Poisson commuting energies for a system of infinitely many bosons

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 Schrodinger hierarchy, the third element of which is the one-dimensional cubic nonlinear Schrodinger equation. This work provides substantial evidence for the GP hierarchy as a new integrable system and is a step towards

Automated summary

This paper studies the cubic Gross-Pitaevskii (GP) hierarchy in one spatial dimension. A series of observables are established such that the corresponding trace functionals, referred to as energies, commute with respect to the weak Lie-Poisson structure defined by the authors in [57]. The Hamiltonian equation associated with the third energy functional precisely corresponds to the GP hierarchy, while the equations of motion corresponding to the remaining energies generalize the well-known nonlinear Schrodinger hierarchy.