## A compactness result for inhomogeneous nonlinear Schrödinger equations.(English)Zbl 1479.35777

Summary: 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 $$( \phi_n )_n$$ of $$H^1$$-functions which converges weakly in $$H^1$$ to a function $$\phi$$, the corresponding solutions with initial data $$\phi_n$$ can be decomposed (up to a remainder term) as a sum of the corresponding solution with initial data $$\phi$$ and solutions to the linear equation.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q41 Time-dependent Schrödinger equations and Dirac equations 35B44 Blow-up in context of PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 78A60 Lasers, masers, optical bistability, nonlinear optics 82D10 Statistical mechanics of plasmas
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