非线性发展方程中解的构造问题
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摘要
在本文中主要应用李群方法、直接对称方法、CK直接方法求解了一个(2+1)-维非线性发展方程,SK-KP方程, MKP-II方程和BK方程,得到了这些方程的对称和显式解,并求得了SK-KP方程,BK方程的守恒律.利用构造方程精确解方法,求解了Burgers方程和KP方程,得到了方程的孤子解和周期解.
在第一章中,利用李群方法,获得了(2+1)-维非线性发展方程的对称约化和精确解,其中包括雅可比椭圆函数解、双曲函数解、三角函数解等精确解.这些精确解可能在解释一些物理问题上起重要作用.
在第二章中,研究了SK-KP方程.在这章中主要利用直接对称方法,首先假设出方程对称的形式,然后通过方程的对称所满足的形式及原方程求出了SK-KP方程的李对称,进而通过对称求出不变量,对方程进行化简、求解.最后求出了该方程的守恒律.
在第三章中,研究了MKP-II方程和BK方程.主要运用CK直接方法,运用此方法,我们不仅得到了方程新旧解之间的关系,而且利用简单变换得到了方程的李对称,最后通过对称求得了方程的不变量,进而化简并求得了许多新的精确解.
在第四章中,讨论了Burgers方程和KP方程.我们首先介绍一种直接构造方程解的方法,然后利用此方法求解了Burgers方程和KP方程,得到了一些新的精确解,包括孤子解和周期解.
In this paper, by using the Lie group method, the direct symmetry method and the modified CK’s direct method to solve a (2+1)-dimensional nonlinear evolution equation, the SK-KP equation, the MKP-II equation and the BK equation, we obtain these equations’symmetries and explicit solutions. We also obtain conservation laws of the SK-KP equation and the BK equation. By using the direct construction method, we get soliton solutions and periodic solutions of the Burgers equation and the KP equation.
In Chapter 1, by applying a direct symmetry method, we obtain the symmetry reduction and some new exact solutions of the (2+1)-dimensional nonlinear evolution equation, which include Jacobi elliptic function, hyperbolic function, trigonometric function and so on. These exact solutions will be helpful to a better understanding of some practical problems in physics.
In Chapter 2, the SK-KP equation is discussed. Firstly, assuming the form of symmetry. Secondly, by using the form of symmetry and the SK-KP equation, we resent the Lie symmetry. Similarity reductions and many new exact solutions of the SK-KP equation are obtained. At last, we also give the conservation laws of SK-KP equation.
In Chapter 3, we obtain the exact solutions of the MKP-II equation and the BK equation through the generalized CK direct method. A relationship is constructed between the new solutions and the old ones of the MKP-II equation and the BK equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the BK equation. Then we get the reductions using the symmetry and give some exact solutions of the BK equation.
In Chapter 4, we study the Burgers equation and the KP equation. We construct a frame of soliton solutions and periodic solutions to nonlinear equations. And we successfully solve the (2+1)-dimensional Burgers and (3+1)-Dimensional KP equation. And a lot of exact traveling solutions to the Burgers and KP equations are obtained.
在第一章中,利用李群方法,获得了(2+1)-维非线性发展方程的对称约化和精确解,其中包括雅可比椭圆函数解、双曲函数解、三角函数解等精确解.这些精确解可能在解释一些物理问题上起重要作用.
在第二章中,研究了SK-KP方程.在这章中主要利用直接对称方法,首先假设出方程对称的形式,然后通过方程的对称所满足的形式及原方程求出了SK-KP方程的李对称,进而通过对称求出不变量,对方程进行化简、求解.最后求出了该方程的守恒律.
在第三章中,研究了MKP-II方程和BK方程.主要运用CK直接方法,运用此方法,我们不仅得到了方程新旧解之间的关系,而且利用简单变换得到了方程的李对称,最后通过对称求得了方程的不变量,进而化简并求得了许多新的精确解.
在第四章中,讨论了Burgers方程和KP方程.我们首先介绍一种直接构造方程解的方法,然后利用此方法求解了Burgers方程和KP方程,得到了一些新的精确解,包括孤子解和周期解.
