CONSTRAINED KP HIERARCHY AND BI-HAMILTONIAN STRUCTURES

The Kadomtsev-Petviashvili (KP) hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions. Constrain ing the KP flows in terms of squared eigenfunctions one obtains 1 + 1- dimensional integrable equations with scattering problems given by pse udo-differential Lax operators. The bi-Hamiltonian nature of these sys tems is shown by a systematic construction of two general Poisson brac kets on the algebra of associated Lax-operators. Gauge transformations provide Miura links to modified equations. These systems are constrai ned flows of the modified KP hierarchy, for which again a general desc ription of their bi-Hamiltonian nature is given. The gauge transformat ions are shown to be Poisson maps relating the bi-Hamiltonian structur es 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.