Schrödinger group on Zhidkov spaces

Abstract

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.