The Multicomponent KP Hierarchy: Differential Fay Identities and Lax Equations

TL;DR

This paper derives the Lax representation and auxiliary linear equations for the N-component KP hierarchy using differential Fay identities, extending the understanding of its integrable structure and charge sector relations.

Contribution

It demonstrates that four differential Fay identities suffice to derive the Lax equations and introduces generalized equations relating different charge sectors.

Findings

Derived Lax equations for N-component KP hierarchy.

Established auxiliary linear equations from Fay identities.

Generalized modified KP hierarchy with charge sector relations.

Abstract

In this article, we show that four sets of differential Fay identities of an $N$-component KP hierarchy derived from the bilinear relation satisfied by the tau function of the hierarchy are sufficient to derive the auxiliary linear equations for the wave functions. From this, we derive the Lax representation for the $N$-component KP hierarchy, which are equations satisfied by some pseudodifferential operators with matrix coefficients. Besides the Lax equations with respect to the time variables proposed in \cite{2}, we also obtain a set of equations relating different charge sectors, which can be considered as a generalization of the modified KP hierarchy proposed in \cite{3}.