On the quantum KP hierarchy and its relation to the non-linear Schrödinger equation

Abstract We establish a relation between the classical non-linear Schrodinger equation and the KP hierarchy, and we extend this relation to the quantum case by defining a quantum KP hierarchy. We also present evidence that an integrable hierarchy of equations is obtained by quantizing the first hamiltonian structure of the KdV equation. The connection between infinite-dimensional algebras and integrable models is discussed.