The $n$-component KP hierarchy and representation theory

Abstract

Starting from free charged fermions we give equivalent definitions of the $n\/$-component KP hierarchy, in terms of $τ\/$-functions $τ_α\/$ (where $α\in M =\/$ root lattice of $sl_n\/$), in terms of $n \times n\/$ matrix valued wave functions $W_α(α\in M)\/$, and in terms of pseudodifferential wave operators $P_α(α\in M)\/$. 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.