非线性方程的精确解及求解方法的分析
|
摘要
随着科学技术的发展及实际应用的需要,人们面临着大量的非线性问题,它们很多可以用非线性方程来刻画,因此,研究非线性方程的求解具有非常重要的意义。经过国内外专家学者长期的努力,许多有效的求解方法已被提出,比如:反散射方法、Backlund变换法、Darboux变换法、Hirota双线性算子法、Lie方法、分离变量法、Painleve展开法等。近年来,随着计算机代数的发展,又出现了各种直接代数方法。比如:齐次平衡法、F-展开法及其扩展形式、双曲函数法及其扩展或修正形式、试探函数法、Jacobi椭圆函数法及其扩展形式、辅助方程法、Riccati方程展开法、投影Riccati方程法、直接约化法、Sine-Cosine方法、广义幂-指函数法等。通过这些方法我们可以得到非线性方程各种形式的精确解。另一个问题产生了:这些形式不同的解是否对应着非线性方程本质上不同的解?
本文研究(2+1)维色散长水波方程组的解,并由解的分析延伸到方法的分析,系统地总结并分析了双曲函数法,给出了双曲函数法求解的一般步骤。
第一部分简要介绍非线性科学的一些发展背景。
第二部分首先介绍齐次平衡法的一般原理及其一些新的应用;然后利用该方法得到二维色散长波方程组的Backlund变换,最后利用Backlund变换得到方程组的四种精确解。
第三部分首先介绍双曲函数法的一般求解步骤,并利用该方法得到(2+1)维色散长波方程组的六组精确解;然后给出了Sirendaoreji等在文献[ 23]中利用辅助方程法求得的四组解,通过恒等变形我们发现这四组解和我们给出的其中四组解是等价的;最后由上节启发,我们发现两种不同的方法得到的解可能是等价的。于是我们系统地分析了Yan,Dai,Huang,Xie, Shang等在文献中提出的拓展的双曲函数法,结果显示这些方法只是数学形式不同,本质上是等价的。
With the development of science and technology and the needs of practical application, people are faced with a large number of non-linear problems, many of which needs to be depicted with nonlinear equation. Therefore, the study of nonlinear equations is very important. Through the long-term efforts of experts and scholars, many effective methods have been raised, such as inverse scattering method, Backlund transformation, Darboux transformation, Hirota bilinear method, Lie method, variable separation method, Painleve expansion and so on. Recently, with the development of computer algebras, some effective and direct methods have been raised. For example the homogeneous balance method, F-expansion method and its extended form, hyperbolic function method and its extension or amendment, trial function method, Jacobi elliptic function method and its extended form, auxiliary equation method, Riccati equation method, projection Riccati equation method, direct reduction method, Sine-Cosine method,the generalized power– exponential function method. Through these methods we can get various exact solutions of nonlinear equations. Another question arises: are the solutions in different mathematics expressions really different in essence to the corresponding nonlinear equation?
In this paper, we study the (2+1)-dimensional dispersive long wave equations, and extend analytical solution to the method analysis and systematically summarize and analyze the hyperbolic function method, give the general steps of hyperbolic function method at last.
The first part briefly introduces the development backgrounds of nonlinear science.
In the second part, we first introduce the general principles and new applications of the homogeneous balance method, then we obtained Backlund transformation of the two-dimensional dispersive long wave equations. Finally, we get four exact solutions of the equations by Backlund transformations.
In the third part, firstly we introduce the general steps of hyperbolic function method, and get six groups solutions of (2+1)-dimensional dispersive long wave equations by this method, then we give four groups solutions by Sirendaoreji in [23] using auxiliary equation method, through identical deformation we find that they are equivalent to the four solutions we have got. Finally we find these solutions are equivalent through two different methods. So we systematically analyze the expansion of the hyperbolic function method proposed in the literature by Yan, Dai, Xie, Huang, Shang. However, the results show that these methods are mathematically different, but essentially equivalent.
