The multicomponent 2D Toda hierarchy: discrete flows and string equations

Abstract

The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated with an infinite-dimensional group. A new set of discrete flows is considered and the corresponding Lax and Zakharov–Shabat equations are characterized. Reductions of block Toeplitz and Hankel bi-infinite matrix types are proposed and studied. Orlov–Schulman operators, string equations and additional symmetries (discrete and continuous) are considered. The continuous-discrete Lax equations are shown to be equivalent to a factorization problem as well as to a set of string equations. A congruence method to derive site-independent equations is presented and used to derive equations in the discrete multicomponent KP sector (and also for its modification) of the theory as well as dispersive Whitham equations.