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

TL;DR

This paper derives the energy-critical 3D quintic nonlinear Schrödinger equation from Bose gases with three-body interactions, using Bogoliubov's approximation to understand fluctuations around the condensate.

Contribution

It introduces a derivation of the 3D energy-critical quintic NLS from Bose gases with three-body interactions, addressing new challenges compared to pair-interaction cases.

Findings

Derivation of 3D quintic NLS as mean field limit

Application of Bogoliubov's approximation to fluctuations

Norm approximation for N-body wave function

Abstract

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