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The asymptotic solutions to a class of nonlinear disturbed evolution equations

  • 1.

    Department of Electronics and Information Engineering, Bozhou College, Bozhou 236800, China

  • 2.

    Department of Mathematics, Anhui Normal University, Wuhu 241003, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2017.09.006
  • Received Date: 11 May 2016
  • Accepted Date: 29 December 2016
  • Rev Recd Date: 29 December 2016
  • Publish Date: 30 September 2017
    Abstract

  • 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.

    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.
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    • References(32)
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    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.
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    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.
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    MO Jiaqi. The singularly perturbed problem for combustion reaction diffusion[J]. Acta Math Appl Sin, 2001, 17(2): 255-259.
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    MO Jiaqi. A class of nonlocal singularly perturbed problems for nonlinear hyperbolic differential equation[J]. Acta Math Appl Sin, 2001, 17(4): 469-474.
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    MO Jiaqi, WANG Hui. A class of nonlinear nonlocal singularly perturbed problems for reaction diffusion equations[J]. J Biomathematics, 2002, 17(2): 143-148.
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    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.
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    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.
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    [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.
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