The n-component KP hierarchy and representation theory

It is the aim of the present article to give all formulations of the n-component KP hierarchy and clarify connections between them. The generalization to the n-component KP hierarchy is important because it contains many of the most popular systems of soliton equations, like the Davey-Stewartson system (for n=2), the two-dimensional Toda lattice (for n=2), the n-wave system (for ngreater than or equal to3) and the Darboux-Egoroff system. It also allows us to construct natural generalizations to the Davey-Stewartson and Toda lattice systems. Of course, the inclusion of all these systems in the n-component KP hierarchy allows us to construct their solutions by making use of vertex operators. (C) 2003 American Institute of Physics.