孤立子理论中非线性发展方程求解研究
摘要
孤立子理论在自然科学的研究中占有非常重要的部分.非线性发展方程的孤立子理论研究是其中一个重要的热点内容.许多有着物理意义背景的非线性发展方程都具有孤立子特性.因此,寻求非线性发展方程的解,在理论研究中有着重要的意义.目前已经提出了很多求解非线性发展方程精确解的方法,如反散射方法,函数展开法,形变映射法,混合指数法,双线性方法和达布变换法.由于求解非线性发展方程还没有普遍使用的统一方法.因此,继续寻找有效的求解非线性发展方程精确解的方法,仍然是有待进一步研究的工作和问题.
本文在对现有的孤立子理论和非线性发展方程的求解方法进行了较为深入和系统的研究基础之上,对几种非线性发展方程精确行波解的求解方法做了应用和改进,求出了非线性方程的几类新的精确解.
本文分为三章：
第一章介绍了孤立子理论的历史背景,进展情况.概述了孤立子可积性,非线性发展方程的研究现状和求解非线性发展方程的几种常用方法.简要阐述了本文的研究内容和研究意义.
第二章基于孤立子理论求解的研究方法,首先扩展了已有的tanh-coth方法,并将该方法应用于广义Zakharov方程,得到了一系列精确解,再利用二次齐次平衡方法,分别以一个微分方程和耦合投影Riccati方程为辅助方程,求解广义Zakharov方程,拓展了原方法所得到的解的结构.其次,运用推广的Jacobi椭圆函数法求解(2+1)维Konopelchenko-Dubrovsky方程,获得了方程新的椭圆函数解.在极限情况下这些解退化为孤子解和三角函数解.最后,利用改进的(G'/G)展开法求解了(2+1)维破裂孤立子方程,得到了丰富的精确解,例如含参数的双曲函数解,三角函数解和有理数解.
第三章是全文的结论部分,主要是对全文内容的总结和对以后研究方向的展望.
Soliton theory in the natural science research plays a very important part. The soliton theory of the nonlinear evolution equations is the important hot spot topic. Some nonlinear evolution equations with physical background have soliton properties. Therefore, the important significance in theory is to get solutions of the nonlinear evolution equations. At present, a number of methods are proposed to look for the exact solutions of the nonlinear evolution equations, for example, inverse scattering transformation method, function expansion method, deformation mapping method, mixing exponential method, bilinear method and Darboux transformation method. As for solving nonlinear evolution equations have not an unified way. As a consequence, it is still a task for further research and issue to go on searching for efficient approaches to solving nonlinear evolution equations.
This dissertation is based on systematic research and on the existing technique of solving nonlinear evolution equations and the existing soliton theory. Some methods for constructing the exact traveling wave solutions of the nonlinear evolution equations are applied and improved, new exact solutions of several types have been obtained.
This dissertation consists of three chapters.
In Chapter 1, we introduce the historical background and the development of the soliton theory, integrability of the soliton, study development of nonlinear evolution equation and several commonly used methods for solving the nonlinear evolution equation. Then briefly describes the content and significance in this dissertation.
In Chapter 2, based on the methods of solving nonlinear evolution equations, we firstly extend the existing tanh-coth method, the method is applied to the generalized Zakharov equation and we obtained a series of exact solutions. By using the second homogeneous balance method with a differential equation and Coupled projected Riccati equations, solving generalized Zakharov equation, extending the structure of the solution obtained of the original method. Secondly, by using the promotion of the Jacobi elliptic function method, we obtain the new elliptic function solutions of the (2+1) dimensional Konopelchenko-Dubrovsky equation. In the limit cases, these solutions degenerate to solition solutions and triangular function solutions. Finally, by using the improved (G'/G) expansion method, solving the (2 +1)-dimensional breaking soliton equation, we find abundant exact solutions. The solutions are expressed by hyperbolic functions, trigonometric functions and rational functions contained arbitrary parameters.
本文在对现有的孤立子理论和非线性发展方程的求解方法进行了较为深入和系统的研究基础之上,对几种非线性发展方程精确行波解的求解方法做了应用和改进,求出了非线性方程的几类新的精确解.
