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

TL;DR

This paper explores the multicomponent 2D Toda hierarchy, introducing new discrete flows, analyzing reductions, and establishing connections to string equations and symmetries within an integrable systems framework.

Contribution

It presents a novel analysis of the hierarchy via a factorization problem, introduces new discrete flows, and studies reductions and symmetries, extending the understanding of integrable multicomponent systems.

Findings

Characterization of new discrete flows and their Lax equations.

Reduction of the hierarchy to block Toeplitz and Hankel matrices.

Derivation of string equations and symmetries in the multicomponent KP sector.

Abstract

The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated to 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.