Darboux-Bäcklund solutions of SL(p, q) KP-KdV hierarchies, constrained generalized Toda lattices, and two-matrix string model

Abstract We present a unifying description of the graded SL( p , q ) KP-KdV hierarchies, including the Wronskian construction of their τ-functions as well as the coefficients of the pertinent Lax operators, obtained via successive applications of special Darboux-Backlund transformations. The emerging Darboux-Backlund structure is identified as a constrained generalized Toda lattice system relevant for the two-matrix string model. It allows simple derivation of the n -soliton solutions of the unconstrained KP system. Also, the exact Wronskian solution for the two-matrix model partition function is found.