本文分为三章：
第一章介绍了孤立子理论的历史背景,进展情况.概述了孤立子可积性,非线性发展方程的研究现状和求解非线性发展方程的几种常用方法.简要阐述了本文的研究内容和研究意义.
第二章基于孤立子理论求解的研究方法,首先扩展了已有的tanh-coth方法,并将该方法应用于广义Zakharov方程,得到了一系列精确解,再利用二次齐次平衡方法,分别以一个微分方程和耦合投影Riccati方程为辅助方程,求解广义Zakharov方程,拓展了原方法所得到的解的结构.其次,运用推广的Jacobi椭圆函数法求解(2+1)维Konopelchenko-Dubrovsky方程,获得了方程新的椭圆函数解.在极限情况下这些解退化为孤子解和三角函数解.最后,利用改进的(G'/G)展开法求解了(2+1)维破裂孤立子方程,得到了丰富的精确解,例如含参数的双曲函数解,三角函数解和有理数解.
第三章是全文的结论部分,主要是对全文内容的总结和对以后研究方向的展望.
Soliton theory in the natural science research plays a very important part. The soliton theory of the nonlinear evolution equations is the important hot spot topic. Some nonlinear evolution equations with physical background have soliton properties. Therefore, the important significance in theory is to get solutions of the nonlinear evolution equations. At present, a number of methods are proposed to look for the exact solutions of the nonlinear evolution equations, for example, inverse scattering transformation method, function expansion method, deformation mapping method, mixing exponential method, bilinear method and Darboux transformation method. As for solving nonlinear evolution equations have not an unified way. As a consequence, it is still a task for further research and issue to go on searching for efficient approaches to solving nonlinear evolution equations.
This dissertation is based on systematic research and on the existing technique of solving nonlinear evolution equations and the existing soliton theory. Some methods for constructing the exact traveling wave solutions of the nonlinear evolution equations are applied and improved, new exact solutions of several types have been obtained.
This dissertation consists of three chapters.
In Chapter 1, we introduce the historical background and the development of the soliton theory, integrability of the soliton, study development of nonlinear evolution equation and several commonly used methods for solving the nonlinear evolution equation. Then briefly describes the content and significance in this dissertation.
In Chapter 2, based on the methods of solving nonlinear evolution equations, we firstly extend the existing tanh-coth method, the method is applied to the generalized Zakharov equation and we obtained a series of exact solutions. By using the second homogeneous balance method with a differential equation and Coupled projected Riccati equations, solving generalized Zakharov equation, extending the structure of the solution obtained of the original method. Secondly, by using the promotion of the Jacobi elliptic function method, we obtain the new elliptic function solutions of the (2+1) dimensional Konopelchenko-Dubrovsky equation. In the limit cases, these solutions degenerate to solition solutions and triangular function solutions. Finally, by using the improved (G'/G) expansion method, solving the (2 +1)-dimensional breaking soliton equation, we find abundant exact solutions. The solutions are expressed by hyperbolic functions, trigonometric functions and rational functions contained arbitrary parameters.
引文
[1]Elwakil S A, El-labany S K, Zahran M A et al. New exact solutions for a generalized variable coeffieients 2D KdV equation. Chaos,Solitons and Fractals,2004:19-1083.
[2]Ablowitz M J, Kaup D J, Newell A C et al. Method for solving the Sine-Gordon equation. Phys Rev Lett,1973,30:1262-1264.
[3]黄景宁,徐济仲,熊吟涛.孤子概念原理和应用.北京：高等教育出版社,2004.
[4]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.
[5]Kay I, Moses H E. The determination of the scattering potential from the spectral measure function. Nuovo Cimeneto,1956,10(3):276-304.
[6]Gu C H, H S, Zhou Z X. Daroux Transformation in soliton theory and its Geometric Applications. Shanghai Science Technical Publishers,1999.
[7]Russell J S. Report on Waves. Fourteen meeting of the British association for the advancement of science, John Murray,London,1844:311-390.
