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

Woocheol Choi; Youngwoo Koh

doi:10.3934/dcds.2021019

Article Contents

2021, Volume 41, Issue 8: 3837-3867. Doi: 10.3934/dcds.2021019

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

Woocheol Choi^1, and
Youngwoo Koh^2,

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

Early access: January 2021

Published: August 2021

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Abstract

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 $.

Keywords:

Nonlinear Schrödinger equations,
splitting method.

Mathematics Subject Classification: Primary:35Q55, 65M15.

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References

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