含扰动非线性波方程的Jacobi椭圆函数展开法及其在KdV类方程中的应用
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摘要
非线性波方程是非线性数学物理,特别是孤立子理论中最前沿的研究课题之一。对非线性波方程的求解有助于我们弄清系统在非线性作用下的运动变化规律。以实际物理问题作为背景的含有参数扰动因素的非线性波方程的研究是当代非线性科学的一个重要研究方向。通过对受扰动因素下非线性波方程的求解和定性研究,更能合理地解释复杂的自然现象,更深刻地描述系统的本质特征,极大推动相关学科的发展。
本文在许多专家学者研究工作的基础上,对现有求解传统非线性波方程的方法进行分析研究,吸取求解变系数偏微分方程行波解的构造方法,基于Jacobi椭圆函数展开法、齐次平衡法和转化的思想,改进了Jacobi椭圆函数展开法,并将方法应用于受扰情形下的变系数KdV方程,变系数组合KdV—mKdV方程和两类非线性耦合KdV方程,获得了一些有意义的新结果,其中包括变速类孤立波解、变速类三角双曲型周期解,以及变速类三角函数型周期解。
本文的方法在求解受扰情形下的非线性波方程中具有普遍性,可推广到其它的非线性波方程及高阶非线性耦合方程组。
Nonlinear wave equations are one of the forefront topics in the studies of nonlinear mathematical physics. The research on finding solutions of nonlinear wave equations can help us understand the motion laws of the nonlinear systems under the nonlinear interactions. The nonlinear wave equations with a spatiotemporal perturbation based on actual physical problems is one of the significant subjects in contemporary study of nonlinear science. The research on finding solutions and analyzing the qualitative behavior of solutions of nonlinear wave equations with a spatiotemporal perturbation can explain the corresponding natural phenomena reasonably, describe the essential properties of the systems more deeply and promote the development of related subjects greatly.
In this dissertation, based on the method of Jacobian elliptic function expansion, homogeneous balance method and transformed thought, the method of Jacobian elliptic function expansion is improved and is applied to KdV equation with variable coefficient, the combined KdV-mKdV equation with variable coefficients and two kinds of the coupled KdV equations. As a result, many helpful conclusions are obtained. In fact, speed-changed similar solitary wave solutions, similar periodic solutions with trigonometric function and periodic solutions with hyperbolic function.
It is worthwhile to mention that the above method can also be applied to some other nonlinear wave equations and higher-order coupled equations.
本文在许多专家学者研究工作的基础上,对现有求解传统非线性波方程的方法进行分析研究,吸取求解变系数偏微分方程行波解的构造方法,基于Jacobi椭圆函数展开法、齐次平衡法和转化的思想,改进了Jacobi椭圆函数展开法,并将方法应用于受扰情形下的变系数KdV方程,变系数组合KdV—mKdV方程和两类非线性耦合KdV方程,获得了一些有意义的新结果,其中包括变速类孤立波解、变速类三角双曲型周期解,以及变速类三角函数型周期解。
本文的方法在求解受扰情形下的非线性波方程中具有普遍性,可推广到其它的非线性波方程及高阶非线性耦合方程组。
Nonlinear wave equations are one of the forefront topics in the studies of nonlinear mathematical physics. The research on finding solutions of nonlinear wave equations can help us understand the motion laws of the nonlinear systems under the nonlinear interactions. The nonlinear wave equations with a spatiotemporal perturbation based on actual physical problems is one of the significant subjects in contemporary study of nonlinear science. The research on finding solutions and analyzing the qualitative behavior of solutions of nonlinear wave equations with a spatiotemporal perturbation can explain the corresponding natural phenomena reasonably, describe the essential properties of the systems more deeply and promote the development of related subjects greatly.
In this dissertation, based on the method of Jacobian elliptic function expansion, homogeneous balance method and transformed thought, the method of Jacobian elliptic function expansion is improved and is applied to KdV equation with variable coefficient, the combined KdV-mKdV equation with variable coefficients and two kinds of the coupled KdV equations. As a result, many helpful conclusions are obtained. In fact, speed-changed similar solitary wave solutions, similar periodic solutions with trigonometric function and periodic solutions with hyperbolic function.
It is worthwhile to mention that the above method can also be applied to some other nonlinear wave equations and higher-order coupled equations.
引文
[1]傅新楚.关于非线性与复杂性[R].上海：上海大学出版社,2004.
[2]E. Infeld. Nonliner waves, Soliton and Chaos[M]. Cambridge:Cambridge University Press,1990.
