代数法求解变形Boussinesq方程
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摘要
求解非线性方程的行波解是一种非常重要和有意义的工作,不管行波解表达式是以显函数的形式给出还是以隐函数的行式给出,我们感兴趣的是它的应用.这些形式的波在其传播过程中将不改变它的形状,也比较容易观察.当然我们最关心的是三种形式的行波解:首先是孤立波解,它是一个局部的行波解,此种行波解在充分远处波形将趋于零;其次是周期波解;最后是冲击波解,这种波从一端到另一端单调减少或者单调增加.
近来出现了很多求解非线性波方程精确行波解的方法,一些重要的方法比如,齐次平衡法、双曲正切函数法、试探函数法、非线性变换法和sine-cosine法等等.但是这些方法大部分只能求得非线性波方程的冲击波解和孤立波解,不能求得非线性波方程的周期解.本文首先介绍和应用Tanh函数法和推广的Tanh函数法去求解变形Boussinesq方程;接着介绍了另一种代数方法叫映射法,并且也介绍了改进的映射法和推广的映射法去获得偏微分方程的精确行波解,并应用椭圆函数展开法去求解Boussinesq方程;最后重点讨论应用一种统一的方法----映射法去求解大量的偏微分方程的精确行波解.映射法包括几种直接的方法作为特例比如:双曲正切函数展开法、Jacobi椭圆函数法等待.总而言之,如果行波解存在用此方法同时可获得冲击波解、周期波解和孤立波解.通过映射法、改进的映射法和推广的映射法可以去获得大量的偏微分方程的精确行波解,但是此时要求所研究的非线性偏微分方程中不能同时存在奇数阶导数和偶数阶导数.
The exact traveling wave solution is one of most important solutions during solving the nonlinear partial differential equations. Traveling waves, whether their solutions are in explicit or implicit forms, are very interesting form the point of view of applications. These types of waves will not change their shapes during propagation and are thus easy to detect, of particular interesting are three types of traveling waves: the solitary waves, which are localized traveling waves, asymptotically zero at large distances; the periodic waves; the kink waves, which rise or decent from one asymptotic state to another. Recently, there many methods for obtain the exact traveling wave solutions of a nonlinear partial differential equation. Some of the most important methods, for instance, tanh-function method, nonlinear transformation method, sine-cosine method, trial function method and so on. However, using these methods does not obtain the periodic wave solutions of the nonlinear wave equations, only get the solutions of the solitary wave and the kink wave solutions. This paper will firstly introduce and apply tanh-function method and extended tanh-function method to obtain the traveling wave solutions of the Boussinesq equation; secondly, we will introduce and apply Jacobi elliptic function expansion method get the traveling wave solutions of the Boussinesq equation. Finally we introduce and apply a unified algebraic method called the mapping method to obtain exact traveling wave solutions for a large variety of nonlinear partial differential equations. This method includes several direct methods as special case, such as tanh-function method, sech-function method and Jacobi elliptic function method. Above all, by means of this method, the solitary wave, the periodic wave and the kink wave solution can, if they exist, be obtained simultaneously to the equation in question without extra efforts. A large variety of exact traveling solutions of nonlinear partial differential equations are obtained by means of the mapping method, the modified mapping method and the extended mapping method, as long as odd-and even-order derivative terms do not coexist in nonlinear partial differential equations under consideration.
近来出现了很多求解非线性波方程精确行波解的方法,一些重要的方法比如,齐次平衡法、双曲正切函数法、试探函数法、非线性变换法和sine-cosine法等等.但是这些方法大部分只能求得非线性波方程的冲击波解和孤立波解,不能求得非线性波方程的周期解.本文首先介绍和应用Tanh函数法和推广的Tanh函数法去求解变形Boussinesq方程;接着介绍了另一种代数方法叫映射法,并且也介绍了改进的映射法和推广的映射法去获得偏微分方程的精确行波解,并应用椭圆函数展开法去求解Boussinesq方程;最后重点讨论应用一种统一的方法----映射法去求解大量的偏微分方程的精确行波解.映射法包括几种直接的方法作为特例比如:双曲正切函数展开法、Jacobi椭圆函数法等待.总而言之,如果行波解存在用此方法同时可获得冲击波解、周期波解和孤立波解.通过映射法、改进的映射法和推广的映射法可以去获得大量的偏微分方程的精确行波解,但是此时要求所研究的非线性偏微分方程中不能同时存在奇数阶导数和偶数阶导数.
