带 Delta 势的非线性 Schrödinger 方程的局部保结构算法

数学物理学报 ›› 2026, Vol. 46 ›› Issue (5): 1857-1883.

带 Delta 势的非线性 Schrödinger 方程的局部保结构算法

汪佳玲^*(), 邹淑慧

南京信息工程大学数学与统计学院南京 210044; 南京信息工程大学江苏省应用数学中心南京 210044; 南京信息工程大学江苏省系统建模与数据分析国际合作联合实验室南京 210044

收稿日期:2025-04-01 修回日期:2025-10-21 出版日期:2026-10-26 发布日期:2026-09-07
通讯作者: 汪佳玲 E-mail:wjl19900724@126.com
基金资助:
江苏省本科高校 "高质量数理类课程教材改革研究" 专项课题(2025JYSLKJ027);国家自然科学基金(11801277);国家自然科学基金(12426524)

Local Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation with Delta Potential

Jialing Wang^*(), Shuhui Zou

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044; Center for Applied Mathematics of Jiangsu Province, Nanjing University of Information Science and Technology, Nanjing 210044; Jiangsu International Joint Laboratory of System Modeling and Data Analysis, Nanjing University of Information Science and Technology, Nanjing 210044

Received:2025-04-01 Revised:2025-10-21 Online:2026-10-26 Published:2026-09-07
Contact: Jialing Wang E-mail:wjl19900724@126.com
Supported by:
Special Project on High-Quality Math & Physics Textbook Reform of Jiangsu Undergraduate Universities(2025JYSLKJ027);NSFC(11801277);NSFC(12426524)

摘要/Abstract

摘要：

该文聚焦于一类特殊的带 Delta 势的非线性 Schrödinger 方程. 作者基于弱多辛形式, 通过复合构造方法系统地给出了弱多辛意义下的局部保结构算法构造的统一框架. 作者成功开发了四类多辛算法和两类局部能量守恒算法, 并讨论了它们在弱多辛意义下的局部和全局守恒律. 这些算法不依赖于边界条件, 可适用于任何满足弱多辛形式的偏微分方程. 数值实验结果表明所提算法优异的计算性能. 相关理论框架与数值算例可直接用于偏微分方程数值解方向的研究生课程教学, 为计算数学专业人才培养提供典型教学案例.

关键词: 带 Delta 势的非线性 Schrödinger 方程, 弱多辛形式, 多辛算法, 局部能量守恒算法

Abstract:

This paper focuses on a special class of nonlinear Schrödinger equation with Delta potential. Based on the weak multisymplectic form, we systematically establish a unified framework for constructing the locall structure-preserving algorithms in the weak multisymplectic sense using the concatenating method. We successfully develop four multi-symplectic algorithms and two local energy-preserving algorithms, whose local and global conservation laws in the weak sense are also discussed afterwards. These algorithms are independent of boundary conditions and can be applied to any partial differential equations satisfying the weak multi-symplectic form. Numerical experiments demonstrate the excellent computational performance of the proposed algorithm. The relevant theoretical framework and numerical examples can be directly applied to postgraduate teaching courses on numerical solutions of partial differential equations, providing typical teaching cases for talent cultivation in computational mathematics.

Key words: nonlinear Schrödinger equation with Delta potential, weak multi-symplectic form, multi-symplectic algorithm, local energy-preserving algorithm

中图分类号:

O241.8

汪佳玲, 邹淑慧. 带 Delta 势的非线性 Schrödinger 方程的局部保结构算法[J]. 数学物理学报, 2026, 46(5): 1857-1883.

Jialing Wang, Shuhui Zou. Local Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation with Delta Potential[J]. Acta mathematica scientia,Series A, 2026, 46(5): 1857-1883.

