一类非线性扰动发展方程的渐近解

doi:10.3969/j.issn.0253-2778.2017.09.006

中国科学技术大学学报 ›› 2017, Vol. 47 ›› Issue (9): 749-754.DOI: 10.3969/j.issn.0253-2778.2017.09.006

一类非线性扰动发展方程的渐近解

冯依虎，刘树德，莫嘉琪

1.亳州学院电子与信息工程系，安徽亳州 236800; 2．安徽师范大学数学系, 安徽芜湖 241003

收稿日期:2016-05-11 修回日期:2016-12-29 接受日期:2016-12-29 出版日期:2023-03-27 发布日期:2016-12-29
通讯作者: 莫嘉琪
作者简介:冯依虎，男，1982年生，硕士/副教授.研究方向：应用数学. E-mail: fengyihubzsz@163.com
基金资助:
国家自然科学基金(41275062,11202106), 安徽省教育厅自然科学重点基金项目 (KJ2015A347, KJ2017A702)和安徽省高校优秀青年人才支持计划重点项目(gxyqZD2016520)资助.

The asymptotic solutions to a class of nonlinear disturbed evolution equations

FENG Yihu, LIU Shude, MO Jiaqi

1. Department of Electronics and Information Engineering, Bozhou College, Bozhou 236800, China; 2. Department of Mathematics, Anhui Normal University, Wuhu 241003, China

Received:2016-05-11 Revised:2016-12-29 Accepted:2016-12-29 Online:2023-03-27 Published:2016-12-29

摘要/Abstract

摘要： 研究了一类非线性发展方程. 首先作行波变换, 讨论了在非扰动情况下的非线性方程, 利用双曲函数待定系数方法, 求得了相应方程的孤立子精确解. 然后利用广义变分迭代方法，求出了原非线性扰动发展方程渐近孤立子行波解. 最后通过举例, 说明了利用本方法求出的渐近孤立子解简单可行, 并有良好的精度.

关键词: 发展方程, 非线性, 扰动

Abstract: A class of nonlinear evolution equations was considered. Firstly, introducing a travelling wave transform, the non-disturbance case was discussed by employing the undetermined coefficient method of the hyperbolic functions and solitary exact solution to the corresponding nonlinear equation was obtained. Then, the solitary travelling wave asymptotic solution to the original nonlinear disturbed evolution equation was founded by using the generalized variational iteration method. Finally, an example was given to show the simplicity and feasibility of the asymptotic solitary solution.

Key words: evolution equation, nonlinear, destabilization

中图分类号:

O175.29

冯依虎, 刘树德, 莫嘉琪. 一类非线性扰动发展方程的渐近解[J]. 中国科学技术大学学报, 2017, 47(9): 749-754.

FENG Yihu, LIU Shude, MO Jiaqi. The asymptotic solutions to a class of nonlinear disturbed evolution equations[J]. Journal of University of Science and Technology of China, 2017, 47(9): 749-754.

