Orbital stability and uniqueness of the ground state for NLS in   dimension one
  We prove that standing-waves solutions to the non-linear Schr\"odinger equation in dimension one whose profiles can be obtained as minima of the energy over the mass, are orbitally stable and non-degenerate, provided the non-linear term $ G $ satisfies a Euler differential inequality. When the non-linear term $ G $ is a combined pure power-type, then there is only one positive, symmetric minimum of prescribed mass. 
Daniele Garrisi, Vladimir Georgiev
math.AP