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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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