In this paper, by using the Lie group method, the direct symmetry method and the modified CK’s direct method to solve a (2+1)-dimensional nonlinear evolution equation, the SK-KP equation, the MKP-II equation and the BK equation, we obtain these equations’symmetries and explicit solutions. We also obtain conservation laws of the SK-KP equation and the BK equation. By using the direct construction method, we get soliton solutions and periodic solutions of the Burgers equation and the KP equation.
In Chapter 1, by applying a direct symmetry method, we obtain the symmetry reduction and some new exact solutions of the (2+1)-dimensional nonlinear evolution equation, which include Jacobi elliptic function, hyperbolic function, trigonometric function and so on. These exact solutions will be helpful to a better understanding of some practical problems in physics.
In Chapter 2, the SK-KP equation is discussed. Firstly, assuming the form of symmetry. Secondly, by using the form of symmetry and the SK-KP equation, we resent the Lie symmetry. Similarity reductions and many new exact solutions of the SK-KP equation are obtained. At last, we also give the conservation laws of SK-KP equation.
In Chapter 3, we obtain the exact solutions of the MKP-II equation and the BK equation through the generalized CK direct method. A relationship is constructed between the new solutions and the old ones of the MKP-II equation and the BK equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the BK equation. Then we get the reductions using the symmetry and give some exact solutions of the BK equation.
In Chapter 4, we study the Burgers equation and the KP equation. We construct a frame of soliton solutions and periodic solutions to nonlinear equations. And we successfully solve the (2+1)-dimensional Burgers and (3+1)-Dimensional KP equation. And a lot of exact traveling solutions to the Burgers and KP equations are obtained.
引文
[1]郭柏灵.非线性演化方程[M].上海:上海科技教育出版社,1995.
[2]刘式适,刘式达.物理学中的非线性方程[M].北京:北京大学出版社,2000.
[3]谷超豪,李大潜,沈玮熙.应用偏微分方程[M].北京:高等教育出版社,1994.
[4]范恩贵.可积系统与计算机代数[M].北京:科学出版社,2004.
[5] Yanguang Li, Artyom V. Yurov. Lax pairs and darboux transformations for Euler equations. Studies in applied mathematics. 2003;111:101.
[6]Bluman G W, Cole J D.The general similarity solution of the heat equation [J].J. Math. Mech. 1969, 18: 1025–1042.
[7]Lie S.Uber die integration darch bestimmte integrale von einer Klasse linear partieller differential gleichung [J]. Arch. for Math. 1981, 6:328–368.
[8] Dong Zhongzhou, Liu Xiqiang, Bai Chenglin. Symmetry reduction, exact solutions and conservation laws of (2+1)-dimensional Burgers Korteweg-deVries equation[J].Commun. Theor. Phys. (Beijing, China), 2006, 46(1) :15-20.
[9]田畴.李群及其在微分方程中的应用[M].北京,科学出版社,2001.
[10] Lou Senyue. Exact solitary wave in a convecting fluid[J]. J. Phys. A: Math. Gen., 1991, 24(11): 587-590.
[11] Liu Xiqiang, Jiang Song. The sech-Tanh method and its applications[J]. Phys. Lett. A, 2002, 298(4):253-258.
[12] Zhang Lihua, Liu Xiqiang, Bai Chenglin. New Multiple Soliton-like and Periodic Solutions for (2+1)-Dimensional Canonical Generalized KP Equation with Variable Coefficients[J]. Commun. Theor. Phys. (Beijing, China), 2006, 46(5):793-798.
[13]谷超豪,胡和生,周子翔.孤立子中的Darboux变换及其几何应用[M].上海:上海科技出版社,1999.
[14] M. J. Ablowitz, P. A. Clarkson. Nonlinear evolution equations and inverse scattering[M]. Cambridge, Cambridge Univ. Press, 1991.
[15] Wang Mingliang, Zhou Yubin, Li Zhibin. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics[J]. Phys. Lett. A, 1996, 216(1):67-75.
[16]刘希强,白成林.耦合KdV方程的新孤立解[J].量子电子学报,1999,16(1):360-364.
[17] Liu Xiqiang. New explicit solutions of the (2+1)-dimensional Broer-Kaup equations[J]. J. Partial Diff. Eqs., 2004, 17(1):1-11.