[3]P. G.Drazin, R.S.Johnson. Solitons[M].Cambridge:Cambridge University Press, 1989.
[4]刘延柱,陈立群.非线性动力学[M].上海：上海交通大学出版社,2000.
[5]Russell J S.The British Association for the Advancement of Science[R].1837. 417-496.
[6]Russell J S.The 14th Meeting of the British Association for the Advancement of Science[R].1844.311-390.
[7]王振东.孤立子与孤立波[J].力学与实践,2005,27(5)：86-88.
[8]Miles J W.The Korteweg-de Vries equation[J]. Journal of Fluid Mechanics,1981, 106:131-147.
[9]Rayleigh L. On Waves [J]. Philosophical Magazine,1876,5(1):257-279.
[10]McCowan J. On the solitary wave [J]. Philosophical Magazine,1891,32(194)45-58.
[11]Korteweg D J, de Vries G.On the change of form of long waves advancing in a regular canal,and on a new type of long stationary waves[J].Philosophical Magazine, 1895,39(5):422-443.
[12]闫振亚.非线性波与可积系统[D].大连：大连理工大学,2002.
[13]Zabusky N J, Kruskal M D. Interaction of solitons in a collisionless plasma and the recurrence of initial states[J].Physical Review Letters,1965,15(6):240-243.
[14]张翼.基于双线性方法的孤子可积系统[D].上海：上海大学,2004.
[15]李翊神.孤子与可积系统[M].上海：上海科技教育出版社,1999.
[16]陈勇.孤立子理论中的若干问题的研究及机械化实现[D].大连：大连理工大学,2003.
[17]谷超豪,胡和生,周子翔.孤立子理论中的达布变换及其几何应用[M].上海：上海科学技术出版社,1999.
[18]Wahlquist H D,Estabrook F B. Backlund Transformation for solitons of the Korteweg-de Vries equation[J]. Physical Review Letters,1973,31:1386-1390.
[19]刘希强.非线性发展方程显式解的研究[D].北京：中国工程物理研究所,2002.
[20]Gardner C S, Green J M, Kruskal M D et al. Method for solving the Korteweg-de Vries equation[J]. Physical Review Letters,1967,19:1095-1097.
[21]倪皖荪,魏荣爵.水槽中的孤波[M].上海：上海科技教育出版社,1997.
[22]Lax P D. Integrals of nonlinear equations of evolution and solitary waves[J]. Communications on Pure and Applied Mathematics,1968,21(5):467-490.
[23]张鸿庆.弹性力学方程组一般解的统一理论[J].大连：大连理工大学学 报,1978,17(3)：23-27.
[24]范恩贵.孤立子和可积系统[D].大连：大连理工大学,1998.
[25]张鸿庆,吴方向.一类偏微分方程组的一般解及其在壳体理论中的应用[J].力学学报,1992,24(6)：700-707.
[26]张鸿庆,吴方向.构造板壳问题基本解的一般方法[J].科学通报,1993,38(7)：671-672.
[27]张鸿庆,冯红.构造弹性力学学位位移函数的机械化算法[J].应用数学和力学,1995,16(4)：315-322.
[28]张鸿庆,冯红.非齐次线性算子方程组一般解的代数构造[J].大连：大连理工大学学报,1994,34(3)：249-255.
[29]闫振亚,张鸿庆.一类非线性演化方程新的显式行波解[J].物理学报,1999,48(1)：1-5.
[30]闫振亚,张鸿庆.具有三个任意函数的变系数KdV-mKdV方程的精确类孤子解[J].物理学报,1999,48(11)：1957-1961.
[31]闫振亚,张鸿庆.非线性浅水波近似方程组的显式精确解[J].物理学报,1999,48(11)：1962-1968.
[32]Dolye P W. Separation of variables for scalar evolutions in one space dimension [J]. Journal of Physics A,1996,29(23):7581-7595.
[33]Dolye P W, Vassiliou P J. Separation of variables for the 1-dimensional nonlinear diffusion equation[J]. International Joural of Nonlinear Mechanics,1998,33(2):315-326.
[34]Zhdanov R Z. Separation of variables in the nonlinear wave equation [J]. Journal of Physics A,1994,27 (9):L291-L297.
[35]Zhdanov R Z, Revenko I V, Fushchych W I. Orthgonal and non-orthogonal separation of variables in the wave equation utt-unxx+V(x)u=0 [J]. Journal of Physics A,1993,26(17):5959-5972.
[36]曹策问.AKNS族的Lax方程组的非线性化[J].中国科学(A辑),1989,32(7)：701-707.
