A new proof of scattering below the ground state for the 3d radial   focusing cubic NLS
  We revisit the scattering result of Holmer and Roudenko on the radial focusing cubic NLS in three space dimensions. Using the radial Sobolev embedding and a virial/Morawetz estimate, we give a simple proof of scattering below the ground state that avoids the use of concentration compactness. 
Benjamin Dodson, Jason Murphy
math.AP