基于计算机符号计算的非线性模型孤子解研究
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摘要
随着信息技术的迅速发展,计算机符号计算作为人工智能的新分支学科之一,也逐渐成熟和完善,并被应用到非线性科学的研究中来。目前,计算机符号计算因其强大而精确的符号计算能力和操作简便、直观,易于实现等特点,已经逐渐发展成为人们从事非线性科学研究必不可少的计算机辅助工具之一,并在此基础上取得了丰硕的研究成果。尤其是近十几年间,基于计算机符号计算的孤子理论研究越来越成熟和发展起来。作为非线性科学的核心问题之一,孤子理论一直是国内外众多学者关注的一个热点领域,其中对于非线性模型孤子解的研究,无论在理论研究还是在实际应用上一直都是重要而又难以解决的问题,几乎在各个学科领域中都会遇到。
在本论文中,作者结合本研究领域的发展现状,基于计算机符号计算对若干有着实际物理应用的非线性模型孤子解进行了研究,主要内容包括两个方面,研究构造非线性模型精确孤子解的方法;研究非线性模型孤子解的时间和空间特性。具体包括如下几个方面:
①基于计算机符号计算将传统的双线性求解方法进行推广,使其适合于求解更复杂的高维和高阶非线性模型。推广的双线性方法借助于齐次平衡思想推导出非线性模型的双线性变换,将目标模型转化成易于求解的齐次方程。同时在求解过程中略去了将齐次方程分块和双线性化的过程,这样就避免了不适当的分块可能产生的对于模型孤子解的结构的限制。同时,利用计算机符号计算给出双线性算子的计算式,大大方便了我们利用双线性方法处理相关问题。
②一般来讲,广义的时间变系数模型大多是不可积的,但是,当各个系数函数满足了一定的约束条件时却是可积的,也就是存在多孤子解,B(a｜¨)cklund变换,Lax对等。本文将推广的双线性方法应用于求解广义的变系数Korteweg-de Vries模型和变系数Kadomtsev-Petviashvili模型的精确多孤子解;基于这些方程的双线性形式,利用双线性算子的性质,推导出其对应解的自B(a｜¨)cklund变换,包括双线性形式和Lax形式的自B(a｜¨)cklund变换;进一步地,从解的双线性形式自B(a｜¨)cklund变换出发,可以得到模型的Lax对,或者解的非线性叠加公式;对于求得的多孤子解,通过绘图分析研究模型的各个变系数对于解的稳定性和传输方式的影响。
③近年来,对于非线性Schr(o｜¨)dinger模型多孤子间的部分相干相互作用的研究引起了人们的兴趣。本文我们借助于计算机符号计算和推广的双线性方法,研究一个在现代光纤通信领域中有着重要应用的广义(1+1)维耦合非线性Schr(o｜¨)dinger模型的孤子解。并从得到的解出发,研究随着时间的发展各个变系数函数对于孤子稳定性的影响,包括传播速度和振幅,重点研究孤子间的相互作用,包括部分相干相互作用和孤波间的成对碰撞。
④部分非线性Schr(o｜¨)dinger模型的孤子间存在一种特殊的形变碰撞,由于在这种碰撞过程中发生了能量再分配,而越来越受到人们的关注。借助于计算机符号计算,我们将研究一个广义耦合高阶非线性Schr(o｜¨)dinger模型间的这种奇妙的非弹性相互作用。该研究在光纤中的一个重要应用就是能量放大器,由于这种放大过程不需要外界的干扰和新能量的补充,因而很稳定,拥有广阔的实际应用前景。
⑤对于非线性模型局域孤子的研究越来越积极和发展起来,各种局域相干结构也被不断地发现。同时,做为孤子间的一种非弹性相互作用,孤子的裂变和聚变也开始引起人们的关注。我们对于(1+1)维孤子解的相互作用已经研究的很深,并取得了很大的成就,但是高维方程的孤子解之间的相互作用确实很复杂的,而且维数越高,相互作用就越复杂。本文借助于计算机符号计算和计算机模拟,对于色散长波模型,在多孤子解的基础上,通过计算机软件模拟绘图的方式集中分析研究不同的局域相干结构间的这种裂变型非弹性相互作用,并且从数学角度简要分析裂变现象产生的原因。
综上所述,本文将计算机符号计算与传统的双线性方法相结合,并将其推广应用于求解复杂的高维、高阶非线性模型中。对于几个有着典型物理意义的非线性模型进行了解析研究,包括构造多孤子解,B(a｜¨)cklund变换,Lax对等;并利用Mathematica计算软件的模拟绘图功能对求得孤子解的时间发展特性,相互之间的弹性和非弹性相互作用,以及它们潜在的物理应用进行了深入分析。希望本文的理论研究结果在将来的实验观察中能够得到证实,并对于相关研究有所帮助。
With the quick development of information technology, symbolic computation, as a new branch of artificial intelligence, has become perfect and ripe gradually, and has been applied to the research of nonlinear science. Nowadays, because of its powerful ability in exactly dealing with the complicated and tedious calculations, and its convenient and direct practicability, symbolic computation has grown into a necessary and wonderful assistant tool for the study of nonlinear science. Especially in the latest decade, much progress has already been made in the further development of soliton theory based on symbolic computation. As a core issue of nonlinear science, the soliton theory has always been the hot research topic, and absorbed the close attention of numerous researchers in the world. Both in theoretical research and in pratical application, the study on soliton solutions of nonlinear models has been an important and difficult issue, which may be encountered in every field of science.