[37]Lou Senyue, Lu Jizong. Special solutions from the variable separation approach:the Davey-Stewartson equation [J]. Journal of Physics A,1996,29(14):4209-4215.
[38]Hirota Ryogo. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons[J].Physical Review Letters,1971,27(18):1192-1194.
[39]Wang Mingliang. Solitary wave solutions for variant Boussinesq equations[J]. Physics Letters A,1995,199(3-4):169-172.
[40]Wang Mingliang. Exact solutions for a compound KdV-Burgers equation[J]. Physics Letters A,1996,213(5-6):279-287.
[41]Tian Bo, Gao Yitian. New families of exact solutions to the integrable dispersive long wave equations in 2+1-dimensional spaces[J]. Journal of Physics A,1996,29(11): 2895-2903.
[42]Fan Engui, Zhang Hongqing. A note on the homogeneous balance method[J]. Physics Letters A,1998,246(5):403-406.
[43]Fan Engui, Zhangqing. New exact solutions to a system of coupled KdV equations[J]. Physics Letters A,1998,245(5):389-392.
[44]闫振亚,张鸿庆.Brusselator反应扩展模型的Auto-Darboux变换和精确解[J].应用数学和力学,2001,22(5)：477-482.
[45]杨磊,杨孔庆.GKS方程的孤立波解[J].兰州大学学报(自然科学版),1998,34(4)：53.
[46]B. Abdel-Hamid. Exact solutions of some nonlinear evolution equations using symbolic computations[J]. Computers and Mathematics with Applications,2000,291-302.
[47]王明亮,李志斌,周宇斌.齐次平衡原则及其应用[J].兰州：兰州大学学报,1999,35(3)：8-11.
[48]J. L. Yang, R. H. Han, T. Z. Li, et al. Solitary wave solutions of nonlinear equations[J]. Physics Letters A,1998,239:359-363.
[49]Liu Shikuo, Fu Zuntao, Liu Shida, et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations[J]. Physics Letters A,2001,289(1):69-74.
[50]翁建平.用Jacobi椭圆函数求非线性方程的解析解[J].安徽大学学报,2005,29(5)：36-39.
[51]肖亚峰,张鸿庆.Modified Improved Boussinesq方程的扩展椭圆函数展开解[J].兰州理工大学学报,2004,30(1)：109-112.
[52]徐昌智,郑春龙.构造非线性波动方程孤波解的一种新方法[J].浙江师范大学学报,2003,26(3)：238-242.
[53]Nikolai A. Kudryashov. Simplest equation method to look for exact solutions of nonlinear differential equations[J]. Chaos, Soliton and Fractals,2005,1212-1231.
[54]Liu Xiqiang. Exact solutions of the variable coefficient KdV and SG type equations[J]. Applied Mathematics JCU,1998, B13:25-30.
[55]Liu Xiqiang, Jiang Song. Exact solutions for general variable coefficient KdV equation[J]. Applied Mathematics JCU,2001, B16:377-380.
[56]R. Scharf and A. R. Bishop.Soliton Chaos in the nonlinear Schrdinger equation with spatially periodic perturbations[J]. Physics Review A,1992,46(6):2973-2976.
[57]洪宝剑,卢殿臣,张大珩.带强迫项变系数组合KdV-Burgers方程的显式精确解[J].广西师范大学学报(自然科学版),2007,25(1)：17-20.
[58]Zuntao Fu, Shikuo Liu, Shida Liu,et al.New Jacobi elliptic expansion and new periodic solutions of nonlinear wave equations[J]. Physics Letters A,2001,72-76.
[59]张善卿,李志斌.Jacobi椭圆函数展开法的新应用[J].物理学报,2003,52(5)：1066-1069.
[60]Grimshaw R H J.Slowly varying solitary waves[J]. Proc Roy Soc Lon A,1979,368(1734):259-375.
[61]洪宝剑,卢殿臣,田立新.带强迫项变系数组合KdV方程的显式精确解[J].物理学报,2006,55(11)：5617-5621.
[62]付遵涛,刘式适,刘式达.一类非线性演化方程的新多级准确解[J].物理学 报,2003,52(12)：2949-2953.
[63]Hirota R, Satsuma J. Soliton solutions of a coupled Korteweg-de-Vries equation[J]. Physics Letters A,1981,85:407-408.
[64]Liu Xiqiang, Bai Chenglin. New soliton solutions of coupled KdV equations[J]. Chinese Journal of Quantum Electromics,1999,16:360-364.
[65]Wang Ruimin,Xu Changzhi. Rational fraction solutions for generalized coupled KdV equations[J]. Journal of Anhui University,2004,27(1):37-40.
