Derivation of the 3D quintic Gross--Pitaevskii equation

We study the time evolution of Bose–Einstein condensates with three-body interactions in the Gross–Pitaevskii regime. We show that Bose–Einstein condensation is preserved under many-body evolution and that the condensate wavefunction evolves according to the quintic Gross–Pitaevskii equation in ${\mathbb{R}}^3$, which is energy critical. In particular, we show that the effective coupling constant is universal and depends only on a three-body scattering hypervolume.