Based on symbolic computation and the development status of this area, the present dissertation is carried out to analytically investigate the soliton solutions of some physically important nonlinear models, such as generalizing the known methods that are used to construct the exact soliton solutions, analyzing the stabilities and structures of solitons, studying the interactions between solitonic waves, including elastic and inelastic ones. The research work of the dissertation mainly includes the following aspects:
①With the help of symbolic computation, the traditional bilinear method is extended to dealing with the more complicated higher-order and higher-dimensional nonlinear models. The main idea of our extended bilinear method is that, through the bilinear transformation derived by employing the balancing-act method, the original model is transformed into a homogeneous equation, which is then solved directly by the format-parameter expansion method. One key point of the extended method is that the unimaginable complicated calculations involved in the solving process with the increase of the order of solution, are easily done via symbolic computation. This extended procedure skips the process of equation splitting and bilinearization so as to avoid the possibility of introducing additional limitation on the structure of the soliton solutions. Besides, the computerized function of bilinear operation is defined in software Mathematica, and this greatly conveniences our calculation.
②Generally speaking, most of the generalized variable-coefficient models are not completely integrable. But when some constraints on the variable-coefficient functions are satisfied, the variable-coefficient models may possess some integrabel properties, such as multi-soliton solutions, Backlund transformation, and Lax pair. With the help of symbolic computation, the multi-soliton solutions of a nonisospectral and variable-coefficient Korteweg-de Vries model and a generalized Kadomtsev-Petviashvili model with variable coefficients are investigated by using the extended bilinear method. Based on their bilinear forms and by means of properties of the bilinear operator, the Backlund transformations in both bilinear form and Lax pair form are derived. Furthermore, starting from the Backlund transformations, the Lax pairs and the nonlinear superposition formula are obtained. Finally, some figures are plotted to analyze the effects of the variable coefficients on the stabilities and propagation characteristics of the solitonic waves.
③In recent years, more and more interests have been attracted on the research of partially coherent interactions of multi-structures for nonlinear Schrodinger models. Hereby, via symbolic computation, a generalized (1+1)-dimensional coupled nonlinear Schrodinger model is investigated. With the obtained multi-soliton solutions, some propagation and interaction properties of the solitons are discussed simultaneously. Moreover, some figures are plotted to graphically analyze the pairwise collisions and partially coherent interactions among three solitons.
④Recent developments have already revealed that soliton collisions encountered in some coupled nonlinear Schrodinger models possess novel features, such as the shape changes with intensity redistributions or energy exchanges. It is a desirable feature of such collisions that solitons during interaction can transform energy, in other words, solitons can be amplified without using other techniques. This type of collision-based amplification has aroused increasing interests because it does not require any external amplification medium, neither does it induce any noise. Motivated by this, the shape-changing collision of solitons in a generalized coupled higher-order nonlinear Schrodinger model is investigated through asymptotic analysis on the various features underlying the desirable collision between two solitons.
