Some energy-preserving relaxation-type schemes for two-dimensional space fractional nonlinear Schrödinger equations

Junjie Wang; Junjie Wang

doi:10.3934/nhm.2026021

Networks and Heterogeneous Media

2026, Volume 21, Issue 2: 446-475. doi: 10.3934/nhm.2026021

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Some energy-preserving relaxation-type schemes for two-dimensional space fractional nonlinear Schrödinger equations

Junjie Wang ^,

School of Mathematics and Statistics, Pu'er University, Yunnan 665000, China

Received: 16 September 2025 Revised: 24 January 2026 Accepted: 12 February 2026 Published: 26 March 2026

In this paper, we develop some energy-preserving relaxation-type schemes to solve two-dimensional space fractional nonlinear Schrödinger equations with periodic boundary conditions. First, we change the original system into an equivalent relaxation form by introducing some new variables, which transforms the energy conservation law of the original system into quadratic invariants, and satisfies the mass conservation law of the original system. Then an implicit relaxation scheme is applied to deal with the time derivative, and the resulting semi discrete system can exactly preserve the mass and energy conservation laws. However, the obtained semi discrete system is nonlinear. Next a linear implicit relaxation scheme is directly used for the modified system to arrive at a semi discrete scheme, and the conservation of the semi discrete system is analyzed. Second, the resulting semi discrete systems are discretized by the Fourier spectral method with periodic boundary conditions, and the efficient iterative algorithms of the fully-discrete systems are given. Finally, numerical experiments of some space fractional nonlinear Schrödinger equations are given to verify the correctness of the theoretical results.
- fractional nonlinear Schrödinger equation,
- conservation law,
- Fourier spectral method,
- relaxation scheme,
- energy-preserving scheme
Citation: Junjie Wang. Some energy-preserving relaxation-type schemes for two-dimensional space fractional nonlinear Schrödinger equations[J]. Networks and Heterogeneous Media, 2026, 21(2): 446-475. doi: 10.3934/nhm.2026021

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Abstract

In this paper, we develop some energy-preserving relaxation-type schemes to solve two-dimensional space fractional nonlinear Schrödinger equations with periodic boundary conditions. First, we change the original system into an equivalent relaxation form by introducing some new variables, which transforms the energy conservation law of the original system into quadratic invariants, and satisfies the mass conservation law of the original system. Then an implicit relaxation scheme is applied to deal with the time derivative, and the resulting semi discrete system can exactly preserve the mass and energy conservation laws. However, the obtained semi discrete system is nonlinear. Next a linear implicit relaxation scheme is directly used for the modified system to arrive at a semi discrete scheme, and the conservation of the semi discrete system is analyzed. Second, the resulting semi discrete systems are discretized by the Fourier spectral method with periodic boundary conditions, and the efficient iterative algorithms of the fully-discrete systems are given. Finally, numerical experiments of some space fractional nonlinear Schrödinger equations are given to verify the correctness of the theoretical results.

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