The Kp Hierarchy

Abstract

As an application of the theory of infinite-dimensional Grassmannians and the representation theory of gl1 we shall study in this chapter certain nonlinear “exactly solvable” systems of differential equations. Exactly solvable means here that the nonlinear system can be transformed to an (infinite-dimensional) linear problem. A prototype of the equations is the Korteweg-de Vries equation $$\frac{{\partial u}}{{\partial t}} = \frac{3}{3}u\frac{{\partial u}}{{\partial x}} + \frac{1}{4}\frac{{{\partial ^3}u}}{{\partial {x^3}}}$$ . It turns out that it is more natural to consider an infinite system of equations like that above, for obtaining explicit solutions. The set of equations is called the KdV hierarchy and it can be derived from another set of equations, the KP (Kadomtsev-Petviashvili) hierarchy. The Grassmannian approach can be more directly applied to the KP hierarchy and therefore we shall mainly consider the KP case.