[8]楼深岳,唐晓艳.非线性数学物理方法.北京：科学出版社,2006：4-313.
[9]李大潜,陈韵梅.非线性发展方程.北京：科学出版社,1999：49-146.
[10]郭玉翠.非线性偏微分方程引论.北京：清华大学出版社,2008.
[11]Fan E. Two new applications of the homogeneous balance method. Phys Lett A,2000(265): 353-7.
[12]GU CHAOHAO. On the backlund Transformations for the Generalized Hierarchies of Compound MKdV-SG Equations. Letters in Mathematical Physics,1986,12:31-41.
[13]Li Yishen. Gauge Transformation, backlund Transformation and the Nonlinear Superposition Formula. Advances in Mathematics.1989,8(18):356-372.
[14]C. Rogers, W. K. Schief. Backlund and Darboux Transformations Geometry and Modern Applca-tions in soliton theory, Cambridge University Press, Cambridge,2002.
[15]李志斌.非线性数学物理方程的行波解.北京：科学出版社,2007.
[16]Fan E G. Extended tanh-fanction method and its applications to nonlinear equations. Phys.Lett. A,2000:277-212.
[17]范恩贵.可积系统与计算机代数.北京：科学出版社,2003.
[18]Luwai Wazzan. A modified tanh-coth method for solving the kdv and the KdV-Burgers' equations. Nonlinear Science and Numerical Simulation,2009,14:443-450.
[19]ZhangHuiqun. New exact complex travelling wave solution to nonlinear Schrodinger equation[J]. Nonlinear Science and Numerical Simulation,2009,14:668-673.
[20]Sirendaoreji. A New Application of the Extended Tanh-Function Method. Journal of Inner Mongolia Normal University,2007,36:391-401.
[21]M. A. Abdou, Zhang Sheng. New periodic wave solutions via extended mapping method. Nonlinear Science and Numerical Simulation,2009,14:2-11.
[22]A. S. Abdel Rady, E. S. Osman, Mohammed Khalfallah. On soliton solutions for a generalized Hirota-Satsuma coupled KdV equation. Commun Nonlinear Sci Numer Simulat,2009,15:264-274.
[23]Mohammed Khalfallah. New exact traveling wave solutions of the (3+1)dimensional Ka-domtsev-Petviashvili equation. Commun Nonlinear Sci Numer Simulat,2009,14:69-75.
[24]WANGM L, ZHOU Y B, L I Z B. Application of homogeneous balance method to exact solutions of nonlinear equations in mathematical physics[J]. Phys LettA,1996,213:67-75.
[25]A. S. Abdel Rady, Mohammed Khalfallah. On soliton solutions for Boussinesq-Burgers equation. Commun Nonlinear Sci Numer Simulat,2010,15:886-894.
[26]Saad Zagloul Rida, Mohammed Khalfallah. New periodic wave and soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrodinger equation. Commun Nonlinear Sci Numer Simulat,2010,15:1818-2827.
[27]杨立波,卢殿臣.Zakharov方程组的一些新精确解.淮阴工学院学报,2010,1(19)：8-13.
[28]李向正,李晓燕,王明亮.Jacobi椭圆函数的四个恒等式.河南科技大学学,2004,3(25)：83-86.
[29]HIROTAR. Exact solution of the Korteweg-de Vries equation formultiple collisions of solitons. Phys Rev lett,1971,27:1192-1194.
[30]Wazwaz AM, Multiple-soliton solutions for the Boussinesq equation. Appl Math Comput 2007;192(2):479-86.
[31]WANGM L, ZHOU Y B, L I Z B. Application of homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys LettA,1996,213:67-75.
[32]Yang Wang, Long Wei. New exact solutions to the (2+1)-dimensional Konopelchenko-Dubro-vsky equation. Commun Nonlinear Sci Numer Simulat,2010,15:216-224.
[33]Fu ZT, Liu SK, Liu SD. Zhao Q. New Jacobi elliptic function expansion and new periodic Solutions of nonlinear wave equations. Phys Lett A 2001,290:72-6.
[34]Wang M L,Li X Z,Zhang J L 2008 Phys,Lett. A372 417.
