A compactness result for inhomogeneous nonlinear Schrödinger equations

We establish a compactness property of the difference between nonlinear and linear operators (or the Duhamel operator) related to the inhomogeneous nonlinear Schrödinger equation. The proof is based on a refined profile decomposition for the equation. More precisely, we prove that any sequence (ϕn)n of H1-functions which converges weakly in H1 to a function ϕ, the corresponding solutions with initial data ϕn can be decomposed (up to a remainder term) as a sum of the corresponding solution with initial data ϕ and solutions to the linear equation.