The exact traveling wave solution is one of most important solutions during solving the nonlinear partial differential equations. Traveling waves, whether their solutions are in explicit or implicit forms, are very interesting form the point of view of applications. These types of waves will not change their shapes during propagation and are thus easy to detect, of particular interesting are three types of traveling waves: the solitary waves, which are localized traveling waves, asymptotically zero at large distances; the periodic waves; the kink waves, which rise or decent from one asymptotic state to another. Recently, there many methods for obtain the exact traveling wave solutions of a nonlinear partial differential equation. Some of the most important methods, for instance, tanh-function method, nonlinear transformation method, sine-cosine method, trial function method and so on. However, using these methods does not obtain the periodic wave solutions of the nonlinear wave equations, only get the solutions of the solitary wave and the kink wave solutions. This paper will firstly introduce and apply tanh-function method and extended tanh-function method to obtain the traveling wave solutions of the Boussinesq equation; secondly, we will introduce and apply Jacobi elliptic function expansion method get the traveling wave solutions of the Boussinesq equation. Finally we introduce and apply a unified algebraic method called the mapping method to obtain exact traveling wave solutions for a large variety of nonlinear partial differential equations. This method includes several direct methods as special case, such as tanh-function method, sech-function method and Jacobi elliptic function method. Above all, by means of this method, the solitary wave, the periodic wave and the kink wave solution can, if they exist, be obtained simultaneously to the equation in question without extra efforts. A large variety of exact traveling solutions of nonlinear partial differential equations are obtained by means of the mapping method, the modified mapping method and the extended mapping method, as long as odd-and even-order derivative terms do not coexist in nonlinear partial differential equations under consideration.
引文
[1]屠规彰.非线性方程的逆散射方法.应用数学与计算科学,1979, 1: 21-42.
[2] Ablowitz M J, Clarkson P A. Solitions, nonlinear evolution equations and inverse scattering[ M]. Cambridge: Cambridge University Press, 1991.
[3] Miura M R. B?cklund Transformation[M]. Berlin:Springer Verlag,1978.
[4]谷超豪,胡和生,周子翔.孤立子理论中的达布变换及几何应用, .上海:上海科学技术出版社,1999.
[5] Masataka L,Hirota R. Soliton solutions of a coupled derivative modified KdV equations. J. Phys. Soc. Jpn,2000, 69: 59-72.
[6] Matveev VB, Salle MA. Darboux Transformations and Solitons[M]. Belin: Springer, 1991.
[7] B.R.Duffy, E.J.Parkes Phys,Lett.A214(1996) 271.
[8] Z B Li. Exact Solitary wave solution of nonlinear evolution equations. In Mathematics Mechanization and Application, Eds. England, Academic Press, 2000.
[9] Fan EG. Extended tanh-functon method and its applications to nonlinear equations.Phys. Len 2000,A277: 212-218.
[10]王明亮,李志斌,周宇斌.齐次平衡原则及其应用.兰州大学学报. 1999,35(3): 8-16.
[11]范恩贵,张鸿庆.齐次平衡法若干新的应用.数学物理学报. 1999,19 :286-292.
[12]李志斌,张善卿.非线性波方程的准确孤立波解的符号计算.数学物理报.1997,17(1):81-89.
[13]吕克璞,石玉仁,段文山等.Kdv-Burgers方程的孤立解.物理学报.2001,50(11):2074-2076.
[14]张卫国.几类具5次强非线性项的发展方程的显式精确孤波解.应用数学学报,1998,21(2):249-256.
