The two--component discrete KP hierarchy

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 $\widehat{sl}_{M+N}=sl_{M+N}[\lambda,\lambda^{-1}]\oplus\mathbb{C}c \ (M,N\geq1)$.