Poisson Commuting Energies for a System of Infinitely Many Bosons

TL;DR

This paper demonstrates that the one-dimensional cubic GP hierarchy possesses an infinite sequence of commuting energies, indicating its integrability and extending the understanding of its mathematical structure.

Contribution

It establishes the existence of an infinite sequence of commuting energies for the GP hierarchy, linking it to integrable systems and generalizing known nonlinear Schrödinger equations.

Findings

Existence of an infinite sequence of commuting energies.

The third energy's Hamiltonian matches the GP hierarchy.

Remaining energies generalize the nonlinear Schrödinger hierarchy.

Abstract

We consider the cubic Gross-Pitaevskii (GP) hierarchy in one spatial dimension. We establish the existence of an infinite sequence of observables such that the corresponding trace functionals, which we call ``energies,'' commute with respect to the weak Lie-Poisson structure defined by the authors in arXiv:1908.03847. The Hamiltonian equation associated to the third energy functional is precisely the GP hierarchy. The equations of motion corresponding to the remaining energies generalize the well-known nonlinear Schr\"odinger hierarchy, the third element of which is the one-dimensional cubic nonlinear Schr\"odinger equation. This work provides substantial evidence for the GP hierarchy as a new integrable system.