二阶非线性SchrÖdinger方程的低正则算法
Low-Regularity Integrator for the Quadratic NLS Equation
DOI: 10.12677/AAM.2022.1110782, PDF, HTML, 下载: 190  浏览: 256  国家自然科学基金支持
作者: 宁 翠：广东金融学院，金融数学与统计学院，广东 广州
关键词: 二次非线性SchrÖdinger方程；低正则；一阶收敛；Quadratic Nonlinear SchrÖdinger Equation； Low-Regularity Integrator； First Order Convergent
摘要: 本文研究二次非线性SchrÖdinger方程具有一阶收敛的一种低正则积分器，这种积分器是显式的并且快速有效。 特别地，我们的算法不需要损失额外的导数就可以实现一阶收敛。 通过严格的误差分析，我们证明了当初值属于Hγ (T) 时，二次非线性SchrÖdinger方程在Hγ (T)上具有一阶收敛，其中γ > ½。
Abstract: In this paper, we introduce a first order low-regularity integrator for the quadratic nonlinear SchrÖdinger equation. The scheme is explicit and efficient to implement. In particular, our scheme does not cost any additional derivative for the first order convergence. By rigorous error analysis, we show that the scheme provides first order accuracy in Hγ (T) for rough initial data in Hγ (T) with γ > ½ .
文章引用：宁翠. 二阶非线性SchrÖdinger方程的低正则算法[J]. 应用数学进展, 2022, 11(10): 7362-7372. https://doi.org/10.12677/AAM.2022.1110782