⑤It is of great significance to study localized coherent structures and their interaction behavior in (2+1)-dimensional integrable nonlinear systems. With the aid of symbolic computation, under investigation is the (2+1)-dimensional dispersive long wave equations. Based on the analytic multi-soliton solutions derived by the extended bilinear method, some novel soliton interaction behaviors for this model are revealed. At the same time, the main propagation properties are studied through the graphical illustration. Our analysis shows that the novel interactions include the fission phenomenon, which is a kind of non-elastic behavior. Compared with the completely elastic interaction of solitons, a simple answer to why the fission phenomenon exists is given from the viewpoint of mathematics.
In conclusion, combined with the technique of symbolic computation, the bilinear method is extended to investigate some complicated higher-order or higher-dimensional models. These models are of practical importance in various branches of physics like fluid dynamics, fiber communications, superconductors, Bose-Einstein condensates, plasma physics, atmosphere and oceans. The study of the models includes deriving soliton solutions, B(a|¨)cklund transformations and Lax pairs, where symbolic computation becomes a necessary and helpful implement. Furthermore, through computer simulation in software Mathematica, the stabilities, propagation characteristics, and the elastic and inelastic interaction properties of the solitonic waves are all graphically analyzed. Moreover, their underlying physical applications are discussed in detail. It is expected that the analytic results and relevant discussions in this dissertation will be observed in the future laboratory experiments, and will be helpful to the future studies.
在本论文中,作者结合本研究领域的发展现状,基于计算机符号计算对若干有着实际物理应用的非线性模型孤子解进行了研究,主要内容包括两个方面,研究构造非线性模型精确孤子解的方法;研究非线性模型孤子解的时间和空间特性。具体包括如下几个方面:
①基于计算机符号计算将传统的双线性求解方法进行推广,使其适合于求解更复杂的高维和高阶非线性模型。推广的双线性方法借助于齐次平衡思想推导出非线性模型的双线性变换,将目标模型转化成易于求解的齐次方程。同时在求解过程中略去了将齐次方程分块和双线性化的过程,这样就避免了不适当的分块可能产生的对于模型孤子解的结构的限制。同时,利用计算机符号计算给出双线性算子的计算式,大大方便了我们利用双线性方法处理相关问题。
②一般来讲,广义的时间变系数模型大多是不可积的,但是,当各个系数函数满足了一定的约束条件时却是可积的,也就是存在多孤子解,B(a｜¨)cklund变换,Lax对等。本文将推广的双线性方法应用于求解广义的变系数Korteweg-de Vries模型和变系数Kadomtsev-Petviashvili模型的精确多孤子解;基于这些方程的双线性形式,利用双线性算子的性质,推导出其对应解的自B(a｜¨)cklund变换,包括双线性形式和Lax形式的自B(a｜¨)cklund变换;进一步地,从解的双线性形式自B(a｜¨)cklund变换出发,可以得到模型的Lax对,或者解的非线性叠加公式;对于求得的多孤子解,通过绘图分析研究模型的各个变系数对于解的稳定性和传输方式的影响。
③近年来,对于非线性Schr(o｜¨)dinger模型多孤子间的部分相干相互作用的研究引起了人们的兴趣。本文我们借助于计算机符号计算和推广的双线性方法,研究一个在现代光纤通信领域中有着重要应用的广义(1+1)维耦合非线性Schr(o｜¨)dinger模型的孤子解。并从得到的解出发,研究随着时间的发展各个变系数函数对于孤子稳定性的影响,包括传播速度和振幅,重点研究孤子间的相互作用,包括部分相干相互作用和孤波间的成对碰撞。
④部分非线性Schr(o｜¨)dinger模型的孤子间存在一种特殊的形变碰撞,由于在这种碰撞过程中发生了能量再分配,而越来越受到人们的关注。借助于计算机符号计算,我们将研究一个广义耦合高阶非线性Schr(o｜¨)dinger模型间的这种奇妙的非弹性相互作用。该研究在光纤中的一个重要应用就是能量放大器,由于这种放大过程不需要外界的干扰和新能量的补充,因而很稳定,拥有广阔的实际应用前景。
⑤对于非线性模型局域孤子的研究越来越积极和发展起来,各种局域相干结构也被不断地发现。同时,做为孤子间的一种非弹性相互作用,孤子的裂变和聚变也开始引起人们的关注。我们对于(1+1)维孤子解的相互作用已经研究的很深,并取得了很大的成就,但是高维方程的孤子解之间的相互作用确实很复杂的,而且维数越高,相互作用就越复杂。本文借助于计算机符号计算和计算机模拟,对于色散长波模型,在多孤子解的基础上,通过计算机软件模拟绘图的方式集中分析研究不同的局域相干结构间的这种裂变型非弹性相互作用,并且从数学角度简要分析裂变现象产生的原因。
综上所述,本文将计算机符号计算与传统的双线性方法相结合,并将其推广应用于求解复杂的高维、高阶非线性模型中。对于几个有着典型物理意义的非线性模型进行了解析研究,包括构造多孤子解,B(a｜¨)cklund变换,Lax对等;并利用Mathematica计算软件的模拟绘图功能对求得孤子解的时间发展特性,相互之间的弹性和非弹性相互作用,以及它们潜在的物理应用进行了深入分析。希望本文的理论研究结果在将来的实验观察中能够得到证实,并对于相关研究有所帮助。
With the quick development of information technology, symbolic computation, as a new branch of artificial intelligence, has become perfect and ripe gradually, and has been applied to the research of nonlinear science. Nowadays, because of its powerful ability in exactly dealing with the complicated and tedious calculations, and its convenient and direct practicability, symbolic computation has grown into a necessary and wonderful assistant tool for the study of nonlinear science. Especially in the latest decade, much progress has already been made in the further development of soliton theory based on symbolic computation. As a core issue of nonlinear science, the soliton theory has always been the hot research topic, and absorbed the close attention of numerous researchers in the world. Both in theoretical research and in pratical application, the study on soliton solutions of nonlinear models has been an important and difficult issue, which may be encountered in every field of science.
