H. Aratyn et al., METHOD OF SQUARED EIGENFUNCTION POTENTIALS IN INTEGRABLE HIERARCHIES OF KP TYPE, Communications in Mathematical Physics, 193(3), 1998, pp. 493-525
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.