灰色系统理论在波包演化近似计算中的应用

doi:10.19596/j.cnki.1001-246x.8361

摘要/Abstract

摘要：

基于灰色系统理论, 对波包分布进行GM(1, 1)模型时间响应函数替换, 通过变分法等数学技术得到波包波函数的模拟演化行为, 并进行误差分析和修正。在实际应用中, 只需要对其发展系数参量进行确定, 就能较好地得到波函数的近似演化行为。以高斯型波包为例, 将数学建模中的灰色系统理论引入物理学的计算当中, 提供了一种近似计算方法, 在涉及波包演化的计算中可得到应用。

关键词: 波包, 演化, 灰色系统, 数学建模

Abstract:

Based on the grey system theory, we replace the time response function of GM (1, 1) model for wave packet distribution, and obtain simulated evolution behavior of wave packet's wave function with mathematical techniques such as variational method, and make error analysis and correction. In practical applications, the approximate evolution behavior of wave function can be obtained with parameters of its development coefficient. Gaussian wave packet is taken as an example to show the method. The grey system theory in mathematical modeling is introduced into the calculation of physics, and an approximate calculation method is provided, which can be used in the calculation of wave packet evolution.

Key words: wave packet, evolution, grey system, mathematical modeling

倪嘉雨, 高德宏, 叶林峰, 冯向东. 灰色系统理论在波包演化近似计算中的应用[J]. 计算物理, 2022, 39(2): 201-211.

Jiayu NI, Dehong GAO, Linfeng YE, Xiangdong FENG. Application of Grey System Theory in Approximate Calculation of Wave Packet Evolution[J]. Chinese Journal of Computational Physics, 2022, 39(2): 201-211.

图/表 8

图1 方差的相对误差函数示意图

Fig.1 A schematic of relative error function of the variance

图2 某种光子的波函数，落在区间[-1000, 1000] 的部分占整体的99.8%

Fig.2 Wave function of a photon in which the part falling in the interval [-1000, 1000] is 99.8% of the whole

图3 灰色波函数与波函数在0、300、500、1000 m处的相对误差函数

Fig.3 Relative error functions of gray wave function and wave function at 0, 300, 500, 1000 m

图4 随着发展系数α的增加，(a) 有效时间迅速减小；(b) 有效时间内的平均相对误差持续减小

Fig.4 (a) The effective time decreases rapidly as the development factor α increases; (b) The average relative error in effective time decreases continuously as the development factor α increases

图5 灰色波函数与波函数的比较(随着时间推移，更“胖”更“矮”的波包由灰色波函数所描述。)

Fig.5 Comparison between gray wave function and wave function (The fatter and shorter wave packets are described with the gray wave function over time.)

图6 有效时间最终时刻t=95.43s时，φG与φ的相对误差

Fig.6 The relative error of φG and φ at the final moment of effective time t=95.43 s

图7 Kawahara波相对误差函数

Fig.7 Relative error function of Kawahara wave

图8 拟合效果图(图中红色的标记是数据点，蓝色和绿色的线是拟合函数。)

Fig.8 The figure of fit effect (The red markers are data points. The blue and green lines are fit functions.)