参考文献

[1] Bejenaru, I. and Tao, T. (2006) Sharp Well-Posedness and Ill-Posedness Results for a Quadratic Non-Linear Schr¨odinger Equation. Journal of Functional Analysis, 233, 228-259.
https://doi.org/10.1016/j.jfa.2005.08.004
[2] Berestycki, H. and Cazenave, T. (1981) Instability of Stationary States in Nonlinear Schro¨dinger and Klein-Gordon Equations. Comptes Rendus de l’Acad´emie des Sciences, 293, 489-492.
[3] Li, Y., Wu, Y. and Xu, G. (2011) Global Well-Posedness for the Mass-Critical Nonlinear Schro¨dinger Equation on T. Journal of Differential Equations, 250, 2715-2736.
https://doi.org/10.1016/j.jde.2011.01.025
[4] Li, Y., Wu, Y. and Xu, G. (2011) Low Regularity Global Solutions for the Focusing Mass- Critical NLS in R∗. SIAM Journal on Mathematical Analysis, 43, 322-340.
https://doi.org/10.1137/090774537
[5] Weinstein, M.I. (1986) Lyapunov Stability of Ground States of Nonlinear Dispersive Evolution Equations. Communications on Pure and Applied Mathematics, 39, 51-68.
https://doi.org/10.1002/cpa.3160390103
[6] Guo, B. and Wu, Y. (1995) Orbital Stability of Solitary Waves for the Nonlinear Derivative Schro¨dinger Equation. Journal of Differential Equations, 123, 35-55.
https://doi.org/10.1006/jdeq.1995.1156
[7] Wu, Y. (2013) Global Well-Posedness of the Derivative Nonlinear Schro¨dinger Equations in Energy Space. Analysis & PDE, 6, 1989-2002.
https://doi.org/10.2140/apde.2013.6.1989
[8] Wu, Y. (2015) Global Well-Posedness on the Derivative Nonlinear Schro¨dinger Equation. Anal- ysis & PDE, 8, 1101-1113.
https://doi.org/10.2140/apde.2015.8.1101
[9] Liu, X., Simpson, G. and Sulem, C. (2013) Stability of Solitary Waves for a Generalized Derivative Nonlinear Schro¨dinger Equation. Journal of Nonlinear Science, 23, 557-583.
https://doi.org/10.1007/s00332-012-9161-2
[10] Le Coz, S. and Wu, Y. (2018) Stability of Multi-Solitons for the Derivative Nonlinear Schro¨dinger Equation. International Mathematics Research Notices, 2018, 4120-4170.
https://doi.org/10.1093/imrn/rnx013
[11] Ning, C. (2020) Instability of Solitary Wave Solutions for Derivative Nonlinear Schro¨dinger Equation in Borderline Case. Nonlinear Analysis, 192, Article ID: 111665.
https://doi.org/10.1016/j.na.2019.111665
[12] Feng, B. and Zhu, S. (2021) Stability and Instability of Standing Waves for the Fractional Nonlinear Schro¨dinger Equations. Journal of Differential Equations, 292, 287-324.
https://doi.org/10.1016/j.jde.2021.05.007
[13] Guo, Q. and Zhu, S. (2018) Sharp Threshold of Blow-Up and Scattering for the Fractional Hartree Equation. Journal of Differential Equations, 264, 2802-2832.
https://doi.org/10.1016/j.jde.2017.11.001
[14] Zhu, S. (2016) Existence and Uniqueness of Global Weak Solutions of the Camassa-Holm Equation with a Forcing. Discrete and Continuous Dynamical Systems, 36, 5201-5221.
https://doi.org/10.3934/dcds.2016026
[15] Zhu, S., Zhang, J. and Yang, H. (2010) Limiting Profile of the Blow-Up Solutions for the Fourth-Order Nonlinear Schro¨dinger Equation. Dynamics of Partial Differential Equations, 7, 187-205.
https://doi.org/10.4310/DPDE.2010.v7.n2.a4
[16] Court´es, C., Lagouti`ere, F. and Rousset, F. (2020) Error Estimates of Finite Difference Schemes for the Korteweg-de Vries Equation. IMA Journal of Numerical Analysis, 40, 628-685.
https://doi.org/10.1093/imanum/dry082
[17] Holden, H., Koley, U. and Risebro, N. (2014) Convergence of a Fully Discrete Finite Difference Scheme for the Korteweg-de Vries Equation. IMA Journal of Numerical Analysis, 35, 1047- 1077.
https://doi.org/10.1093/imanum/dru040
[18] Aksan, E. and O¨ zde¸s, A. (2006) Numerical Solution of Korteweg-de Vries Equation by Galerkin B-Spline Finite Element Method. Applied Mathematics and Computation, 175, 1256-1265.
https://doi.org/10.1016/j.amc.2005.08.038
[19] Dutta, R., Koley, U. and Risebro, N.H. (2015) Convergence of a Higher Order Scheme for the Korteweg-de Vries Equation. SIAM Journal on Numerical Analysis, 53, 1963-1983.
https://doi.org/10.1137/140982532
[20] Holden, H., Karlsen, K.H., Risebro, N.H. and Tang, T. (2011) Operator Splitting for the KdV Equation. Mathematics of Computation, 80, 821-846.
https://doi.org/10.1090/S0025-5718-2010-02402-0
[21] Holden, H., Lubich, C. and Risebro, N.H. (2012) Operator Splitting for Partial Differential Equations with Burgers Nonlinearity. Mathematics of Computation, 82, 173-185.
https://doi.org/10.1090/S0025-5718-2012-02624-X
[22] Ma, H. and Sun, W. (2001) Optimal Error Estimates of the Legendre-Petrov-Galerkin Method for the Korteweg-de Vries Equation. SIAM Journal on Numerical Analysis, 39, 1380-1394.
https://doi.org/10.1137/S0036142900378327
[23] Shen, J. (2003) A New Dual-Petrov-Galerkin Method for Third and Higher Odd-Order Dif- ferential Equations: Application to the KdV Equation. SIAM Journal on Numerical Analysis, 41, 1595-1619.
https://doi.org/10.1137/S0036142902410271
[24] Yan, J. and Shu, C.W. (2002) A Local Discontinuous Galerkin Method for KdV Type Equa- tions. SIAM Journal on Numerical Analysis, 40, 769-791.
https://doi.org/10.1137/S0036142901390378
[25] Liu, H. and Yan, J. (2006) A Local Discontinuous Galerkin Method for the Kortewegde Vries Equation with Boundary Effect. Journal of Computational Physics, 215, 197-218.
https://doi.org/10.1016/j.jcp.2005.10.016
[26] Hochbruck, M. and Ostermann, A. (2010) Exponential Integrators. Acta Numerica, 19, 209-286.
https://doi.org/10.1017/S0962492910000048
[27] Hofmanov´a, M. and Schratz, K. (2017) An Exponential-Type Integrator for the KdV Equation. Numerische Mathematik, 136, 1117-1137.
https://doi.org/10.1007/s00211-016-0859-1
[28] Lubich, C. (2008) On Splitting Methods for Schro¨dinger-Poisson and Cubic Nonlinear Schro¨dinger Equations. Mathematics of Computation, 77, 2141-2153.
https://doi.org/10.1090/S0025-5718-08-02101-7
[29] Ostermann, A. and Schratz, K. (2018) Low Regularity Exponential-Type Integrators for Semi- linear Schr¨odinger Equations. Foundations of Computational Mathematics, 18, 731-755.
https://doi.org/10.1007/s10208-017-9352-1
[30] Wu, Y. and Yao, F. (2022) A First-Order Fourier Integrator for the Nonlinear Schro¨dinger Equation on T without Loss of Regularity. Mathematics of Computation, 91, 1213-1235.
https://doi.org/10.1090/mcom/3705
[31] Kato, T. and Ponce, G. (1988) Commutator Estimates and the Euler and Navier-Stokes E- quations. Communications on Pure and Applied Mathematics, 41, 891-907.
https://doi.org/10.1002/cpa.3160410704

为你推荐