Darboux-Backlund Solutions of SL(p,q) KP-KdV Hierarchies, Constrained Generalized Toda Lattices, and Two-Matrix String Model

TL;DR

This paper unifies the description of graded SL(p,q) KP-KdV hierarchies, constructs their tau-functions via Darboux-Bäcklund transformations, and links these to constrained Toda lattices and two-matrix string models, providing exact solutions.

Contribution

It introduces a unified framework for SL(p,q) KP-KdV hierarchies, employing Darboux-Bäcklund transformations to derive tau-functions and connect to two-matrix models.

Findings

Darboux-Bäcklund transformations generate tau-functions for hierarchies.

The structure relates to constrained Toda lattice systems.

Exact Wronskian solutions for two-matrix model partition functions.

Abstract

We present an unifying description of the graded $SL(p,q)$ KP-KdV hierarchies, including the Wronskian construction of their tau-functions as well as the coefficients of the pertinent Lax operators, obtained via successive applications of special Darboux-B\"{a}cklund transformations. The emerging Darboux-B\"{a}cklund structure is identified as a constrained generalized Toda lattice system relevant for the two-matrix string model. Also, the exact Wronskian solution for the two-matrix model partition function is found.