On the splitting method for the nonlinear Schrödinger equation with initial data in $H^1$

Abstract

In this paper, we establish a convergence result for the operator splitting scheme $Z_{\tau }$ introduced by Ignat, with initial data in $H^1$, for the nonlinear Schrödinger equation : $${\partial }_{t}u=i\Delta u+i\lambda |u{|}^{p}u,u(x,0)=\varphi (x),$$ where $(x,t)\in {\mathbb{R}}^d\times [0,\infty )$, with $0<p<4/(d-2)$ for $d\ge 3$ and $0<p<\infty$ for $d=1,2$. We prove the $L^2$ convergence of order $\mathcal{O}({\tau }^{1/2})$ for this scheme with initial data in the space $H^1({\mathbb{R}}^d)$.