带有PT对称势的非线性薛定谔方程的两类反问题
Two Types of Inverse Problems of the Nonlinear Schr?dinger Equation with PT Symmetric Potentials
摘要: 本文对带有PT对称势三阶五阶幂律非线性薛定谔方程提出了关于参数和势函数反演的两类反问题。对于参数反演问题，我们分别采用PINNs (Physics Informed Neural Networks)和传统的结合有限差分法与优化算法求解的方法进行比较。计算结果显示，在求解反问题时，传统方法每步参数优化需要数值求解非线性薛定谔方程，计算量较大。而PINNs的方法无需重复求解薛定谔方程，计算效率更高。对于PT对称势函数反演问题，通过在PINNs中嵌入自适应基函数，从而反演得到PT对称势。数值实验显示PINNs在算法计算反问题效率上优于传统微分数值求解和优化相结合的方法。
Abstract: This paper proposes two types of inverse problems about parameter and potential function inversion for the third-order and fifth-order power-law nonlinear Schrödinger equation with PT-symmetric potential. For the parameter inversion problem, PINNs are used respectively, compared with the traditional method that combines the finite difference and optimization algorithms to solve the problem. The calculation results show that when solving the inverse problem, the traditional method needs to numerically solve the nonlinear Schrödinger equation for each step of parameter optimization, which requires a large amount of calculation. The PINNs method does not require repeatedly solving the Schrödinger equation, and the calculation efficiency is higher. For the inversion problem of the PT-symmetric potential function, the PT-symmetric potential is inverted by embedding adaptive basis functions in PINNs. Numerical experiments show that PINNs are superior to traditional methods that combine differential numerical solving and optimization in algorithmic calculation of inverse problems.
文章引用：张坤. 带有PT对称势的非线性薛定谔方程的两类反问题[J]. 理论数学, 2024, 14(3): 117-134. https://doi.org/10.12677/pm.2024.143091