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参考文献 38

[1]	Wang Z, Zhu J, Wang C. Finite difference alternative unequal-sized weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Physics of Fluids, 2022, 34: 1-29
[2]	Elgindi T, Masmoudi N. $L_{\infty}$ ill-posedness for a class of equations arising in hydrodynamics. Archive for Rational Mechanics and Analysis, 2020, 235: 1979-2025 doi: 10.1007/s00205-019-01457-7
[3]	Kundu P, Almusawa H, Fahim M. Linear and nonlinear effects analysis on wave profiles in optics and quantum physics. Results in Physics, 2021, 23: 1-9
[4]	DeWitt S, Thornton K. Phase field modeling of microstructural evolution. Computational Materials System Design, 2018, 22: 67-87
[5]	Feng K, Qin M. The symplectic methods for the computation of Hamiltonian equations. Numerical Methods for Partial Differential Equations, 2006, 1297: 1-37
[6]	Bridges T, Reich S. Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Physics Letters A, 2001, 284: 184-193 doi: 10.1016/S0375-9601(01)00294-8
[7]	Marsden J, Patrick G, Shkoller S. Multisymplectic geometry, variational integrators, and nonlinear PDEs. Communications in Mathematical Physics, 1998, 199(2): 351-395 doi: 10.1007/s002200050505
[8]	Reich S. Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations. Journal of Computational Physics, 2000, 157: 473-499 doi: 10.1006/jcph.1999.6372
[9]	Cohen D, Hairer E, Lubich C. Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations. Numerische Mathematik, 2008, 110: 113-143 doi: 10.1007/s00211-008-0163-9
[10]	Chen Y, Sun Y, Tang Y. Energy-preserving numerical methods for Landau-Lifshitz equation. Journal of Physics A, 2011, 44: 307-329
[11]	Wang T, Guo B, Xu Q. Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions. Journal of Computational Physics, 2013, 243: 382-399 doi: 10.1016/j.jcp.2013.03.007
[12]	Yi N, Liu H. An energy conserving local discontinuous Galerkin method for a nonlinear variational wave equation. Communications in Computational Physics, 2018, 23: 747-772 doi: 10.4208/cicp.OA-2016-0189
[13]	Cai J, Wang Y, Jiang C. Local structure-preserving algorithms for general multi-symplectic Hamiltonian PDEs. Communications in Computational Physics, 2019, 235: 210-220
[14]	Yusuf A, Alshomrani A, Sulaiman T. Extended classical optical solitons to a nonlinear Schrodinger equation expressing the resonant nonlinear light propagation through isolated flaws in optical waveguides. Optical and Quantum Electronics, 2022, 54: 1-13 doi: 10.1007/s11082-021-03373-1
[15]	Peregrine D. Water wave, nonlinear Schrödinger equations and their solutions. J Austral Math Soc Ser B, 1983, 25(1): 16-43 doi: 10.1017/S0334270000003891
[16]	Du D, Wu Y, Zhang K. On the blow-up for nonlinear Schrödinger equation with inhomogeneous perturbation. Mathematical Methods in the Applied Sciences, 2024, 47(7): 5664-5676 doi: 10.1002/mma.v47.7
[17]	Li Z, Tian S, Yang J. Riemann-Hilbert approach and soliton solutions for the higher-order dispersive nonlinear Schrödinger equation with nonzero boundary conditions. East Asian Journal on Applied Mathematics, 2021, 11(2): 369-388 doi: 10.4208/eajam
[18]	Bao W, Cai Y. Uniform error estimates offinite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM Journal on Numerical Analysis, 2012, 50: 492-521 doi: 10.1137/110830800
[19]	Wang T, Guo B, Xu Q. Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions. Journal of Computational Physics, 2013, 243: 382-399 doi: 10.1016/j.jcp.2013.03.007
[20]	Akrivis G, Dougalis V, Karakashian Q. On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. Numerische Mathematik, 1991, 59: 31-53 doi: 10.1007/BF01385769
