One reduction of the modified Toda hierarchy

TL;DR

This paper introduces a new reduction of the modified Toda hierarchy, called the generalized bigraded modified Toda hierarchy, and explores its relation to the Toda reduction and tau function formulations.

Contribution

It constructs a novel two-component hierarchy reduction of the mToda hierarchy and relates it to existing Toda reductions and tau function representations.

Findings

Defined the generalized bigraded modified Toda hierarchy.

Established relations with the Toda reduction.

Presented tau function formulations for the reductions.

Abstract

The modified Toda (mToda) hierarchy is a two-component generalization of the 1-st modified KP (mKP) hierarchy, which connects the Toda hierarchy via Miura links and has two tau functions. Based on the fact that the mToda and 1-st mKP hierarchies share the same fermionic form, we firstly construct the reduction of the mToda hierarchy $L_1(n)^M=L_2(n)^N+\sum_{l\in\mathbb{Z}}\sum_{i=1}^{m}q_{i,n}\Lambda^lr_{i,n+1}\Delta$ and $(L_1(n)^M+L_2(n)^N)(1)=0$, called the generalized bigraded modified Toda hierarchy, which can be viewed as a new two-component generalization of the constrained mKP hierarchy $\mathfrak{L}^k=(\mathfrak{L}^k)_{\geq 1}+\sum_{i=1}^m \mathfrak{q}_i\partial^{-1}\mathfrak{r}_i\partial$. Next the relation with the Toda reduction $\mathcal{L}_1(n)^M=\mathcal{L}_2(n)^{N}+\sum_{l\in \mathbb{Z}}\sum_{i=1}^{m}\tilde{q}_{i,n}\Lambda^l\tilde{r}_{i,n}$ is discussed. Finally we give equivalent formulations of the Toda and mToda reductions in terms of tau functions.