[15]刘式适,付遵涛,刘式达等.求某些非线性偏微分方程特解的一个简洁方法.应用数学和力学,2001,22(3):281-286.
[16]闻振亚,张鸿庆,范恩贵.一类非线性演化方程新的显式行波解.物理学报,1999,48(1):1-5.
[17]同振亚,张鸿庆.非线性浅水近似方程组的显式精确解.物理学报,1999, 48(11) :1962-1968.
[18]徐桂琼,李志斌.构造非线性发展方程孤波解的棍合指数方法.物理学报,2002,51(5) :946-950.
[19]徐桂琼,李志斌.扩展的混合指数方法及其应用.物理学报,2002,51(7):l 424-1427.
[20] Fan E G,Zbang H Q. New exact solutions to a system of coupld equations[J]. Physics Letters A,1998,(245):389-392.
[21]张解放.长水波近似方程的多孤子解.物理学报,1998,47(9):1 416-1 420.
[22]那仁满都拉,乌恩宝音,王克协.具5次强非线性项的波方程新的孤波解.物理学报,2004,53(1):11-14.
[23]刘成仕,杜兴华.耦合Klein-Gordon-Schrodinger方程新的精确解.物理学报,2005,54(3) :1039-1043.
[24]刘成仕.试探方程法及其在非线性发展方程中的应用.物理学报,2005,54(6) :2 505-2 509.
[25]刘式适,付遵涛,刘式达等.Jacobi椭圆函数展开法及其在求解非线性波动方程中的应用.物理学报,2001,50(11):2 068-2 073.
[26]刘官厅,范天佑.一般变换下的Jacobi椭圆函数展开法及应用.物理学报,2004,53(3):676-679.
[27]刘式适,付遵涛,刘式达等.变系数非线性方程的Jacobi椭圆函数展开解.物理学报,2002,51(9) :1923-1926.
[28]张桂戊,李志斌,段一士.非线性波方程的精确孤立波解.中国科学(A辑),2000,30(12):1 103-1 108.
[29] Lin J,Lou S Y,Wang K L. High-dimensional Virasoro integrable models and exact solution[J]. Phys Lett,2001,(A287):257.
[30]张解放,陈芳跃.截断展开法和广义变系数KdV方程新的精确类孤子解.物理学报,2001,50(9):l 648-1 651.
[31] Lou S Y,Lin J,Yu J. (3+1)-dimensional integrable models under the meaning that they possess infinite dimensional Virasoro-type symmetry[J]. Phys Lett,1995 , (A201) .47.
[32]刘式适,刘式达.物理学中的非线性方程.北京.北京大学出版社2000
[33]朱加民,马正义,方建平等.General jacobian elliptic function expansion method and its applications [J]. Chinese Physics, 2004,13(6):798一804.
[34] Peng YZ. Exact periodic wave solutions to a new Hamiltonian amplitude equation. Journal of the Physical Society of Japan, 2003, 72(6): 1356 -1359.
[35] Peng YZ.Exact Periodic Wave Solutions of Two Types of Modified Boussinesq Equtions.Acta Phys.Pol.A . 2003,103:417-421
[36] Peng YZ.A mapping method for obtaining exact traveling wave solutions to nonlinear evolution.Chinese Journal of Physics 41, 2003,103-110.
[37] Peng Yan-ze.International Journal of Theoretical Physics,vol.42,No.4, 2003.
[38] Peng Y.Z.,Exact Periodic Wave Solutions to the Nizhnik-Novikov-Veselov Eqution.Acta Phys.Pol.A Vol. 105 .2004.
[39] Peng YZ. Exact travelling wave solutions to the (3+1)-Dimensional Kadomtsev-Petviashvili equation. Acta Physica Polonica A, 2005,108: 421-424.
[40] Y.Z.Peng,Exact Periodic Wave Solutions of Two Types of Modified Boussinesq Equtions.Acta Phys.Pol.A Vol. 103(2003)
[41] Krishnan EV, Peng YZ. Exact solutions to the combined KdV-mKdV equation bythe extended mapping method. Institute of Physics, 2006.73: 405-409.