[18] Fan Engui. Two new applications of the homogeneous balance method[J]. Phys. Lett. A, 2000,265(5):353-357.
[19]田畴.李群及其在微分方程中的应用[M].北京,科学出版社,2001.
[20] P. A. Clarkson, M. D. Kruskal. New similarity solutions of the Boussinesq equation[J]. J. Math. Phys., 1989, 30(10):2201-2213.
[21] Ma Hongcai. Generating Lie point symmetry groups of (2+1)-dimensional Broer-Kaup equation via a simple direct method[J]. Commun. Theor. Phys. (Beijing, China), 2005, 43(6):1047-1052.
[22] Lou Senyue, Ma Hongcai. Non-Lie symmetry groups of (2+1)-dimensional nonlinear systems obtained from a simple direct method[J]. J. Phys. A: Math. Gen., 2005, 38:129-137.
[23]Wang Mingliang, Li Xiangzheng, Zhang Jinliang.The ( G′G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics[J]. Phys. Lett. A 2008, 372:417–423.
[24]谷超豪.孤立子理论及其应用[M].杭州:浙江科技出版社,1990.
[25]Wu J.P, N-soliton solution,generalized double Wronskian determinant solution and rational solution for a (2+1)-dimensional nonlinear evolution equation[J], Phys. Lett. A. 2008, 373: 83-88.
[26]Geng X.G, Cao C. W,Dai H. H. Quasi-periodic solutions for some (2+1)-dimensional integrable models generated by the Jaulent-Miodek hierarchy[J], J. Phys. A: Math. Gen. 2001, 34: 989-992.
[27]Hu J.Q. An algebraic method exactly solving two high-dimensional nonlinear evolution equations[J]. Chaos, Solitons and Fractals. 2005, 23: 91-98.
[28]Abdul-Majid Wazwaz. Multiple kink solutions and multiple singular kink solutions for the (2 + 1)-dimensional Burgers equations. Appl. Math. Comput. 2008, 204: 817-823.
[29] S. Zhang, T. C. Xia. Variable-coefficient extended mapping method for nonlinear evolution equations. Phys. Lett .A. 2008,372 :1741-1749. [30 ] M. L. Wang, X. Z . Li, J. L, Zhang, The GG′-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A .2008,372: 417-423.
[31] Hai-Ling Lü, Xi-Qiang Liu, Lei Niu. A generalized GG′-expansion method and its applications to nonlinear evolution equations. Appl. Math. Comput. 2010,215:3811–3816
[32] Nail H. Ibragimov. A new conservation theorem. J. Math. Anal. Appl. 2007,333: 311–328.
[33] Meltem L Y and Lee J H. Dissipative hierarchies and resonance solitons for KP-II and MKP-II Appl. Math. Comput. 2007,74 : 323-332.
[34] Halim A A . Soliton solutions of the (2 + 1)-dimensional Harry Dym equation via Darboux transformation. Chaos. Solitons and Fractals 2008,36:646-653.
[35] M. V. Prabhakar, H. Bhate, Exact Solutions of the Bogoyavlensky–Konoplechenko Equation Lett. Math. Phys. 2003,64:1-6.
[36] H.C. Hu, New positon, negaton and complexiton solutions for the Bogoyavlensky–Konoplec-henko equation.Phys. Lett. A. 2009,373: 1750-1753.
[37] F. Calogero, A method to generate solvable nonlinear evolution equations. Lett. Nuovo Cimento 1975,14: 443-447.
[38] O.I. Bogoyavlenskii, Overturning solitons in new two-dimensional integrable equations. Izv. Akad. Nauk SSSR Ser. Mat. 1989,53: 243-257.
[39] K. Toda, S. J. Yu, A study of the construction of equations in (2+1)-dimensions. Inverse Problems 2001,17: 1053-1060.
[40] B.G. Konopelchenko, Solitons in Multidimensions, World Scientific, Singapore,1993.
[41] Guo-cheng Wu, Tie-cheng Xia, A new method for constructing soliton solutions and periodic solutions of nonlinear evolution equations[J]. Phys. Lett. A .2008,372 : 604–609.
