura link to the Kaup-Broer hierarchy.