Derivation of the Cubic Non-linear Schr\"odinger Equation from Quantum   Dynamics of Many-Body Systems
  We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic non-linear Schr\"odinger equation in a suitable scaling limit. The result is extended to $k$-particle density matrices for all positive integer $k$. 
Laszlo Erdos, Benjamin Schlein, Horng-Tzer Yau