参考文献

［1］
MCPHADEN M J, ZHANG D. Slowdown of the meridional overturning circulation in the upper Pacific Ocean [J]. Nature, 2002, 415: 603-608.
[2] GU Daifang, PHILANDER S G H. Interdecadal climate fluctuations that depend on exchanges between the tropics and extratropics[J]. Science, 1997, 275 (7): 805-807.
[3] 刘式适, 赵强，付遵涛, 等. 一类非线性方程的新周期解[J]. 物理学报, 2002, 51(1): 10-14.
LIU Shikuo, ZHAO Qiang, FU Zuntao, et al. New periodic solutions to a kind of nonlinear wave equations[J]. Acta Phys Sin, 2002, 51(1): 10-14.
[4] 左伟明, 潘留仙, 颜家壬. Landau-Ginzburg-Higgs方程的微扰理论[J]. 物理学报, 2005, 54 (1): 1-5.
ZUO Weiming, PAN Liuxian, YAN Jiaren. The theory of the perturbation for Landau-Ginzburg-Higgs equation[J]. Acta Phys Sin, 2005, 54 (1): 1-5.
[5] 陈观伟, 马世旺. 带有边界势和一般时间频率的非周期离散非线性Schrodinger方程：无穷多个孤立子[J]. 中国科学:数学, 2014, 44(8): 843-856.
CHEN Guanwei, MA Shiwang. Non-periodic discrete nonlinear Schrodinger equations with unbounded potential and general temporal frequencies: Infinitely many solitons[J]. Scientia Sinica (Mathematica), 2014, 44(8): 843-856.
[6] LIN Yihua, ZENG Qingcun. Kelvin waves in a simple air-sea coupled model in the tropics[J]. Progress in Natural Sci, 1999, 9 (3): 211-215.
[7] LIN Yihua, JI Zhongzhen, ZENG Qingcun. Meridional wind forced low frequency disturbances in the tropical ocean[J]. Progress in Natural Sci, 1999, 9 (7): 532-538.
[8] 封国林, 戴新刚, 王爱慧, 等. 混沌系统中可预报性研究[J]. 物理学报, 2001, 50(4): 606-611.
FENG Guolin, DAI Xingang, WANG Aihui, et al. On numerical predictability in the chaos system[J]. Acta Phys Sin, 2001, 50(4): 606-611.
[9] PAN Liuxian, YAN Jiaren, ZHOU Cuanghui. Direct approach for soliton perturbations based on the Green’s function[J]. Chin Phys, 2003, 10 (7): 594-598.
[10] 范恩贵, 张鸿庆. 非线性波动方程的孤波解[J]. 物理学报, 1997, 46 (7): 1254-1258.
FAN Engui, ZHANG Hongqing. The solitary wave solutions for a class of nonlinear wave equations[J]. Acta Phys Sin, 1997, 46 (7): 1254-1258.
[11] MO Jiaqi. Singular perturbation for a class of nonlinear reaction diffusion systems[J]. Science in China, Ser A, 1989, 32(11): 1306-1315.
[12] MO Jiaqi, LIN Wantao, ZHU Jiang. The perturbed solution of sea-air oscillator for ENSO model[J]. Progress in Natural Sci, 2004, 14(6): 550-552.
[13] MO Jiaqi, LIN Wantao. The variational iteration solving method for sea-air oscillator model of interdecadal climate fluctuations[J]. Chin Phys B, 2005, 14 (5): 875-878.
[14] MO Jiaqi, WANG Hui, LIN Wantao, et al. Varitional iteration method for solving the mechanism of the equatorial Eastern Pacific El Nio-Southern Oscillation[J]. Chin Phys B, 2006, 15(4): 671-675.
[15] MO Jiaqi. The singularly perturbed problem for combustion reaction diffusion[J]. Acta Math Appl Sin, 2001, 17(2): 255-259.
[16] MO Jiaqi, SHAO S S L. The singularly perturbed boundary value problems for higher-order semilinear elliptic equations[J]. Advances in Math, 2001, 30 (2): 141-148.
[17] MO Jiaqi. A class of nonlocal singularly perturbed problems for nonlinear hyperbolic differential equation[J]. Acta Math Appl Sin, 2001, 17(4): 469-474.
[18] MO Jiaqi, WANG Hui. A class of nonlinear nonlocal singularly perturbed problems for reaction diffusion equations[J]. J Biomathematics, 2002, 17(2): 143-148.
[19] MO Jiaqi. Homotopiv mapping solving method for gain fluency of a laser pulse amplifier[J]. Science in China， Ser G, 2009, 59 (7): 1007-1010.
[20] MO Jiaqi, LIN Sutong. The homotopic mapping solution for the solitary wave for a generalized nonlinear evolution equation[J]. Chin Phys B, 2009, 18(9): 3628-3631.
[21] MO Jiaqi. Approximate solution of homotopic mapping to solitary wave for generalized nomlinear KdV system[J]. Chin Phys Lett, 2009, 26 (1): 010204.
[22] MO Jiaqi. Generalized variational iteration solution of soliton for disturbed KdV equation[J]. Commun Theor Phys, 2010, 53 (3): 440-442.

[23] MO Jiaqi, CHEN Xianfeng. Homotopic mapping method of solitary wave solutions for generalized complex Burgers equation[J]. Chin Phys B, 2010, 19(10): 100203.
[24] MO Jiaqi. A singularly perturbed reaction diffusion problem for the nonlinear boundary condition with two parameters[J]. Chin Phys B, 2010, 18 (1): 010203.
[25] 莫嘉琪. 一类非线性扰动发展方程的广义迭代解[J]. 物理学报, 2011, 60 (2): 020202.
MO Jiaqi. The variational iteration solution method for a class of nonlinear disturbed evolution equations[J]. Acta Phys Sin, 2011, 60 (2): 020202.
[26] FENG Yihu, MO Jiaqi. The shock asymptotic solution for nonlinear elliptic equation with two parameters[J]. Math Appl, 2015, 27(3): 579-585.
[27] FENG Yihu, LIU Shude. Spike layer solutions of some quadratic singular perturbation problems with high-order turning points[J]. Math Appl, 2014, 27 (1): 50-51.
[28] 冯依虎, 石兰芳, 汪维刚, 等. 一类广义非线性强阻尼扰动发展方程的行波解[J]. 应用数学和力学, 2015, 36(3): 315-324.
FENG Yihu, SHI Lanfang, WANG Weigang, et al. The traveling wave solution for a class of generalized nonlinear strong damping disturbed evolution equations[J]. Appl Math Mech, 2015, 36(3): 315-324.

