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A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation

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Abstract

In this paper, we propose a conservative linearized difference scheme for the nonlinear fractional Schrödinger equation. The scheme efficiently avoids the time consuming iteration procedure necessary for the nonlinear scheme and thus is time saving relatively. It is rigorously proved that the scheme is mass conservative and uniquely solvable. Then employing mathematical induction, we further show that the proposed scheme is convergent at the order of O(τ 2 + h 2) in the l 2 norm with time step τ and mesh size h. Moreover, an extension to coupled nonlinear fractional Schrödinger systems is presented. Finally, numerical tests are carried out to corroborate the theoretical results and investigate the impact of the fractional order α on the collision of two solitons.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Pengde Wang & Chengming Huang

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  1. Pengde Wang
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  2. Chengming Huang
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Correspondence to Pengde Wang.

Additional information

This work was supported by NSF of China (Nos. 91130003 and 11371157) and the Fundamental Research Funds for the Central Universities (No. 2013TS137).

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Wang, P., Huang, C. A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation. Numer Algor 69, 625–641 (2015). https://doi.org/10.1007/s11075-014-9917-x

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