参考文献 27

1	CHOU C C . Hydrodynamic analysis of the Schrödinger-Langevin equation for wave packet dynamics[J]. Physics Letters A, 2017, 381 (39): 3384- 3390. DOI
2	CAVALIERI A V , JORDAN P , LESSHAFFT L . Wave-packet models for jet dynamics and sound radiation[J]. Applied Mechanics Reviews, 2019, 71 (2): 802.
3	HADER C , FASEL H F . Three-dimensional wave packet in a Mach 6 boundary layer on a flared cone[J]. Journal of Fluid Mechanics, 2020, 885, R3. DOI
4	WANG L , ZONG J , WANG X L , et al. Solitary waves with double kinks of mBBM equation and their dynamical stabilities[J]. Chinese Journal of Computaional Physics, 2016, 33 (2): 212- 220.
5	NESTOR S , HOUWE A , REZAZADEH H , et al. New solitary waves for the Klein-Gordon-Zakharov equations[J]. Modern Physics Letters B, 2020, 34 (23): 205- 216.
6	VERNIERO J L , HOWES G G . The Alfvénic nature of energy transfer mediation in localized, strongly nonlinear Alfvén wavepacket collisions[J]. Journal of Plasma Physics, 2018, 84 (1): 94- 109.
7	KUNNUS K , VACHER M , HARLANG T C , et al. Vibrational wavepacket dynamics in Fe carbene photosensitizer determined with femtosecond X-ray emission and scattering[J]. Nature Communications, 2020, 11 (1): 1- 11. DOI
8	VILLENEUVE D M , HOCKETT P , VRAKKING M J , et al. Coherent imaging of an attosecond electron wave packet[J]. Science, 2017, 356 (6343): 1150- 1153. DOI
9	TUTUNNIKOV I , RAJITHA K V , VORONIN A Y , et al. Impulsively excited gravitational quantum states: Echoes and time-resolved spectroscopy[J]. Physical Review Letters, 2021, 126 (17): 170403. DOI
10	SHU C C , HONG Q Q , GUO Y , et al. Orientational quantum revivals induced by a single-cycle terahertz pulse[J]. Physical Review A, 2020, 102 (6): 063124. DOI
11	YAR A , KHAN S U . Chaotic transport of electron wave packet in Weyl semimetal slab[J]. Physical Review E, 2019, 99 (5): 052213. DOI
12	LI X . A modified surface hopping method for Schrödinger equation with conical crossings[J]. Chinese Journal of Computaional Physics, 2019, 36 (1): 113- 126.
13	LIU S F . Emergence and development of grey system theory and its forward trends[J]. Journal of Nanjing University of Aeronautics and Astronautics, 2004, (02): 267- 272.
14	LIU S F, FORREST J, YANG Y. A brief introduction to grey systems theory[C]. IEEE International Conference on Grey Systems and Intelligent Services, 2011: 1-9.
15	倪嘉雨, 涂缘. 基于高阶AGO生成的GM (1, 1)模型[EB/OL]. http://www.chinaxiv.org/ChinaXiv:202105.00041.pdf.
16	WANG L , ZHU G W , YANG W Q , et al. Forecasting of mid-long-term daily load curve based on grey neural network and grey relational degree[J]. Engineering Journal of Wuhan University, 2019, 52 (01): 58- 70.
17	LIU S F , ZENG B , LIU J F , et al. Study on several basic forms and application scope of GM (1, 1) model[J]. Systems Engineering, 2014, 36 (03): 501- 508.
18	LI Y J . Lecture notes on functional analysis[M]. Beijing: Science Press, 2011.
19	SRIDHAR V , ELLIOTT R L . On the development of a simple downwelling longwave radiation scheme[J]. Agricultural and Forest Meteorology, 2002, 112 (3/4): 237- 243.
20	TIMOTHY S. Numerical analysis[M]. PEI Y, MA G, Transl. Beijing: China Machine Press, 2014.
21	MAO S S . Advanced mathematical statistics[M]. Beijing: Higher Education Press, 2006.
22	ORON A . Three-dimensional nonlinear dynamics of thin liquid films[J]. Physical Review Letters, 2000, 85 (10): 2108. DOI
23	CAO Q , XIAO D , YANG X , WANG J . Numerical simulation on nonlinear evolution of magneto-Rayleigh-Taylor instability[J]. Chinese Journal of Computaional Physics, 2021, 38 (1): 5- 15.
24	KAWAHARA T . Oscillatory solitary waves in dispersive media[J]. Journal of the Physical Society of Japan, 1972, 33 (1): 260- 264. DOI
25	CHI L M G , NA R M D L , HAN Y C . Numerical study on stable propagation of solitary wave in viscous fluid media[J]. Journal of Inner Mongolia University for Nationalities, 2017, 32 (02): 101- 105.
26	DELKHOSH , RAHMANZADEH M . Solving nonlinear differential equations in astrophysics and fluid mechanics using the generalized pseudo spectral method[J]. SeMA Journal, 2021, 1- 18.
27	WANG L , LI H . Numerical solution of the modified Kawahara equation using sinc collocation method[J]. International Journal of Nonlinear Science, 2020, 30 (2/3): 153- 162.

[1]	詹泓飞, 蔡振宁, 胡光辉. 基于虚时间演化与谱方法的一类基态Wigner函数计算方法[J]. 计算物理, 2022, 39(6): 651-665.
[2]	詹杰民, 扁诗奇, 罗莹莹, 胡文清, 龚也君. 雷诺数为15 000方腔环流的基准解和周期性分析[J]. 计算物理, 2020, 37(4): 413-421.
[3]	赵国忠, 蔚喜军, 郭虹平, 董自明. 求解含有高阶导数偏微分方程的局部间断Petrov-Galerkin方法[J]. 计算物理, 2019, 36(5): 517-532.
[4]	王彬彬, 韩永昌. HeH⁺体系光缔合反应的动力学研究[J]. 计算物理, 2018, 35(3): 343-349.
[5]	郭玮, 白静, 李月华. 飞秒激光参数对非绝热耦合分子态布居的影响[J]. 计算物理, 2017, 34(1): 119-125.
[6]	李月华, 郭玮. 飞秒激光参数对四态阶跃型分子光电子能谱的影响[J]. 计算物理, 2016, 33(4): 453-459.
[7]	梁颖, 王军, 樊锡君. 开放Ⅴ型系统相对位相对LWI增益瞬态演化的影响[J]. 计算物理, 2012, 29(5): 775-780.
[8]	裴宪军, 巩春志, 汪志健, 田修波, 杨士勤. 梯形管内壁等离子体离子注入鞘层扩展MATLAB-PIC数值模拟[J]. 计算物理, 2012, 29(2): 221-226.
[9]	梁变, 刘中波, 贾克宁, 梁颖, 樊锡君. V型稠密介质中脉宽对超短脉冲和频谱及其CEP依赖性的影响[J]. 计算物理, 2011, 28(6): 889-894.
[10]	梁变, 贾克宁, 梁颖, 仝殿民, 樊锡君. 稠密V型介质中双色飞秒脉冲及频谱演化的RCEP调制[J]. 计算物理, 2011, 28(4): 583-588.
[11]	王晨星, 唐维军, 程军波. 应用多介质PPM方法计算斜激波与物质交界面的相互作用[J]. 计算物理, 2004, 21(6): 531-537.
[12]	刘林, 季江徽, 廖新浩. 小行星轨道演化研究中的数值方法[J]. 计算物理, 2001, 18(2): 185-188.
[13]	徐静雯, 刘莲君, 毛有东, 赵力, 李元香. H^-与He外场特性的演化求解[J]. 计算物理, 2001, 18(1): 43-46.
[14]	熊盛武, 李元香, 康立山, 陈毓屏. 抛物型方程的演化参数识别方法[J]. 计算物理, 2000, 17(5): 511-517.
[15]	刘林, 廖新浩, 季江徽. 辛算法在近地小行星轨道演化数值研究中的应用[J]. 计算物理, 1997, 14(S1): 649-651.