Gallo, Clément Schrödinger group on Zhidkov spaces. (English) Zbl 1103.35093 Adv. Differ. Equ. 9, No. 5-6, 509-538 (2004). 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. Cited in 25 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 46N20 Applications of functional analysis to differential and integral equations Keywords:local well-posedness; global well-posedness; Gross-Pitaevskii equation; Cauchy problem; nonlinear Schrödinger equations PDFBibTeX XMLCite \textit{C. Gallo}, Adv. Differ. Equ. 9, No. 5--6, 509--538 (2004; Zbl 1103.35093)