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We derive the 3D quintic NLS as the mean field limit of a Bose gas withthree-body interactions. The quintic NLS is energy-critical, leading to severalnew difficulties in comparison with the cubic NLS which emerges from Bose gaseswith pair-interactions. Our method is based on Bogoliubov's approximation,which also provides the information on the fluctuations around the condensatein terms of a norm approximation for the N-body wave function.

Derivation of 3D energy-critical nonlinear Schr\"odinger equation and Bogoliubov excitations for Bose gases