本文研究(2+1)维色散长水波方程组的解,并由解的分析延伸到方法的分析,系统地总结并分析了双曲函数法,给出了双曲函数法求解的一般步骤。
第一部分简要介绍非线性科学的一些发展背景。
第二部分首先介绍齐次平衡法的一般原理及其一些新的应用;然后利用该方法得到二维色散长波方程组的Backlund变换,最后利用Backlund变换得到方程组的四种精确解。
第三部分首先介绍双曲函数法的一般求解步骤,并利用该方法得到(2+1)维色散长波方程组的六组精确解;然后给出了Sirendaoreji等在文献[ 23]中利用辅助方程法求得的四组解,通过恒等变形我们发现这四组解和我们给出的其中四组解是等价的;最后由上节启发,我们发现两种不同的方法得到的解可能是等价的。于是我们系统地分析了Yan,Dai,Huang,Xie, Shang等在文献中提出的拓展的双曲函数法,结果显示这些方法只是数学形式不同,本质上是等价的。
With the development of science and technology and the needs of practical application, people are faced with a large number of non-linear problems, many of which needs to be depicted with nonlinear equation. Therefore, the study of nonlinear equations is very important. Through the long-term efforts of experts and scholars, many effective methods have been raised, such as inverse scattering method, Backlund transformation, Darboux transformation, Hirota bilinear method, Lie method, variable separation method, Painleve expansion and so on. Recently, with the development of computer algebras, some effective and direct methods have been raised. For example the homogeneous balance method, F-expansion method and its extended form, hyperbolic function method and its extension or amendment, trial function method, Jacobi elliptic function method and its extended form, auxiliary equation method, Riccati equation method, projection Riccati equation method, direct reduction method, Sine-Cosine method,the generalized power– exponential function method. Through these methods we can get various exact solutions of nonlinear equations. Another question arises: are the solutions in different mathematics expressions really different in essence to the corresponding nonlinear equation?
In this paper, we study the (2+1)-dimensional dispersive long wave equations, and extend analytical solution to the method analysis and systematically summarize and analyze the hyperbolic function method, give the general steps of hyperbolic function method at last.
The first part briefly introduces the development backgrounds of nonlinear science.
In the second part, we first introduce the general principles and new applications of the homogeneous balance method, then we obtained Backlund transformation of the two-dimensional dispersive long wave equations. Finally, we get four exact solutions of the equations by Backlund transformations.
In the third part, firstly we introduce the general steps of hyperbolic function method, and get six groups solutions of (2+1)-dimensional dispersive long wave equations by this method, then we give four groups solutions by Sirendaoreji in [23] using auxiliary equation method, through identical deformation we find that they are equivalent to the four solutions we have got. Finally we find these solutions are equivalent through two different methods. So we systematically analyze the expansion of the hyperbolic function method proposed in the literature by Yan, Dai, Xie, Huang, Shang. However, the results show that these methods are mathematically different, but essentially equivalent.
引文
[1] Ablowitz M Jand Clarkson P A. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge, Cambridge University Press, 1991.
[2]谷超豪等.孤立子理论及其应用.杭州:浙江科学技术出版社,1990.
[3]范恩贵.可积系统与计算机代数.北京:科学出版社,2004.
[4]李志斌.非线性数学物理方程的行波解.北京:科学出版社,2006.
[5] N.J.Zabusky, M.D.Kruskal. Interation of solitons in a collisionless plasma and the recurrence of intial states. Phys Rev Lett 15 (1965), 240-243.
[6]C.H.Gu H.S.Hu. A unified explicit form of Backlund transformations for generalized hierarchies of the KdV equation. Lett. Math Phys 11 (1986), 325- 335.
[7] R.Hirota. Exact solution of the KdV equation formultiple collisions of solitons. Phys Rev Lett 27 (1971), 1192-1194.
