Generalized Bigraded Toda Hierarchy

TL;DR

This paper introduces a generalized bigraded Toda hierarchy extending the classical hierarchy, derives its bilinear equations via fermionic and bosonic methods, and establishes its Lax structure, enriching the theory of integrable systems.

Contribution

It generalizes the bigraded Toda hierarchy by incorporating additional terms, connecting it with constrained KP hierarchies, and derives its fundamental equations and Lax structure using fermionic and bosonic formalisms.

Findings

Derived bilinear equations for the generalized hierarchy.

Established the Lax structure of the hierarchy.

Connected the hierarchy with constrained KP systems.

Abstract

Bigraded Toda hierarchy $L_1^M(n)=L_2^N(n)$ is generalized to $L_1^M(n)=L_2^{N}(n)+\sum_{j\in \mathbb Z}\sum_{i=1}^{m}q^{(i)}_n\Lambda^jr^{(i)}_{n+1}$, which is the analogue of the famous constrained KP hierarchy $L^{k}= (L^{k})_{\geq0}+\sum_{i=1}^{m}q_{i}\partial^{-1}r_i$. It is known that different bosonizations of fermionic KP hierarchy will give rise to different kinds of integrable hierarchies. Starting from the fermionic form of constrained KP hierarchy, bilinear equation of this generalized bigraded Toda hierarchy (GBTH) are derived by using 2--component boson--fermion correspondence. Next based upon this, the Lax structure of GBTH is obtained. Conversely, we also derive bilinear equation of GBTH from the corresponding Lax structure.