The $n$-component KP hierarchy and representation theory

TL;DR

This paper provides new equivalent definitions of the n-component KP hierarchy using fermions, tau-functions, matrix wave functions, and pseudodifferential operators, and explores their connections to integrable systems and solutions.

Contribution

It introduces multiple equivalent formulations of the n-component KP hierarchy and links it to known integrable systems like the Davey-Stewartson and n-wave equations.

Findings

Equivalent definitions of the n-component KP hierarchy are established.

The 2-component KP hierarchy contains the Davey-Stewartson system.

The n-component KP hierarchy generalizes the n-wave interaction equations.

Abstract

Starting from free charged fermions we give equivalent definitions of the $n\/$-component KP hierarchy, in terms of $\tau\/$-functions $\tau_\alpha\/$ (where $\alpha \in M =\/$ root lattice of $sl_n\/$), in terms of $n \times n\/$ matrix valued wave functions $W_\alpha(\alpha\in M)\/$, and in terms of pseudodifferential wave operators $P_\alpha(\alpha\in M)\/$. These imply the deformation and the zero curvature equations. We show that the 2-component KP hierarchy contains the Davey-Stewartson system and the $n\geq3\/$ component KP hierarchy continues the $n\/$-wave interaction equations. This allows us to construct theis solutions.