The multicomponent 2D Toda hierarchy: Dispersionless limits

In this paper we discuss the dispersionless limit of the multicomponent 2D Toda hierarchy. The discrete flows of the hierarchy are used to define charge preserving Lax and Orlov–Schulman operators. This construction allows us to perform two types of dispersionless limits, one type leads to the 0-genus universal Whitham hierarchy while the other leads to a dispersionless hierarchy which contains the dispersionless 2D Toda hierarchy as one of its components. Thus, we refer to this last hierarchy as the multicomponent dispersionless 2D Toda hierarchy. We also discuss additional symmetries and string equations in the dispersive context and show that their dispersionless counterparts correspond to string equations for the Whitham and multicomponent dispersionless 2D Toda hierarchy.