Poisson commuting energies for a system of infinitely many bosons

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 [57]. 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 Schrödinger hierarchy, the third element of which is the one-dimensional cubic nonlinear Schrödinger equation. This work provides substantial evidence for the GP hierarchy as a new integrable system and is a step towards understanding the origins of the integrability of the NLS in terms of a scaling limit of a quantum integrable system.