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Schrödinger group on Zhidkov spaces. (English) Zbl 1103.35093
Summary: We consider the Cauchy problem for nonlinear Schrödinger equations on \(\mathbb{R}^n\) with nonzero boundary condition at infinity, a situation which occurs in stability studies of dark solitons. We prove that the Schrödinger operator generates a group on Zhidkov spaces \(X^k (\mathbb{R}^n)\) for \(k>n/2\), and that the Cauchy problem for NLS is locally well-posed on the same Zhidkov spaces. We justify the conservation of classical invariants which implies in some cases the global well-posedness of the Cauchy problem.

35Q55 NLS equations (nonlinear Schrödinger equations)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
46N20 Applications of functional analysis to differential and integral equations