[42]范恩贵.可积系统与计算机代数.北京:科学出版社,2004.
[43] Wang ML. Solitary wave solutions for variant Boussinesq equations. Phys. Lett. A, 1995: 169-172.
[44] Wang M L.Solitary wave solution for Boussineq equations ..Phys Lett A 1995, 199:162-172.
[45]谷超豪,李大潜.应用偏微分方程,北京:高等教育出版社,1993.
[46] J Weiss,M Tabor and G Carnevale.J Math Phys,24(1983)13.
[47] J Satsuma.Topics in soliton theory and exactly solvable nonlinear equations.ed.M Abolowitz,B Fuchssteiner and M Kruskal.Singapore:World Scientific,1987.
[48] F.Bowman.Introducttion to Elliptic Functions,Universities,London,1959.
[49] V.Prasolov,Y.Solovyev,Elliptic Functions and Elliptic Integrals,American Mathematical Society,providence,1997.
[50] G.B. Whitham,Linear and Nonlear Waves,Wiley,New York 1973.
[2] Ablowitz M J, Clarkson P A. Solitions, nonlinear evolution equations and inverse scattering[ M]. Cambridge: Cambridge University Press, 1991.
[3] Miura M R. B?cklund Transformation[M]. Berlin:Springer Verlag,1978.
[4]谷超豪,胡和生,周子翔.孤立子理论中的达布变换及几何应用, .上海:上海科学技术出版社,1999.
[5] Masataka L,Hirota R. Soliton solutions of a coupled derivative modified KdV equations. J. Phys. Soc. Jpn,2000, 69: 59-72.
[6] Matveev VB, Salle MA. Darboux Transformations and Solitons[M]. Belin: Springer, 1991.
[7] B.R.Duffy, E.J.Parkes Phys,Lett.A214(1996) 271.
[8] Z B Li. Exact Solitary wave solution of nonlinear evolution equations. In Mathematics Mechanization and Application, Eds. England, Academic Press, 2000.
[9] Fan EG. Extended tanh-functon method and its applications to nonlinear equations.Phys. Len 2000,A277: 212-218.
[10]王明亮,李志斌,周宇斌.齐次平衡原则及其应用.兰州大学学报. 1999,35(3): 8-16.
[11]范恩贵,张鸿庆.齐次平衡法若干新的应用.数学物理学报. 1999,19 :286-292.
[12]李志斌,张善卿.非线性波方程的准确孤立波解的符号计算.数学物理报.1997,17(1):81-89.
[13]吕克璞,石玉仁,段文山等.Kdv-Burgers方程的孤立解.物理学报.2001,50(11):2074-2076.
[14]张卫国.几类具5次强非线性项的发展方程的显式精确孤波解.应用数学学报,1998,21(2):249-256.
[15]刘式适,付遵涛,刘式达等.求某些非线性偏微分方程特解的一个简洁方法.应用数学和力学,2001,22(3):281-286.
[16]闻振亚,张鸿庆,范恩贵.一类非线性演化方程新的显式行波解.物理学报,1999,48(1):1-5.
[17]同振亚,张鸿庆.非线性浅水近似方程组的显式精确解.物理学报,1999, 48(11) :1962-1968.
[18]徐桂琼,李志斌.构造非线性发展方程孤波解的棍合指数方法.物理学报,2002,51(5) :946-950.
[19]徐桂琼,李志斌.扩展的混合指数方法及其应用.物理学报,2002,51(7):l 424-1427.
[20] Fan E G,Zbang H Q. New exact solutions to a system of coupld equations[J]. Physics Letters A,1998,(245):389-392.
[21]张解放.长水波近似方程的多孤子解.物理学报,1998,47(9):1 416-1 420.
[22]那仁满都拉,乌恩宝音,王克协.具5次强非线性项的波方程新的孤波解.物理学报,2004,53(1):11-14.