[21]	Chen J, Qin M. Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation. Electronic Transactions on Numerical Analysis, 2001, 12: 193-204
[22]	Wang Y, Wang B, Qin M. Local structure-preserving algorithms for partial differential equations. Science in China Series A: Mathematics, 2008, 51: 2115-2136
[23]	Cai J, Wang Y, Liang H. Local energy-preserving and momentum-preserving algorithms for coupled nonlinear Schrödinger system. Journal of Computational Physics, 2013, 239: 30-50 doi: 10.1016/j.jcp.2012.12.036
[24]	Wang J, Wang Y, Liang D. Construction of local structure-preserving algorithms for the general multi-symplectic Hamiltonian system. Communications in Computational Physics, 2020, 27: 828-860 doi: 10.4208/cicp
[25]	Wang J, Zhou Z, Wang Y. Local structure-preserving algorithms for the Klein-Gordon-Zakharov equation. Acta Mathematica Scientia, 2023, 43: 1211-1238 doi: 10.1007/s10473-023-0313-2
[26]	Bridges T, Reich S. Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Physics Letters A, 2001, 284: 184-193 doi: 10.1016/S0375-9601(01)00294-8
[27]	Dalfovo F, Giorgini S, Pitaevskii L. Theory of Bose-Einstein condensation in trapped gases. Review of Modern Physics, 1999, 71: 463-512 doi: 10.1103/RevModPhys.71.463
[28]	Tornberg A, Engquist B. Regularization techniques for numerical approximation of PDEs with singularities. Journal of Scientific Computing, 2003, 19: 527-552 doi: 10.1023/A:1025332815267
[29]	Qian X, Fu H, Song S. Conservative modified Crank-Nicolson and time-splitting wavelet methods for modeling Bose-Einstein condensates in delta potentials. Applied Mathematics and Computation, 2017, 307: 1-16 doi: 10.1016/j.amc.2017.02.037
[30]	Zhou X, Cai Y, Cai X. Accurate and efficient numerical methods for the nonlinear Schrödinger equation with Dirac delta potential. Calcolo, 2023, 60(4): 1-28 doi: 10.1007/s10092-022-00496-z
[31]	LeVeque R, Li Z. The immersed interface method for elliptic equation with discontinuous coefficients and singular sources. SIAM Journal on Numerical Analysis, 1994, 31: 1019-1044 doi: 10.1137/0731054
[32]	Rutka V, Li Z. An explicit jump immersed interface method for two-phase Navier-Stokes equations with interfaces. Computer Methods in Applied Mechanics and Engineering, 2008, 197: 2317-2328 doi: 10.1016/j.cma.2007.12.016
[33]	Cheng B, Ya M, Chuan F. Nonlinear Schrödinger equation with a Dirac delta potential: finite difference method. Communications in Theoretical Physics, 2020, 72: 1-6
[34]	Bai J, Wang L. EJIIM for the stationary Schrödinger equations with delta potential wells. Applied Mathematics and Computation, 2015, 254: 113-124 doi: 10.1016/j.amc.2014.12.095
[35]	Bai J, Li C, Liu X. Weak multi-symplectic reformulation and geometric numerical integration for the nonlinear Schrödinger equations with delta potentials. IMA Journal of Numerical Analysis, 2018, 38(1): 399-429 doi: 10.1093/imanum/drw062
[36]	Bai J. Multi-symplectic Runge-Kutta-Nyström methods for nonsmooth nonlinear Schrödinger equations. Journal of Mathematical Analysis and Applications, 2016, 444: 721-736 doi: 10.1016/j.jmaa.2016.06.060
[37]	Bai J, Ullah H, Li C. Energy-preserving methods for non-smooth nonlinear Schrödinger equations. Applied Numerical Mathematics, 2023, 185: 188-202 doi: 10.1016/j.apnum.2022.11.017
[38]	Witthaut D, Mossmann S, Korsch H. Bound and resonance states of the nonlinear Schrödinger equation in simple modle systems. Journal of Physics A, 2005, 38: 1777-1792 doi: 10.1088/0305-4470/38/8/013