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 Schr\"odinger equation. Our proof is based on, and extends, methods of Erd\"os-Schlein-Yau, Klainerman-Machedon, and Kirkpatrick-Schlein-Staffilani.