[66]于西昌,董仲周.非线性耦合KdV方程组的多种行波解[J].聊城大学学报,2005,18(3)：31-34.
[67]化存才.非线性波动问题的若干时空动力学研究[D].上海交通大学博士后论文,2002.
[2]E. Infeld. Nonliner waves, Soliton and Chaos[M]. Cambridge:Cambridge University Press,1990.
[3]P. G.Drazin, R.S.Johnson. Solitons[M].Cambridge:Cambridge University Press, 1989.
[4]刘延柱,陈立群.非线性动力学[M].上海：上海交通大学出版社,2000.
[5]Russell J S.The British Association for the Advancement of Science[R].1837. 417-496.
[6]Russell J S.The 14th Meeting of the British Association for the Advancement of Science[R].1844.311-390.
[7]王振东.孤立子与孤立波[J].力学与实践,2005,27(5)：86-88.
[8]Miles J W.The Korteweg-de Vries equation[J]. Journal of Fluid Mechanics,1981, 106:131-147.
[9]Rayleigh L. On Waves [J]. Philosophical Magazine,1876,5(1):257-279.
[10]McCowan J. On the solitary wave [J]. Philosophical Magazine,1891,32(194)45-58.
[11]Korteweg D J, de Vries G.On the change of form of long waves advancing in a regular canal,and on a new type of long stationary waves[J].Philosophical Magazine, 1895,39(5):422-443.
[12]闫振亚.非线性波与可积系统[D].大连：大连理工大学,2002.
[13]Zabusky N J, Kruskal M D. Interaction of solitons in a collisionless plasma and the recurrence of initial states[J].Physical Review Letters,1965,15(6):240-243.
[14]张翼.基于双线性方法的孤子可积系统[D].上海：上海大学,2004.
[15]李翊神.孤子与可积系统[M].上海：上海科技教育出版社,1999.
[16]陈勇.孤立子理论中的若干问题的研究及机械化实现[D].大连：大连理工大学,2003.
[17]谷超豪,胡和生,周子翔.孤立子理论中的达布变换及其几何应用[M].上海：上海科学技术出版社,1999.
[18]Wahlquist H D,Estabrook F B. Backlund Transformation for solitons of the Korteweg-de Vries equation[J]. Physical Review Letters,1973,31:1386-1390.
[19]刘希强.非线性发展方程显式解的研究[D].北京：中国工程物理研究所,2002.
[20]Gardner C S, Green J M, Kruskal M D et al. Method for solving the Korteweg-de Vries equation[J]. Physical Review Letters,1967,19:1095-1097.
[21]倪皖荪,魏荣爵.水槽中的孤波[M].上海：上海科技教育出版社,1997.
[22]Lax P D. Integrals of nonlinear equations of evolution and solitary waves[J]. Communications on Pure and Applied Mathematics,1968,21(5):467-490.
[23]张鸿庆.弹性力学方程组一般解的统一理论[J].大连：大连理工大学学 报,1978,17(3)：23-27.
[24]范恩贵.孤立子和可积系统[D].大连：大连理工大学,1998.
[25]张鸿庆,吴方向.一类偏微分方程组的一般解及其在壳体理论中的应用[J].力学学报,1992,24(6)：700-707.
[26]张鸿庆,吴方向.构造板壳问题基本解的一般方法[J].科学通报,1993,38(7)：671-672.
[27]张鸿庆,冯红.构造弹性力学学位位移函数的机械化算法[J].应用数学和力学,1995,16(4)：315-322.
[28]张鸿庆,冯红.非齐次线性算子方程组一般解的代数构造[J].大连：大连理工大学学报,1994,34(3)：249-255.
[29]闫振亚,张鸿庆.一类非线性演化方程新的显式行波解[J].物理学报,1999,48(1)：1-5.
[30]闫振亚,张鸿庆.具有三个任意函数的变系数KdV-mKdV方程的精确类孤子解[J].物理学报,1999,48(11)：1957-1961.
[31]闫振亚,张鸿庆.非线性浅水波近似方程组的显式精确解[J].物理学报,1999,48(11)：1962-1968.
[32]Dolye P W. Separation of variables for scalar evolutions in one space dimension [J]. Journal of Physics A,1996,29(23):7581-7595.
[33]Dolye P W, Vassiliou P J. Separation of variables for the 1-dimensional nonlinear diffusion equation[J]. International Joural of Nonlinear Mechanics,1998,33(2):315-326.
[34]Zhdanov R Z. Separation of variables in the nonlinear wave equation [J]. Journal of Physics A,1994,27 (9):L291-L297.