[35]李二强,王明亮.(G'/G)方法及组合KdV-Burgers方程的行波解.河南科技大学学报,2008,5(29)：80-83.
[36]Mingliang Wang Exact Solutions for a Compound Kdv-Burgers Equation. Physics Letters A,1997,229:217-220.
[37]Sudao Bilige, Temuer Chaolu. Two Generalizations of the (G'/G)-Expansion Method and Their Applications. Journal of Inner Mongolia University.2010,2(41):144-152.
[38]李帮庆,马玉兰.(G'/G)展开法和(2+1)维非对称Nizhnik-Novikov-Veselov系统的新精确解.物理学报.2009,7(58)：4374-4378.
[39]XU Bin. New Solitons of the Generalized Burgers-KP Equation. Journal of Liao cheng University.2009,2(22):9-19.
[40]李自田.(3+1)维Jimbo-Miwa方程的精确扭结解孤子解和周期形式解.曲靖师范学院学报.2010,3(29)：30-32.
[41]李灵晓,李保安.利用推广的(G'/G)展开法求解Kononpelchenko-Dubrovsky方程.河南科技大学学报.2009,1(30)：75-77.
[42]Wazwaz AM, Multiple-soliton solutions for the Boussinesq equation. Appl Math Comput 2007;192(2):479-86.
[43]王军民,(2+1)维破切孤子方程的Jacobi椭圆函数周期解.河南师范大学学报,2009,5(37)：8-10.
[44]牛艳霞,李二强,张金良.利用(G'/G)展开法求解(2+1)维破裂孤子方程组.河南科技大学报,2008,5(29)：73-76.
[2]Ablowitz M J, Kaup D J, Newell A C et al. Method for solving the Sine-Gordon equation. Phys Rev Lett,1973,30:1262-1264.
[3]黄景宁,徐济仲,熊吟涛.孤子概念原理和应用.北京：高等教育出版社,2004.
[4]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.
[5]Kay I, Moses H E. The determination of the scattering potential from the spectral measure function. Nuovo Cimeneto,1956,10(3):276-304.
[6]Gu C H, H S, Zhou Z X. Daroux Transformation in soliton theory and its Geometric Applications. Shanghai Science Technical Publishers,1999.
[7]Russell J S. Report on Waves. Fourteen meeting of the British association for the advancement of science, John Murray,London,1844:311-390.
[8]楼深岳,唐晓艳.非线性数学物理方法.北京：科学出版社,2006：4-313.
[9]李大潜,陈韵梅.非线性发展方程.北京：科学出版社,1999：49-146.
[10]郭玉翠.非线性偏微分方程引论.北京：清华大学出版社,2008.
[11]Fan E. Two new applications of the homogeneous balance method. Phys Lett A,2000(265): 353-7.
[12]GU CHAOHAO. On the backlund Transformations for the Generalized Hierarchies of Compound MKdV-SG Equations. Letters in Mathematical Physics,1986,12:31-41.
[13]Li Yishen. Gauge Transformation, backlund Transformation and the Nonlinear Superposition Formula. Advances in Mathematics.1989,8(18):356-372.
[14]C. Rogers, W. K. Schief. Backlund and Darboux Transformations Geometry and Modern Applca-tions in soliton theory, Cambridge University Press, Cambridge,2002.
[15]李志斌.非线性数学物理方程的行波解.北京：科学出版社,2007.
[16]Fan E G. Extended tanh-fanction method and its applications to nonlinear equations. Phys.Lett. A,2000:277-212.
[17]范恩贵.可积系统与计算机代数.北京：科学出版社,2003.
[18]Luwai Wazzan. A modified tanh-coth method for solving the kdv and the KdV-Burgers' equations. Nonlinear Science and Numerical Simulation,2009,14:443-450.
[19]ZhangHuiqun. New exact complex travelling wave solution to nonlinear Schrodinger equation[J]. Nonlinear Science and Numerical Simulation,2009,14:668-673.
[20]Sirendaoreji. A New Application of the Extended Tanh-Function Method. Journal of Inner Mongolia Normal University,2007,36:391-401.
