# American Institute of Mathematical Sciences

August  2021, 41(8): 3837-3867. doi: 10.3934/dcds.2021019

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

 1 Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea 2 Department of Mathematics Education, Kongju National University, Kongju 32588, Republic of Korea

Received  July 2019 Revised  December 2020 Published  August 2021 Early access  January 2021

In this paper, we establish a convergence result for the operator splitting scheme
 $Z_{\tau}$
introduced by Ignat [12], with initial data in
 $H^1$
, for the nonlinear Schrödinger equation:
 $\partial_t u = i \Delta u + i\lambda |u|^{p} u,\qquad u (x,0) = \phi (x),$
where
 $p >0$
,
 $\lambda \in \{-1,1\}$
and
 $(x,t) \in \mathbb{R}^d \times [0,\infty)$
. We prove the
 $L^2$
convergence of order
 $\mathcal{O}(\tau^{1/2})$
for the scheme with initial data in the space
 $H^1 (\mathbb{R}^d)$
for the energy-subcritical range of
 $p$
.
Citation: Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $H^1$. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019
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