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柱面非线性麦克斯韦方程组的行波解

胡亮 罗懋康

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柱面非线性麦克斯韦方程组的行波解

胡亮, 罗懋康

Traveling wave solutions of the cylindrical nonlinear Maxwell's equations

Hu Liang, Luo Mao-Kang
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  • 柱面电磁波在各种非均匀非线性介质中的传播问题具有非常重要的研究价值.对描述该问题的柱面非线性麦克斯韦方程组进行精确求解,则是最近几年新兴的研究热点.但由于非线性偏微分方程组的极端复杂性,针对任意初边值条件的精确求解在客观上具有极高的难度,已有工作仅解决了柱面电磁波在指数非线性因子的非色散介质中的传播情况.因此,针对更为确定的物理场景,寻求能够精确描述其中更为广泛的物理性质的解,是一种更为有效的处理方法.本文讨论了具有任意非线性因子与幂律非均匀因子的非色散介质中柱面麦克斯韦方程组的行波精确解,理论分析表明这种情况下柱面电磁波的电场分量E已不存在通常形如E=g(r-kt)的平面行波解;继而通过适当的变量替换与求解相应的非线性常微分方程,给出电场分量E=g(lnr-kt)形式的广义行波解,并以例子展示所得到的解中蕴含的类似于自陡效应的物理现象.
    Study on propagation of cylindrical electromagnetic waves in various inhomogeneous and nonlinear media is of fundamental importance, which can be described by the cylindrical nonlinear Maxwell's equations. In recent years, finding exact solutions for these equations has emerged as a popular research topic. The exact solutions play an irreplaceable role in understanding and predicting physical phenomena, and developing numerical calculation methods, and so on. However, due to the extreme complexity of nonlinear partial differential equations, exact solutions of the cylindrical Maxwell's equations were only able to be obtained in a nonlinear and nondispersive medium whose dielectric function is an exponential function in previous researches. Actually, there is no general method at present which can exactly solve arbitrary cylindrical nonlinear Maxwell's equations. Therefore, finding physically admissible solutions meeting certain particular condition for the cylindrical nonlinear Maxwell's equations might be feasible. In this paper, we discuss the traveling wave solutions which are very important in electromagnetic theory, especially in solitary wave theory. To our knowledge, research on obtaining traveling wave solutions of the cylindrical nonlinear Maxwell's equations is still lacking. The main conclusions in this paper are listed as follows. Firstly, we introduce the cylindrical nonlinear Maxwell's equations mentioned in some previous publications, which can describe cylindrical electromagnetic waves propagation in inhomogeneous nonlinear and nondispersive media. In this paper, we focus on the nondispersive media with arbitrary nonlinearity and power-law inhomogeneity. Secondly, we point out that the electric field component E of the model has no plane traveling wave solutions E=g(r-kt), after theoretical analysis and study. Then generalized traveling wave solutions in form of E=g(lnr-kt) for the electric field component are obtained by finding correct variable substitution and solving second-order nonlinear ordinary differential equation.Finally, we provide two examples to show the physical meanings of our generalized traveling wave solutions. We find that the transmitting speeds of vibrations vary with different points of the electric field. Actually, the transmitting speed of the vibration of a certain point closer to the cylinder center is lower. As a result, we observed a physical phenomenon similar to that of self-steepening. Our work can be used to analyze the electromagnetic properties of ferroelectric materials and new materials. Theoretically, it can also provide an approach to studying the cylindrical nonlinear Maxwell's equations.
      Corresponding author: Luo Mao-Kang, makaluo@scu.edu.cn
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    Xiong H, Si L G, Yang X X, Wu Y 2015 Sci. Reports 5 11071

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    Shi L F, Zhu M, Zhou X C, Wang W G, Mo J Q 2008 Phys. Lett. A 372 417

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    Xu Y H, Han X L, Shi L F, Mo J Q 2014 Acta Phys.Sin. 63 090204 (in Chinese) [许永红, 韩祥临, 石兰芳, 莫嘉琪2014 物理学报63 090204]

    [24]