[8] R.Hirota. Exact N-soliton of the wave equation of long waves in shallow water and in nonlinear lattices. J Math Phys 14 (1973), 810-814.
[9]M.Musette, R.Conto. Algorithmic method for deriving Lax pairs from the invariant Painleve analysis of nonlinear partial differential equations. J Math Phys 32 (1991), 1450-1457.
[10]Wadati M. Wave propagation in nonlinear lattic. J.Phys. Soc. Jan. 38(1975), 673-680.
[11]Y. S. Li, J.E.Zhang. Darboux Transformation of Classical Boussinesq System and its Multi-Solition Solutions.Phys Lett A 284(2001),253-258.
[12] M L Wang et al.. Application of homogeneous balance method to exact solutions of nonlinear evolution equation in mathematical physics. Phy. Lett. A 216(1996), 67-73.
[13]M L Wang. Exact solutions for a compound KdV-Burgers equation. Phy.Lett.A 213(1996),279-287.
[14] M X Wang, S L Xiong, Q X Ye. Explicit wave front solutions of Noyes-Field systems for the Belousov-Zhabotinskii reaction. J Math Anai Appl, 82 (1994) ,705-717.
[15] Sirendaoreji, A new auxiliary equation and exact traveling wave solutions of nonlinear equations. Phy. Lett. A, 336 (2006), 124–130.
[16]范恩贵,张鸿庆.非线性孤子方程的齐次平衡法.物理学报, 47 (1998) :353-361.
[17]范恩贵,张鸿庆.齐次平衡法若干新的应用.数学物理学报,19(1999),286-292.
[18]范恩贵,张鸿庆.Whitham Broer Kaup浅水波方程的Backlund变换和精确解.应用数学和力学,19(1998),667-670.
[19]闫振亚,张鸿庆.(2+1)维非线性色散长波方程的相似约化和解析解.数学物理学报,21A(2001):384-390.
[20]闫振亚,张鸿庆.两个非线性发展方程组精确解析解的研究.应用数学和力学,22(2001):825-833.
[21] Chun Ping Liu, Xiao Ping Liu. A Note on the Auxiliary Equation Method for Solving Nonlinear Partial Differential Equations .Physics Letter A, 348 (2006):222-227.
[22] R Conte, M Musette.Link between solitary waves and projective Riccati equations. J. Phys .A:Math.Gen,25(1992),5609-5623.
[23] Sirendaoreji, Sun Jiong. Auxiliary equation method for solving nonlinear partial differential equations. Phy. Lett. A, 309 (2003), 387–396.
[24] Zhenya Yan.Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres. Chaos.Solitons and Fractals,16(2003)759-766.
[25] Chunping Liu, Exact solutions for the higher-order nonlinear Schordinger equation in nonlinear optical fibres. Chaos solitons and Fractals 23(2005)949-955.
[26]黄定江,张鸿庆.扩展的双曲函数法和Zakharov方程组的新精确解孤立波解.物理学报, 53(2004),2434-2437.
[27] Fuding Xie, New travelling wave solutions of the generalized coupled Hirota-Satsuma KdV system. Chaos solitons and Fractals 20(2004)1005-1012.
[28] Chaoqing Dai, Jiamin Zhu, Jiefang Zhang. New exact solutions to the mKdVequation with variable coefficients. Chaos solitons and Fractals 27(2006)881-886.
[29] Yadong Shang, Yong Huang, Wenjun Yuan. The extended hyperbolic functions method and new exact solutions to the Zakharov equations. Applied Mathematics and Computation 200(2008)110-122.
[2]谷超豪等.孤立子理论及其应用.杭州:浙江科学技术出版社,1990.
[3]范恩贵.可积系统与计算机代数.北京:科学出版社,2004.
[4]李志斌.非线性数学物理方程的行波解.北京:科学出版社,2006.
[5] N.J.Zabusky, M.D.Kruskal. Interation of solitons in a collisionless plasma and the recurrence of intial states. Phys Rev Lett 15 (1965), 240-243.
