Global well-posedness of the Gross--Pitaevskii and cubic-quintic   nonlinear Schr\"odinger equations with non-vanishing boundary conditions
  We consider the Gross--Pitaevskii equation on $\R^4$ and the cubic-quintic nonlinear Schr\"odinger equation (NLS) on $\R^3$ with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the energy-critical NLS, we prove that they are globally well-posed in their energy spaces. In particular, we prove unconditional uniqueness in the energy spaces for these equations. 
Rowan Killip, Tadahiro Oh, Oana Pocovnicu, and Monica Visan
math.AP math-ph math.MP