    Harko T, Mak M K 2015 J. Math. Phys. 56 111501

    [25]

    Sardar A, Husnine S M, Rizvi S T R, Younis M, Ali K 2015 Nonlinear Dynam. 82 1317

    [26]

    Wang M L, Li X Z, Zhang J L 2008 Phys. Lett. A 372 417

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    Shu Y Q 2011 Ph. D. Dissertation (Lanzhou:Lanzhou University) (in Chinese)[舒雅琴 2011 博士学位论文 (兰州:兰州大学)]

    [28]

    Petrov E Y, Kudrin A V 2012 Phys. Rev. E 85 055202

    [29]

    Silva A, Monticone F, Castaldi G, Galdi V, Al A, Engheta N 2014 Science 343 160

    [30]

    Eidini M, Paulino G H 2015 Sci. Adv. 1

    [31]

    Ma G, Sheng P 2016 Sci. Adv. 2

    [32]

    Fan W, Yan B, Wang Z B, Wu L M 2016 Sci. Adv. 2

    [33]

    Chen H W, Yang C R, Fu C L, Zhao L, Gao Z Q 2006 Appl. Surf. Sci. 252 4171

    [34]

    Fong D D, Stephenson G B, Streiffer S K, Eastman J A, Auciello O, Fuoss P H, Thompson C 2004 Science 304 1650

    [35]

    Han S H, Park Q H 2011 Phys. Rev. E 83 066601

    [36]

    Shafeeque Ali A K, Porsezian K, Uthayakumar T 2014 Phys. Rev. E 90 042910

    [37]

    Reed E J, Soljačić M, Gee R, Joannopoulos J D 2007 Phys. Rev. B 75 174302

    [38]

    Grischkowsky D, Duling Ⅲ I N, Chen J C, Chi C C 1987 Phys. Rev. Lett. 59 1663

  • [1]

    Ye P X 2007 Nonlinear Optical Physics (Vol. 1) (Beijing:Peking University Press) pp17-18 (in Chinese)[叶佩弦 2007 非线性光学物理 (北京:北京大学出版社) 第17-18页]

    [2]

    Yao B, Zheng Q H, Peng J H, Zhong R N, Xiang T, Xu W S 2011 Chin. Phys. Lett. 28 118401

    [3]

    Zhang M, Li L S, Zheng N, Shi Q F 2013 Chin. Phys. Lett. 30 077802

    [4]

    Chew W C 1990 Waves and Fields in Inhomogeneous Media (New York:Van Nostrand Reinhold) p161

    [5]

    Ertrk V B, Rojas R G 2003 IEEE Trans. Antenn. Propag. 51 739

    [6]

    Petrov E Y, Kudrin A V 2010 JETP 110 537

    [7]

    Kudrin A V, Petrov E Y 2010 JETP 110 537

    [8]

    Xiong H, Si L G, Huang P, Yang X 2010 Phys. Rev. E 82 057602

    [9]

    Xiong H, Si L G, Ding C, L X Y, Yang X, Wu Y 2011 Phys. Rev. A 84 043841

    [10]

    Xiong H, Si L G, Ding C, Yang X, Wu Y 2011 Phys.Rev. A 84 043841

    [11]

    Xiong H, Si L G, Guo J F, L X Y, Yang X 2014 Chin. Phys. B 23 060304

    [12]

    Xiong H, Si L G, Ding C, Yang X, Wu Y 2012 Phys.Rev. E 85 016606

    [13]

    Chen S Y, Li T, Xie J B, Xie H, Zhou P, Tian Y F,Xiong H, Si L G 2013 Phys. Rev. E 88 035202

    [14]

    Ranjbar M, Bahari A 2016 Opt. Commun. 375 19

    [15]

    Zhang S Y, Ma X R, Zhang S G, Chen L, Wang X Y,Mu K L, Wang S 2014 Chin. Phys. B 23 060304

    [16]

    Zhang S Y, Ma X R, Zhang S G, Chen L, Wang X Y,Mu K L, Wang S 2014 Chin. Phys. B 23 060304

