Method of Squared Eigenfunction Potentials in Integrable Hierarchies of KP Type

Abstract:The method of squared eigenfunction potentials (SEP) is developed systematically to describe and gain new information about the Kadomtsev–Petviashvili (KP) hierarchy and its reductions. Interrelation to the τ-function method is discussed in detail. The principal result, which forms 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 expressed in terms of SEP. In fact, the spectral representations of the (adjoint) BA functions can, in turn, be considered as defining equations for the KP hierarchy. The SEP method is subsequently used to show how the reduction of the full KP hierarchy to the constrained KP (cKPrm) hierarchies can be given entirely in terms of linear constraint equations on the pertinent τ-functions. The concept of SEP turns out to be crucial in providing a description of cKPrm hierarchies in the language of the universal Sato Grassmannian and finding the non-isospectral Virasoro symmetry generators acting on the underlying τ-functions. The SEP method is used to write down generalized binary Darboux-Bäcklund transformations for constrained KP hierarchies whose orbits 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 τ-functions generalizing the known Wronskian (multi-soliton) solutions. Finally, applications to random matrix models in condensed matter physics are briefly discussed.