[42] Qi Wang, Yong Chen, Hong-qing Zhang, A new Riccati equation rational expansion method and its application to (2 + 1)-dimensional Burgers equation[J]. Chaos, Solitons, Fractals. 2005,25 : 1019–1028.
[43] Senatorski A, Infeld E. Simulations of two-dimensional Kadomtsev–Petviashvili soliton dynamics in three-dimensional space[J]. PhyS. Rev. Lett. 1996,77:2855–8.
[44] Yong Chen, Zhen-ya Yan, Honging Zhang. New explicit solitary wave solutions for (2+ 1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation[J]. Phys. Lett. A .2003,307 :107–113.
[45] Fu-ding Xie, Ying Zhang, Zhuo-sheng Lu. Symbolic computation in non-linear evolution equation: application to (3+1)-dimensional Kadomtsev–Petviashvili equation[J]. Chaos, Solitons. Fractals.2005, 24: 257–263.
[2]刘式适,刘式达.物理学中的非线性方程[M].北京:北京大学出版社,2000.
[3]谷超豪,李大潜,沈玮熙.应用偏微分方程[M].北京:高等教育出版社,1994.
[4]范恩贵.可积系统与计算机代数[M].北京:科学出版社,2004.
[5] Yanguang Li, Artyom V. Yurov. Lax pairs and darboux transformations for Euler equations. Studies in applied mathematics. 2003;111:101.
[6]Bluman G W, Cole J D.The general similarity solution of the heat equation [J].J. Math. Mech. 1969, 18: 1025–1042.
[7]Lie S.Uber die integration darch bestimmte integrale von einer Klasse linear partieller differential gleichung [J]. Arch. for Math. 1981, 6:328–368.
[8] Dong Zhongzhou, Liu Xiqiang, Bai Chenglin. Symmetry reduction, exact solutions and conservation laws of (2+1)-dimensional Burgers Korteweg-deVries equation[J].Commun. Theor. Phys. (Beijing, China), 2006, 46(1) :15-20.
[9]田畴.李群及其在微分方程中的应用[M].北京,科学出版社,2001.
[10] Lou Senyue. Exact solitary wave in a convecting fluid[J]. J. Phys. A: Math. Gen., 1991, 24(11): 587-590.
[11] Liu Xiqiang, Jiang Song. The sech-Tanh method and its applications[J]. Phys. Lett. A, 2002, 298(4):253-258.
[12] Zhang Lihua, Liu Xiqiang, Bai Chenglin. New Multiple Soliton-like and Periodic Solutions for (2+1)-Dimensional Canonical Generalized KP Equation with Variable Coefficients[J]. Commun. Theor. Phys. (Beijing, China), 2006, 46(5):793-798.
[13]谷超豪,胡和生,周子翔.孤立子中的Darboux变换及其几何应用[M].上海:上海科技出版社,1999.
[14] M. J. Ablowitz, P. A. Clarkson. Nonlinear evolution equations and inverse scattering[M]. Cambridge, Cambridge Univ. Press, 1991.
[15] Wang Mingliang, Zhou Yubin, Li Zhibin. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics[J]. Phys. Lett. A, 1996, 216(1):67-75.
[16]刘希强,白成林.耦合KdV方程的新孤立解[J].量子电子学报,1999,16(1):360-364.
[17] Liu Xiqiang. New explicit solutions of the (2+1)-dimensional Broer-Kaup equations[J]. J. Partial Diff. Eqs., 2004, 17(1):1-11.
[18] Fan Engui. Two new applications of the homogeneous balance method[J]. Phys. Lett. A, 2000,265(5):353-357.
[19]田畴.李群及其在微分方程中的应用[M].北京,科学出版社,2001.
[20] P. A. Clarkson, M. D. Kruskal. New similarity solutions of the Boussinesq equation[J]. J. Math. Phys., 1989, 30(10):2201-2213.
[21] Ma Hongcai. Generating Lie point symmetry groups of (2+1)-dimensional Broer-Kaup equation via a simple direct method[J]. Commun. Theor. Phys. (Beijing, China), 2005, 43(6):1047-1052.
[22] Lou Senyue, Ma Hongcai. Non-Lie symmetry groups of (2+1)-dimensional nonlinear systems obtained from a simple direct method[J]. J. Phys. A: Math. Gen., 2005, 38:129-137.
