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arXiv · 1910.06959

Poisson Commuting Energies for a System of Infinitely Many Bosons

Dana Mendelson·Andrea R. Nahmod·Nataša Pavlović·Matthew Rosenzweig·Gigliola Staffilani
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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.

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Dana Mendelson, Andrea R. Nahmod, Nataša Pavlović, Matthew Rosenzweig, Gigliola Staffilani. 2019-10-15. Poisson Commuting Energies for a System of Infinitely Many Bosons. https://arxiv.org/abs/1910.06959

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