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 $Z_τ$ introduced by Ignat, with initial data in $H^1$, for the nonlinear Schrödinger equation : $$ \partial_t u = i Δu + iλ|u|^{p} u,\qquad u (x,0) =ϕ(x), $$ where $(x,t) \in \mathbb{R}^d \times [0,\infty)$, with $0< p < 4/(d-2)$ for $d\geq3$ and $0< p<\infty$ for $d=1,2$. We prove the $L^2$ convergence of order $\mathcal{O}(τ^{1/2})$ for this scheme with initial data in the space $H^1 (\mathbb{R}^d)$.