METHOD OF SQUARED EIGENFUNCTION POTENTIALS IN INTEGRABLE HIERARCHIES OF KP TYPE

The method of squared eigenfunction potentials (SEP) is developed syst ematically to describe and gain new information about the Kadomtsev-Pe tviashvili (KP) hierarchy and its reductions. Interrelation to the tau -function method is discussed in detail. The principal result, which f orms the basis of our SEP method, is the proof that any eigenfunction of the general KP hierarchy can be represented as a spectral integral over the Baker-Akhiezer (BA) wave function with a spectral density exp ressed in terms of SEP. In fact, the spectral representations of the ( adjoint) BA functions can, in turn, be considered as defining equation s for the KP hierarchy. The SEP method is subsequently used to show ho w the reduction of the full KP hierarchy to the constrained KP (cKP(ta u,m)) hierarchies can be given entirely in terms of linear constraint equations on the pertinent tau-functions. The concept of SEP turns out to be crucial in providing a description of cKP(tau,m) hierarchies in the language of the universal Sato Grassmannian and finding the non-i sospectral Virasoro symmetry generators acting on the underlying tau-f unctions. The SEP method is used to write down generalized binary Darb oux-Backlund transformations for constrained KP hierarchies whose orbi ts are shown to correspond to a new Toda model on a square lattice. As a result, we obtain a series of new determinant solutions for the tau -functions generalizing the known Wronskian (multi-soliton) solutions. Finally, applications to random matrix models in condensed matter phy sics are briefly discussed.