Structure-Preserving Numerical Methods for Fractional Nonlinear Schrödinger Equations with Wave Operators

Mengnan Zhang, Xinyu Zhou and Cuicui Liao ()
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Mengnan Zhang: College of Science, Jiangnan University, Wuxi 214122, China
Xinyu Zhou: College of Science, Jiangnan University, Wuxi 214122, China
Cuicui Liao: College of Science, Jiangnan University, Wuxi 214122, China

Mathematics, 2025, vol. 13, issue 19, 1-21

Abstract: This main focus of this work is the fractional-order nonlinear Schrödinger equation with wave operators. First, a conservative difference scheme is constructed. Then, the discrete energy and mass conservation formulas are derived and maintained by the difference scheme constructed in this paper. Through rigorous theoretical analysis, it is proved that the constructed difference scheme is unconditionally stable and has second-order precision in both space and time. Due to the completely implicit property of the differential scheme proposed, a linearized iterative algorithm is proposed to implement the conservative differential scheme. Numerical experiments including one example with the fractional boundary conditions were studied. The results effectively demonstrate the long-term numerical behaviors of the fractional nonlinear Schrödinger equations with wave operators.

Keywords: fractional Schrödinger equation; energy-conserving methods; conservation laws; wave operators; convergence; stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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