The Extended Toda Hierarchy

TL;DR

This paper introduces an extended Toda hierarchy, develops its Lax pair formalism, defines a tau function, and links it to the $CP^1$ topological sigma model and an extension of the nonlinear Schrödinger hierarchy.

Contribution

It provides a new extension of the Toda hierarchy, formalizes its Lax pair and tau function, and establishes connections with topological sigma models and nonlinear Schrödinger hierarchy.

Findings

Lax pair formalism for the extended Toda hierarchy

Explicit tau function formulation

Relationship with $CP^1$ topological sigma model and extended NLS hierarchy

Abstract

We present the Lax pair formalism for certain extension of the continuous limit of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an explicit formulation of the relationship between the $CP^1$ topological sigma model and the extended Toda hierarchy. We also establish an equivalence of the latter with certain extension of the nonlinear Schr\"odinger hierarchy.