H. Aratyn et al., REDUCTION OF TODA LATTICE HIERARCHY TO GENERALIZED KDV HIERARCHIES AND THE 2-MATRIX MODEL, International journal of modern physics A, 10(17), 1995, pp. 2537-2577
Toda lattice hierarchy and the associated matrix formulation of the 2M
-boson KP hierarchies provide a framework for the Drinfeld-Sokolov red
uction scheme realized through Hamiltonian action within the second KP
Poisson bracket. By working with free currents, which Abelianize the
second KP Hamiltonian structure, we are able to obtain a unified forma
lism for the reduced SL(M + 1, M - k) KdV hierarchies interpolating be
tween the ordinary KP and KdV hierarchies. The corresponding Lax opera
tors are given as superdeterminants of graded SL(M + 1, M - k) matrice
s in the diagonal gauge and we describe their bracket structure and fi
eld content. In particular, we provide explicit free field representat
ions of the associated W(M, M - k) Poisson bracket algebras generalisi
ng the familiar nonlinear W-M+1 algebra. Discrete Backlund transformat
ions for SL(M + 1, M - k) KdV are generated naturally from lattice tra
nslations in the underlying Toda-like hierarchy. As an application we
demonstrate the equivalence of the two-matrix string model to the SL(M
+ 1, 1) KdV hierarchy.