An infinite sequence of conserved quantities for the cubic Gross-Pitaevskii hierarchy on $\mathbb{R}$

TL;DR

This paper constructs an infinite sequence of conserved quantities for the cubic Gross-Pitaevskii hierarchy on the real line, extending the integrability properties known for the cubic nonlinear Schrödinger equation to this hierarchy.

Contribution

It introduces a novel set of operators that generate conserved quantities for the GP hierarchy, highlighting its integrable structure.

Findings

Established an infinite sequence of conserved operators.

Linked the GP hierarchy's conserved quantities to those of the cubic NLS.

Enhanced understanding of the hierarchy's integrability properties.

Abstract

We consider the (de)focusing cubic Gross-Pitaevskii (GP) hierarchy on $\mathbb{R}$, which is an infinite hierarchy of coupled linear inhomogeneous PDE which appears in the derivation of the cubic nonlinear Schr\"{o}dinger (NLS) equation from quantum many-particle systems. Motivated by the fact that the cubic NLS on $\mathbb{R}$ is an integrable equation which admits infinitely many conserved quantities, we exhibit an infinite sequence of operators which generate analogous conserved quantities for the GP hierarchy.