The multicomponent 2D Toda hierarchy: dispersionless limit

TL;DR

This paper investigates the dispersionless limit of the multicomponent 2D Toda hierarchy, introducing a dispersive Whitham hierarchy and exploring its symmetries, string equations, and connections to KP and Toda frameworks.

Contribution

It introduces a dispersive Whitham hierarchy based on scalar Lax and Orlov--Schulman operators and analyzes its symmetries and string equations in the dispersionless limit.

Findings

Emergence of KP and Toda pictures in the dispersionless limit

Development of a dispersive Whitham hierarchy framework

Analysis of additional symmetries and string equations

Abstract

The factorization problem of the multi-component 2D Toda hierarchy is used to analyze the dispersionless limit of this hierarchy. A dispersive version of the Whitham hierarchy defined in terms of scalar Lax and Orlov--Schulman operators is introduced and the corresponding additional symmetries and string equations are discussed. Then, it is shown how KP and Toda pictures of the dispersionless Whitham hierarchy emerge in the dispersionless limit. Moreover, the additional symmetries and string equations for the dispersive Whitham hierarchy are studied in this limit.