[6]C.H.Gu H.S.Hu. A unified explicit form of Backlund transformations for generalized hierarchies of the KdV equation. Lett. Math Phys 11 (1986), 325- 335.
[7] R.Hirota. Exact solution of the KdV equation formultiple collisions of solitons. Phys Rev Lett 27 (1971), 1192-1194.
[8] R.Hirota. Exact N-soliton of the wave equation of long waves in shallow water and in nonlinear lattices. J Math Phys 14 (1973), 810-814.
[9]M.Musette, R.Conto. Algorithmic method for deriving Lax pairs from the invariant Painleve analysis of nonlinear partial differential equations. J Math Phys 32 (1991), 1450-1457.
[10]Wadati M. Wave propagation in nonlinear lattic. J.Phys. Soc. Jan. 38(1975), 673-680.
[11]Y. S. Li, J.E.Zhang. Darboux Transformation of Classical Boussinesq System and its Multi-Solition Solutions.Phys Lett A 284(2001),253-258.
[12] M L Wang et al.. Application of homogeneous balance method to exact solutions of nonlinear evolution equation in mathematical physics. Phy. Lett. A 216(1996), 67-73.
[13]M L Wang. Exact solutions for a compound KdV-Burgers equation. Phy.Lett.A 213(1996),279-287.
[14] M X Wang, S L Xiong, Q X Ye. Explicit wave front solutions of Noyes-Field systems for the Belousov-Zhabotinskii reaction. J Math Anai Appl, 82 (1994) ,705-717.
[15] Sirendaoreji, A new auxiliary equation and exact traveling wave solutions of nonlinear equations. Phy. Lett. A, 336 (2006), 124–130.
[16]范恩贵,张鸿庆.非线性孤子方程的齐次平衡法.物理学报, 47 (1998) :353-361.
[17]范恩贵,张鸿庆.齐次平衡法若干新的应用.数学物理学报,19(1999),286-292.
[18]范恩贵,张鸿庆.Whitham Broer Kaup浅水波方程的Backlund变换和精确解.应用数学和力学,19(1998),667-670.
[19]闫振亚,张鸿庆.(2+1)维非线性色散长波方程的相似约化和解析解.数学物理学报,21A(2001):384-390.
[20]闫振亚,张鸿庆.两个非线性发展方程组精确解析解的研究.应用数学和力学,22(2001):825-833.
[21] Chun Ping Liu, Xiao Ping Liu. A Note on the Auxiliary Equation Method for Solving Nonlinear Partial Differential Equations .Physics Letter A, 348 (2006):222-227.
[22] R Conte, M Musette.Link between solitary waves and projective Riccati equations. J. Phys .A:Math.Gen,25(1992),5609-5623.
[23] Sirendaoreji, Sun Jiong. Auxiliary equation method for solving nonlinear partial differential equations. Phy. Lett. A, 309 (2003), 387–396.
[24] Zhenya Yan.Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres. Chaos.Solitons and Fractals,16(2003)759-766.
[25] Chunping Liu, Exact solutions for the higher-order nonlinear Schordinger equation in nonlinear optical fibres. Chaos solitons and Fractals 23(2005)949-955.
[26]黄定江,张鸿庆.扩展的双曲函数法和Zakharov方程组的新精确解孤立波解.物理学报, 53(2004),2434-2437.
[27] Fuding Xie, New travelling wave solutions of the generalized coupled Hirota-Satsuma KdV system. Chaos solitons and Fractals 20(2004)1005-1012.
[28] Chaoqing Dai, Jiamin Zhu, Jiefang Zhang. New exact solutions to the mKdVequation with variable coefficients. Chaos solitons and Fractals 27(2006)881-886.
[29] Yadong Shang, Yong Huang, Wenjun Yuan. The extended hyperbolic functions method and new exact solutions to the Zakharov equations. Applied Mathematics and Computation 200(2008)110-122.