参考文献

[1] Bender, C.M. and Boettcher, S. (1998) Real Spectra in Non-Hermitian Hamiltonians Having P T Symmetry. Physical Review Letters, 80, 5243. [Google Scholar] [CrossRef] 
[2] Aslan, E.C., Tchier, F. and Inc, M. (2017) On Optical Solitons of the Schrödinger-Hirota Equation with Power Law Nonlinearity in Optical Fibers. Superlattices and Microstructures, 105, 48-55. [Google Scholar] [CrossRef] 
[3] Ma, Y.L. (2020) Abundant Excited Optical Breathers for a Nonlinear Schrödinger Equation with Variable Dispersion and Nonlinearity Terms in Inhomogenous Fiber Optics. Optik, 201, Article 162821. [Google Scholar] [CrossRef] 
[4] Abdelwahed, H.G., El-Shewy, E.K., Abdelrahman, M.A.E., et al. (2021) On the Physical Nonlinear (N 1)-Dimensional Schrödinger Equation Applications. Results in Physics, 21, Article 103798. [Google Scholar] [CrossRef] 
[5] Wazwaz, A.M. and Kaur, L. (2019) Optical Solitons for Nonlinear Schrödinger (NLS) Equation in Normal Dispersive Regimes. Optik, 184, 428-435. [Google Scholar] [CrossRef] 
[6] Özdemir, Ş.K., Rotter, S., Nori, F., et al. (2019) Parity-Time Symmetry and Exceptional Points in Photonics. Nature Materials, 18, 783-798. [Google Scholar] [CrossRef] [PubMed]
[7] Hogan, C.C., Murphy, J. and Grow, D. (2023) Recovery of a Cubic Nonlinearity for the Nonlinear Schrödinger Equation. Journal of Mathematical Analysis and Applications, 127016. [Google Scholar] [CrossRef] 
[8] Murphy, J. (2022) Recovery of a Spatially-Dependent Coefficient from the NLS Scattering Map. ArXiv Preprint ArXiv:2209.07680,.
[9] Avdonin, S.A., Mikhaylov, A.S., Mikhaylov, V.S., et al. (2021) Inverse Problem for the Schrödinger Equation with Non-Self-Adjoint Matrix Potential. Inverse Problems, 37, 035002. [Google Scholar] [CrossRef] 
[10] Baudouin, L. and Puel, J.P. (2002) Uniqueness and Stability in an Inverse Problem for the Schrödinger Equation. Inverse Problems, 18, 1537. [Google Scholar] [CrossRef] 
[11] Hermann, J., Schätzle, Z. and Noé, F. (2020) Deep-Neural-Network Solution of the Electronic Schrödinger Equation. Nature Chemistry, 12, 891-897. [Google Scholar] [CrossRef] [PubMed]
[12] Raissi, M., Perdikaris, P. and Karniadakis, G.E. (2019) Physics-Informed Neural Networks: A Deep Learning Framework for Solving forward and Inverse Problems Involving Nonlinear Partial Differential Equations. Journal of Computational Physics, 378, 686-707. [Google Scholar] [CrossRef] 
[13] Zhou, Z. and Yan, Z. (2021) Solving forward and Inverse Problems of the Logarithmic Nonlinear Schrödinger Equation with PT-Symmetric Harmonic Potential via Deep Learning. Physics Letters A, 387, Article 127010. [Google Scholar] [CrossRef] 
[14] Song, J. and Yan, Z. (2023) Deep Learning Soliton Dynamics and Complex Potentials Recognition for 1D and 2D PT-Symmetric Saturable Nonlinear Schrödinger Equations. Physica D: Nonlinear Phenomena, 448, Article 133729. [Google Scholar] [CrossRef] 
[15] Göksel, İ., Antar, N. and Bakırtaş, İ. (2015) Solitons of (1 1) D Cubic-Quintic Nonlinear Schrödinger Equation with PT-Symmetric Potentials. Optics Communications, 354, 277-285. [Google Scholar] [CrossRef] 
[16] Marini, F. and Walczak, B. (2015) Particle Swarm Optimization (PSO). A Tutorial. Chemometrics and Intelligent Laboratory Systems, 149, 153-165. [Google Scholar] [CrossRef] 
[17] 附录
[18] 下面给出带有初边值条件的NLSE方程(1)的有限差分格式。
[19] 为了用差分格式求解(1)~(2)，将求解区域作剖分，取正整数。将作M等分，将作N等分。，；，；，；，，，记，，令
[20] 设，引进如下记号：
[21] 设引进如下记号：
[22] 设为上的网格函数，则为上的网格函数，为上的网格函数。
[23] 在点处考虑(1)，于是有
[24] (11)
[25] 由泰勒公式展开
[26] (12)
[27] (13)
[28] 由微分公式可得
[29] (14)
[30] 其中
[31] 可以通过方程(1)结合初值条件(2)可求得。在点处考虑(1)，有
[32] (15)
[33] 将(12)和(13)代入(15)中，可得
[34] (16)
[35] 由初边值条件(2)，并且忽略(14)和(16)小量项，对问题(1)~(2)建立如下线性差分格式
[36] (17)
[37] 首先根据初始和边界条件计算的值，记
[38] 由初值条件可知第0层上的值，根据差分格式
[39] 则关于第1层值的差分格式为
[40] (18)
[41] 在(18)中，将第1层写在方程左边，第0层写在方程右边，令
[42] 可写成如下矩阵形式
[43] 求解以上代数系统，可以得到关于u的第1层数值解。其中通过结合初值条件可求得。
[44] 计算出后，接下来我们建立u的三层格式。假设已确定出了第层的值。根据差分格式为
[45] 则关于第层值的差分格式为
[46] 将第层写在方程左边，层及j层写在方程右边，可写成如下矩阵形式
[47] (19)
[48] 由此我们得到求解本文中NLSE方程的有限差分格式，格式中步长取值越小，数值解越精确。

为你推荐