[23]刘成仕,杜兴华.耦合Klein-Gordon-Schrodinger方程新的精确解.物理学报,2005,54(3) :1039-1043.
[24]刘成仕.试探方程法及其在非线性发展方程中的应用.物理学报,2005,54(6) :2 505-2 509.
[25]刘式适,付遵涛,刘式达等.Jacobi椭圆函数展开法及其在求解非线性波动方程中的应用.物理学报,2001,50(11):2 068-2 073.
[26]刘官厅,范天佑.一般变换下的Jacobi椭圆函数展开法及应用.物理学报,2004,53(3):676-679.
[27]刘式适,付遵涛,刘式达等.变系数非线性方程的Jacobi椭圆函数展开解.物理学报,2002,51(9) :1923-1926.
[28]张桂戊,李志斌,段一士.非线性波方程的精确孤立波解.中国科学(A辑),2000,30(12):1 103-1 108.
[29] Lin J,Lou S Y,Wang K L. High-dimensional Virasoro integrable models and exact solution[J]. Phys Lett,2001,(A287):257.
[30]张解放,陈芳跃.截断展开法和广义变系数KdV方程新的精确类孤子解.物理学报,2001,50(9):l 648-1 651.
[31] Lou S Y,Lin J,Yu J. (3+1)-dimensional integrable models under the meaning that they possess infinite dimensional Virasoro-type symmetry[J]. Phys Lett,1995 , (A201) .47.
[32]刘式适,刘式达.物理学中的非线性方程.北京.北京大学出版社2000
[33]朱加民,马正义,方建平等.General jacobian elliptic function expansion method and its applications [J]. Chinese Physics, 2004,13(6):798一804.
[34] Peng YZ. Exact periodic wave solutions to a new Hamiltonian amplitude equation. Journal of the Physical Society of Japan, 2003, 72(6): 1356 -1359.
[35] Peng YZ.Exact Periodic Wave Solutions of Two Types of Modified Boussinesq Equtions.Acta Phys.Pol.A . 2003,103:417-421
[36] Peng YZ.A mapping method for obtaining exact traveling wave solutions to nonlinear evolution.Chinese Journal of Physics 41, 2003,103-110.
[37] Peng Yan-ze.International Journal of Theoretical Physics,vol.42,No.4, 2003.
[38] Peng Y.Z.,Exact Periodic Wave Solutions to the Nizhnik-Novikov-Veselov Eqution.Acta Phys.Pol.A Vol. 105 .2004.
[39] Peng YZ. Exact travelling wave solutions to the (3+1)-Dimensional Kadomtsev-Petviashvili equation. Acta Physica Polonica A, 2005,108: 421-424.
[40] Y.Z.Peng,Exact Periodic Wave Solutions of Two Types of Modified Boussinesq Equtions.Acta Phys.Pol.A Vol. 103(2003)
[41] Krishnan EV, Peng YZ. Exact solutions to the combined KdV-mKdV equation bythe extended mapping method. Institute of Physics, 2006.73: 405-409.
[42]范恩贵.可积系统与计算机代数.北京:科学出版社,2004.
[43] Wang ML. Solitary wave solutions for variant Boussinesq equations. Phys. Lett. A, 1995: 169-172.
[44] Wang M L.Solitary wave solution for Boussineq equations ..Phys Lett A 1995, 199:162-172.
[45]谷超豪,李大潜.应用偏微分方程,北京:高等教育出版社,1993.
[46] J Weiss,M Tabor and G Carnevale.J Math Phys,24(1983)13.
[47] J Satsuma.Topics in soliton theory and exactly solvable nonlinear equations.ed.M Abolowitz,B Fuchssteiner and M Kruskal.Singapore:World Scientific,1987.
[48] F.Bowman.Introducttion to Elliptic Functions,Universities,London,1959.
[49] V.Prasolov,Y.Solovyev,Elliptic Functions and Elliptic Integrals,American Mathematical Society,providence,1997.
[50] G.B. Whitham,Linear and Nonlear Waves,Wiley,New York 1973.