    [17]

    Zhang C Q, Ruan C J, Zhao D, Wang S Z, Yang X D 2014 Chin. Phys. B 23 088401

    [18]

    Liu L X, Shao C G 2012 Chin. Phys. Lett. 29 111401

    [19]

    Es'kin V A, Kudrin A V, Petrov E Y 2001 Nature 414 716

    [20]

    Xiong H, Si L G, Yang X X, Wu Y 2015 Sci. Reports 5 11071

    [21]

    Grenfell B T, Bjornstad O N, Kappey J 2001 Nature 414 716

    [22]

    Shi L F, Zhu M, Zhou X C, Wang W G, Mo J Q 2008 Phys. Lett. A 372 417

    [23]

    Xu Y H, Han X L, Shi L F, Mo J Q 2014 Acta Phys.Sin. 63 090204 (in Chinese) [许永红, 韩祥临, 石兰芳, 莫嘉琪2014 物理学报63 090204]

    [24]

    Harko T, Mak M K 2015 J. Math. Phys. 56 111501

    [25]

    Sardar A, Husnine S M, Rizvi S T R, Younis M, Ali K 2015 Nonlinear Dynam. 82 1317

    [26]

    Wang M L, Li X Z, Zhang J L 2008 Phys. Lett. A 372 417

    [27]

    Shu Y Q 2011 Ph. D. Dissertation (Lanzhou:Lanzhou University) (in Chinese)[舒雅琴 2011 博士学位论文 (兰州:兰州大学)]

    [28]

    Petrov E Y, Kudrin A V 2012 Phys. Rev. E 85 055202

    [29]

    Silva A, Monticone F, Castaldi G, Galdi V, Al A, Engheta N 2014 Science 343 160

    [30]

    Eidini M, Paulino G H 2015 Sci. Adv. 1

    [31]

    Ma G, Sheng P 2016 Sci. Adv. 2

    [32]

    Fan W, Yan B, Wang Z B, Wu L M 2016 Sci. Adv. 2

    [33]

    Chen H W, Yang C R, Fu C L, Zhao L, Gao Z Q 2006 Appl. Surf. Sci. 252 4171

    [34]

    Fong D D, Stephenson G B, Streiffer S K, Eastman J A, Auciello O, Fuoss P H, Thompson C 2004 Science 304 1650

    [35]

    Han S H, Park Q H 2011 Phys. Rev. E 83 066601

    [36]

    Shafeeque Ali A K, Porsezian K, Uthayakumar T 2014 Phys. Rev. E 90 042910

    [37]

    Reed E J, Soljačić M, Gee R, Joannopoulos J D 2007 Phys. Rev. B 75 174302

    [38]

    Grischkowsky D, Duling Ⅲ I N, Chen J C, Chi C C 1987 Phys. Rev. Lett. 59 1663

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出版历程
  • 收稿日期:  2017-02-03
  • 修回日期:  2017-04-16
  • 刊出日期:  2017-07-05

柱面非线性麦克斯韦方程组的行波解

摘要: 柱面电磁波在各种非均匀非线性介质中的传播问题具有非常重要的研究价值.对描述该问题的柱面非线性麦克斯韦方程组进行精确求解,则是最近几年新兴的研究热点.但由于非线性偏微分方程组的极端复杂性,针对任意初边值条件的精确求解在客观上具有极高的难度,已有工作仅解决了柱面电磁波在指数非线性因子的非色散介质中的传播情况.因此,针对更为确定的物理场景,寻求能够精确描述其中更为广泛的物理性质的解,是一种更为有效的处理方法.本文讨论了具有任意非线性因子与幂律非均匀因子的非色散介质中柱面麦克斯韦方程组的行波精确解,理论分析表明这种情况下柱面电磁波的电场分量E已不存在通常形如E=g(r-kt)的平面行波解;继而通过适当的变量替换与求解相应的非线性常微分方程,给出电场分量E=g(lnr-kt)形式的广义行波解,并以例子展示所得到的解中蕴含的类似于自陡效应的物理现象.

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