[21]M. A. Abdou, Zhang Sheng. New periodic wave solutions via extended mapping method. Nonlinear Science and Numerical Simulation,2009,14:2-11.
[22]A. S. Abdel Rady, E. S. Osman, Mohammed Khalfallah. On soliton solutions for a generalized Hirota-Satsuma coupled KdV equation. Commun Nonlinear Sci Numer Simulat,2009,15:264-274.
[23]Mohammed Khalfallah. New exact traveling wave solutions of the (3+1)dimensional Ka-domtsev-Petviashvili equation. Commun Nonlinear Sci Numer Simulat,2009,14:69-75.
[24]WANGM L, ZHOU Y B, L I Z B. Application of homogeneous balance method to exact solutions of nonlinear equations in mathematical physics[J]. Phys LettA,1996,213:67-75.
[25]A. S. Abdel Rady, Mohammed Khalfallah. On soliton solutions for Boussinesq-Burgers equation. Commun Nonlinear Sci Numer Simulat,2010,15:886-894.
[26]Saad Zagloul Rida, Mohammed Khalfallah. New periodic wave and soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrodinger equation. Commun Nonlinear Sci Numer Simulat,2010,15:1818-2827.
[27]杨立波,卢殿臣.Zakharov方程组的一些新精确解.淮阴工学院学报,2010,1(19)：8-13.
[28]李向正,李晓燕,王明亮.Jacobi椭圆函数的四个恒等式.河南科技大学学,2004,3(25)：83-86.
[29]HIROTAR. Exact solution of the Korteweg-de Vries equation formultiple collisions of solitons. Phys Rev lett,1971,27:1192-1194.
[30]Wazwaz AM, Multiple-soliton solutions for the Boussinesq equation. Appl Math Comput 2007;192(2):479-86.
[31]WANGM L, ZHOU Y B, L I Z B. Application of homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys LettA,1996,213:67-75.
[32]Yang Wang, Long Wei. New exact solutions to the (2+1)-dimensional Konopelchenko-Dubro-vsky equation. Commun Nonlinear Sci Numer Simulat,2010,15:216-224.
[33]Fu ZT, Liu SK, Liu SD. Zhao Q. New Jacobi elliptic function expansion and new periodic Solutions of nonlinear wave equations. Phys Lett A 2001,290:72-6.
[34]Wang M L,Li X Z,Zhang J L 2008 Phys,Lett. A372 417.
[35]李二强,王明亮.(G'/G)方法及组合KdV-Burgers方程的行波解.河南科技大学学报,2008,5(29)：80-83.
[36]Mingliang Wang Exact Solutions for a Compound Kdv-Burgers Equation. Physics Letters A,1997,229:217-220.
[37]Sudao Bilige, Temuer Chaolu. Two Generalizations of the (G'/G)-Expansion Method and Their Applications. Journal of Inner Mongolia University.2010,2(41):144-152.
[38]李帮庆,马玉兰.(G'/G)展开法和(2+1)维非对称Nizhnik-Novikov-Veselov系统的新精确解.物理学报.2009,7(58)：4374-4378.
[39]XU Bin. New Solitons of the Generalized Burgers-KP Equation. Journal of Liao cheng University.2009,2(22):9-19.
[40]李自田.(3+1)维Jimbo-Miwa方程的精确扭结解孤子解和周期形式解.曲靖师范学院学报.2010,3(29)：30-32.
[41]李灵晓,李保安.利用推广的(G'/G)展开法求解Kononpelchenko-Dubrovsky方程.河南科技大学学报.2009,1(30)：75-77.
[42]Wazwaz AM, Multiple-soliton solutions for the Boussinesq equation. Appl Math Comput 2007;192(2):479-86.
[43]王军民,(2+1)维破切孤子方程的Jacobi椭圆函数周期解.河南师范大学学报,2009,5(37)：8-10.
[44]牛艳霞,李二强,张金良.利用(G'/G)展开法求解(2+1)维破裂孤子方程组.河南科技大学报,2008,5(29)：73-76.