The quintic NLS as the mean field limit of a boson gas with three-body interactions

We investigate the dynamics of a boson gas with three-body interactions in dimensions d=1,2. We prove that in the limit of infinite particle number, the BBGKY hierarchy of k-particle marginals converges to a limiting (Gross–Pitaevskii (GP)) hierarchy for which we prove existence and uniqueness of solutions. Factorized solutions of the GP hierarchy are shown to be determined by solutions of a quintic nonlinear Schrödinger equation. Our proof is based on, and extends, methods of Erdös–Schlein–Yau, Klainerman–Machedon, and Kirkpatrick–Schlein–Staffilani.