[23]Wang Mingliang, Li Xiangzheng, Zhang Jinliang.The ( G′G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics[J]. Phys. Lett. A 2008, 372:417–423.
[24]谷超豪.孤立子理论及其应用[M].杭州:浙江科技出版社,1990.
[25]Wu J.P, N-soliton solution,generalized double Wronskian determinant solution and rational solution for a (2+1)-dimensional nonlinear evolution equation[J], Phys. Lett. A. 2008, 373: 83-88.
[26]Geng X.G, Cao C. W,Dai H. H. Quasi-periodic solutions for some (2+1)-dimensional integrable models generated by the Jaulent-Miodek hierarchy[J], J. Phys. A: Math. Gen. 2001, 34: 989-992.
[27]Hu J.Q. An algebraic method exactly solving two high-dimensional nonlinear evolution equations[J]. Chaos, Solitons and Fractals. 2005, 23: 91-98.
[28]Abdul-Majid Wazwaz. Multiple kink solutions and multiple singular kink solutions for the (2 + 1)-dimensional Burgers equations. Appl. Math. Comput. 2008, 204: 817-823.
[29] S. Zhang, T. C. Xia. Variable-coefficient extended mapping method for nonlinear evolution equations. Phys. Lett .A. 2008,372 :1741-1749. [30 ] M. L. Wang, X. Z . Li, J. L, Zhang, The GG′-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A .2008,372: 417-423.
[31] Hai-Ling Lü, Xi-Qiang Liu, Lei Niu. A generalized GG′-expansion method and its applications to nonlinear evolution equations. Appl. Math. Comput. 2010,215:3811–3816
[32] Nail H. Ibragimov. A new conservation theorem. J. Math. Anal. Appl. 2007,333: 311–328.
[33] Meltem L Y and Lee J H. Dissipative hierarchies and resonance solitons for KP-II and MKP-II Appl. Math. Comput. 2007,74 : 323-332.
[34] Halim A A . Soliton solutions of the (2 + 1)-dimensional Harry Dym equation via Darboux transformation. Chaos. Solitons and Fractals 2008,36:646-653.
[35] M. V. Prabhakar, H. Bhate, Exact Solutions of the Bogoyavlensky–Konoplechenko Equation Lett. Math. Phys. 2003,64:1-6.
[36] H.C. Hu, New positon, negaton and complexiton solutions for the Bogoyavlensky–Konoplec-henko equation.Phys. Lett. A. 2009,373: 1750-1753.
[37] F. Calogero, A method to generate solvable nonlinear evolution equations. Lett. Nuovo Cimento 1975,14: 443-447.
[38] O.I. Bogoyavlenskii, Overturning solitons in new two-dimensional integrable equations. Izv. Akad. Nauk SSSR Ser. Mat. 1989,53: 243-257.
[39] K. Toda, S. J. Yu, A study of the construction of equations in (2+1)-dimensions. Inverse Problems 2001,17: 1053-1060.
[40] B.G. Konopelchenko, Solitons in Multidimensions, World Scientific, Singapore,1993.
[41] Guo-cheng Wu, Tie-cheng Xia, A new method for constructing soliton solutions and periodic solutions of nonlinear evolution equations[J]. Phys. Lett. A .2008,372 : 604–609.
[42] Qi Wang, Yong Chen, Hong-qing Zhang, A new Riccati equation rational expansion method and its application to (2 + 1)-dimensional Burgers equation[J]. Chaos, Solitons, Fractals. 2005,25 : 1019–1028.
[43] Senatorski A, Infeld E. Simulations of two-dimensional Kadomtsev–Petviashvili soliton dynamics in three-dimensional space[J]. PhyS. Rev. Lett. 1996,77:2855–8.
[44] Yong Chen, Zhen-ya Yan, Honging Zhang. New explicit solitary wave solutions for (2+ 1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation[J]. Phys. Lett. A .2003,307 :107–113.
[45] Fu-ding Xie, Ying Zhang, Zhuo-sheng Lu. Symbolic computation in non-linear evolution equation: application to (3+1)-dimensional Kadomtsev–Petviashvili equation[J]. Chaos, Solitons. Fractals.2005, 24: 257–263.