The two--component discrete KP hierarchy

TL;DR

This paper introduces the two-component discrete KP hierarchy, deriving its Lax and bilinear equations, and explores reductions related to loop algebra structures, advancing the understanding of multi-component integrable systems.

Contribution

The paper develops the two-component discrete KP hierarchy, deriving its Lax and bilinear formulations, and analyzes its reductions based on loop algebra, which is a novel extension of the discrete KP framework.

Findings

Derived the Lax equation from the bilinear form using scalar Lax operators.

Established the existence of the tau function for the 2dKP hierarchy.

Discussed the reduction of the hierarchy related to specific loop algebra structures.

Abstract

The discrete KP hierarchy is also known as the $(l-l')$--th modified KP hierarchy. Here in this paper, we consider the corresponding two--component generalization, called the two--component discrete KP (2dKP) hierarchy. Firstly, starting from the bilinear equation of the 2dKP hierarchy, we derive the corresponding Lax equation by the Shiota method, this is using scalar Lax operators involving two difference operators $\Lambda_1$ and $\Lambda_2$. Then starting from the 2dKP Lax equation, we obtain the corresponding bilinear equation, including the existence of the tau function. From above discussions, we can determine which are essential in the 2dKP Lax formulation. Finally, we discuss the reduction of the 2dKP hierarchy corresponding to the loop algebra $\widehat{sl}_{M+N}=sl_{M+N}[\lambda,\lambda^{-1}]\oplus\mathbb{C}c \ (M,N\geq1)$.