On the quantum KP hierarchy and its relation to the non-linear Schr\"odinger equation

TL;DR

This paper explores the connection between classical and quantum integrable systems, specifically linking the non-linear Schrödinger equation to the KP hierarchy and extending this to a quantum framework, revealing new algebraic structures.

Contribution

It introduces a quantum KP hierarchy derived from the quantization of the first Hamiltonian structure of the KdV equation, expanding the understanding of quantum integrable models.

Findings

Established a relation between classical NLS and KP hierarchy.

Extended the relation to a quantum KP hierarchy.

Provided evidence for a new integrable hierarchy via quantization.

Abstract

We establish a relation between the classical non-linear Schr\"odinger equation and the KP hierarchy, and we extend this relation to the quantum case by defining a quantum KP hierarchy. We 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.