On the global well-posedness of the quadratic NLS on $L^2(\mathbb{R})+H^1(\mathbb{T})$

Abstract

We study the one dimensional nonlinear Schrödinger equation with power nonlinearity $|u{|}^{\alpha -1}u$ for $\alpha \in [1,5]$ and initial data $u_0\in L^2(\mathbb{R})+H^1(\mathbb{T})$. We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity ($\alpha =2$) we obtain global well-posedness in the space $C(\mathbb{R},L^2(\mathbb{R})+H^1(\mathbb{T}))$ via Gronwall's inequality.

Fixed typos and Theorem 2 has been extended to contain all $\alpha \in [1,5]$. In addition, the Appendix of the current version includes quadratic and sub-quadratic nonlinearities for the periodic NLS