[29] 何吉欢. 工程和科学计算中的近似非线性分析方法[M]. 郑州: 河南科学技术出版社, 2002.
[30] HE J H, WU X H. Construction of solitary solution and compacton-like solution by variational iteration method[J]. Chaos Solitions & Fractals, 2006, 29 (1): 108-113.
[31] DE JAGER E M, JIANG Furu. The Theory of Singular Perturbation[M]. Amsterdam, Netherlands: North- Holland Publishing Co,1996.
[32] BARBU L, MOROSANU G. Singularly Perturbed Boundary-Value Problems[M]. Basel, Switzerland: Birkhauser Verlag AG, 2007.

（上接第737页）

[4] CVETKOVIC′ D M, DOOB M, SACKS H. Spectra of Graphs: Theory and Applications[M]. New York: Academic Press, 1979.
[5] MERRIS R. A note on Laplacian graph eigenvalues[J]. Linear Algebra Appl, 1998, 285: 33-35.
[6] CVETKOVID M, DOOB M, SACHS H. Spectra of Graphs: Theory and Application[M]. Heidelberg: Johann Ambrosius Barth Verlag, 1995.
[7] YUAN X Y. A note on the Laplacian spectral radii of bicyclic graphs[J]. Advances in Mathematics, 2010, 6: 703-708.

[1]	张桂洁, 张晓林, 周长勇, 黄金元, 张启跃, 胡世学, 文芠, 黄勰敏. 早-中三叠世高分辨率的δ¹³C_carb新记录:来自华南天生桥剖面的研究[J]. 中国科学技术大学学报, 2021, 51(6): 459-467.
[2]	胡贝贝, 张玲, 方芳, 张宁. 混合耦合非线性Schrödinger方程的Riemann-Hilbert方法及其孤子解[J]. 中国科学技术大学学报, 2021, 51(3): 196-201.
[3]	张志远，叶五一. 基于MIDAS-Expectile回归模型的加密货币风险测度[J]. 中国科学技术大学学报, 2020, 50(6): 860-872.
[4]	朱红宝，陈松林，杜亚洁. 一类非线性时滞奇异摄动边值问题的激波解[J]. 中国科学技术大学学报, 2020, 50(2): 115-119.
[5]	徐建中，汪维刚，莫嘉琪. 非线性非局部奇摄动分数阶方程Cauchy问题的激波解[J]. 中国科学技术大学学报, 2019, 49(8): 620-624.
[6]	占德胜，储晶. 时间相依的超前倒向随机发展方程[J]. 中国科学技术大学学报, 2019, 49(3): 203-209.
[7]	朱红宝. 一类非线性奇摄动时滞边值问题的激波解[J]. 中国科学技术大学学报, 2018, 48(5): 357-360.
[8]	杨萌，何亚轩，闫德立，张勇，王伟明. 基于弧形修正滤波的柱面AR码识别算法[J]. 中国科学技术大学学报, 2018, 48(1): 1-6.
[9]	罗焕丽，王相綦，李想，唐靖宇，杨征，谢凯. 反对称简易十二极场磁铁的设计[J]. 中国科学技术大学学报, 2017, 47(6): 465-473.
[10]	陈怀军，石兰芳，莫嘉琪. 非线性奇摄动问题的广义内部冲击层解[J]. 中国科学技术大学学报, 2015, 45(8): 638-642.
[11]	刘文超，姚军，李爱芬，孙致学，黄朝琴. 考虑压敏效应的低渗透油藏非线性流动特征[J]. 中国科学技术大学学报, 2012, 42(4): 279-288.
[12]	沈慧，张祖德，胡刚. 利用溴酸钠-苹果酸-[CuL](ClO4)2-硫酸化学振荡体系分析测定酪氨酸[J]. 中国科学技术大学学报, 2010, 40(7): 686-693.
[13]	孙树计，陈春，丁宗华，班盼盼，刘云，奚迪龙. 电场对暴时低纬电离层foF2变化的影响[J]. 中国科学技术大学学报, 2009, 39(7): 683-687.

一类非线性扰动发展方程的渐近解

The asymptotic solutions to a class of nonlinear disturbed evolution equations

PDF (PC)

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 13

编辑推荐

Metrics

本文评价