[35]Zhdanov R Z, Revenko I V, Fushchych W I. Orthgonal and non-orthogonal separation of variables in the wave equation utt-unxx+V(x)u=0 [J]. Journal of Physics A,1993,26(17):5959-5972.
[36]曹策问.AKNS族的Lax方程组的非线性化[J].中国科学(A辑),1989,32(7)：701-707.
[37]Lou Senyue, Lu Jizong. Special solutions from the variable separation approach:the Davey-Stewartson equation [J]. Journal of Physics A,1996,29(14):4209-4215.
[38]Hirota Ryogo. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons[J].Physical Review Letters,1971,27(18):1192-1194.
[39]Wang Mingliang. Solitary wave solutions for variant Boussinesq equations[J]. Physics Letters A,1995,199(3-4):169-172.
[40]Wang Mingliang. Exact solutions for a compound KdV-Burgers equation[J]. Physics Letters A,1996,213(5-6):279-287.
[41]Tian Bo, Gao Yitian. New families of exact solutions to the integrable dispersive long wave equations in 2+1-dimensional spaces[J]. Journal of Physics A,1996,29(11): 2895-2903.
[42]Fan Engui, Zhang Hongqing. A note on the homogeneous balance method[J]. Physics Letters A,1998,246(5):403-406.
[43]Fan Engui, Zhangqing. New exact solutions to a system of coupled KdV equations[J]. Physics Letters A,1998,245(5):389-392.
[44]闫振亚,张鸿庆.Brusselator反应扩展模型的Auto-Darboux变换和精确解[J].应用数学和力学,2001,22(5)：477-482.
[45]杨磊,杨孔庆.GKS方程的孤立波解[J].兰州大学学报(自然科学版),1998,34(4)：53.
[46]B. Abdel-Hamid. Exact solutions of some nonlinear evolution equations using symbolic computations[J]. Computers and Mathematics with Applications,2000,291-302.
[47]王明亮,李志斌,周宇斌.齐次平衡原则及其应用[J].兰州：兰州大学学报,1999,35(3)：8-11.
[48]J. L. Yang, R. H. Han, T. Z. Li, et al. Solitary wave solutions of nonlinear equations[J]. Physics Letters A,1998,239:359-363.
[49]Liu Shikuo, Fu Zuntao, Liu Shida, et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations[J]. Physics Letters A,2001,289(1):69-74.
[50]翁建平.用Jacobi椭圆函数求非线性方程的解析解[J].安徽大学学报,2005,29(5)：36-39.
[51]肖亚峰,张鸿庆.Modified Improved Boussinesq方程的扩展椭圆函数展开解[J].兰州理工大学学报,2004,30(1)：109-112.
[52]徐昌智,郑春龙.构造非线性波动方程孤波解的一种新方法[J].浙江师范大学学报,2003,26(3)：238-242.
[53]Nikolai A. Kudryashov. Simplest equation method to look for exact solutions of nonlinear differential equations[J]. Chaos, Soliton and Fractals,2005,1212-1231.
[54]Liu Xiqiang. Exact solutions of the variable coefficient KdV and SG type equations[J]. Applied Mathematics JCU,1998, B13:25-30.
[55]Liu Xiqiang, Jiang Song. Exact solutions for general variable coefficient KdV equation[J]. Applied Mathematics JCU,2001, B16:377-380.
[56]R. Scharf and A. R. Bishop.Soliton Chaos in the nonlinear Schrdinger equation with spatially periodic perturbations[J]. Physics Review A,1992,46(6):2973-2976.
[57]洪宝剑,卢殿臣,张大珩.带强迫项变系数组合KdV-Burgers方程的显式精确解[J].广西师范大学学报(自然科学版),2007,25(1)：17-20.
[58]Zuntao Fu, Shikuo Liu, Shida Liu,et al.New Jacobi elliptic expansion and new periodic solutions of nonlinear wave equations[J]. Physics Letters A,2001,72-76.
[59]张善卿,李志斌.Jacobi椭圆函数展开法的新应用[J].物理学报,2003,52(5)：1066-1069.
[60]Grimshaw R H J.Slowly varying solitary waves[J]. Proc Roy Soc Lon A,1979,368(1734):259-375.
[61]洪宝剑,卢殿臣,田立新.带强迫项变系数组合KdV方程的显式精确解[J].物理学报,2006,55(11)：5617-5621.
[62]付遵涛,刘式适,刘式达.一类非线性演化方程的新多级准确解[J].物理学 报,2003,52(12)：2949-2953.
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