Lax formulation of 3--component KP hierarchy by Shiota construction

TL;DR

This paper uses Shiota's method to construct the Lax structure of the 3-component KP hierarchy, providing insights into multi-component integrable systems and their reductions.

Contribution

It introduces a novel approach to derive Lax structures for multi-component KP hierarchies using Shiota's method, simplifying relations among discrete variables.

Findings

Constructed Lax structure for 3-component KP hierarchy

Introduced shift operators to relate discrete variables

Provided a framework applicable to general multi-component KP systems

Abstract

It is quite basic in integrable systems to deriving Lax equations from bilinear equations. For multi--component KP theory, corresponding Lax structures are mainly constructed by matrix pseudo-differential operators for fixed discrete variables, or by matrix difference operators for even-component cases. Here we use Shiota method to construct Lax structure of 3-component KP hierarchy and its reduction by introducing two shift operators $\Lambda_1$ and $\Lambda_2$, where relations among different discrete variables can be easily found. We believe the results here are quite typical for general multi-component KP theory, which may be helpful for general cases.