Based on symbolic computation and the development status of this area, the present dissertation is carried out to analytically investigate the soliton solutions of some physically important nonlinear models, such as generalizing the known methods that are used to construct the exact soliton solutions, analyzing the stabilities and structures of solitons, studying the interactions between solitonic waves, including elastic and inelastic ones. The research work of the dissertation mainly includes the following aspects:
①With the help of symbolic computation, the traditional bilinear method is extended to dealing with the more complicated higher-order and higher-dimensional nonlinear models. The main idea of our extended bilinear method is that, through the bilinear transformation derived by employing the balancing-act method, the original model is transformed into a homogeneous equation, which is then solved directly by the format-parameter expansion method. One key point of the extended method is that the unimaginable complicated calculations involved in the solving process with the increase of the order of solution, are easily done via symbolic computation. This extended procedure skips the process of equation splitting and bilinearization so as to avoid the possibility of introducing additional limitation on the structure of the soliton solutions. Besides, the computerized function of bilinear operation is defined in software Mathematica, and this greatly conveniences our calculation.
②Generally speaking, most of the generalized variable-coefficient models are not completely integrable. But when some constraints on the variable-coefficient functions are satisfied, the variable-coefficient models may possess some integrabel properties, such as multi-soliton solutions, Backlund transformation, and Lax pair. With the help of symbolic computation, the multi-soliton solutions of a nonisospectral and variable-coefficient Korteweg-de Vries model and a generalized Kadomtsev-Petviashvili model with variable coefficients are investigated by using the extended bilinear method. Based on their bilinear forms and by means of properties of the bilinear operator, the Backlund transformations in both bilinear form and Lax pair form are derived. Furthermore, starting from the Backlund transformations, the Lax pairs and the nonlinear superposition formula are obtained. Finally, some figures are plotted to analyze the effects of the variable coefficients on the stabilities and propagation characteristics of the solitonic waves.
③In recent years, more and more interests have been attracted on the research of partially coherent interactions of multi-structures for nonlinear Schrodinger models. Hereby, via symbolic computation, a generalized (1+1)-dimensional coupled nonlinear Schrodinger model is investigated. With the obtained multi-soliton solutions, some propagation and interaction properties of the solitons are discussed simultaneously. Moreover, some figures are plotted to graphically analyze the pairwise collisions and partially coherent interactions among three solitons.
④Recent developments have already revealed that soliton collisions encountered in some coupled nonlinear Schrodinger models possess novel features, such as the shape changes with intensity redistributions or energy exchanges. It is a desirable feature of such collisions that solitons during interaction can transform energy, in other words, solitons can be amplified without using other techniques. This type of collision-based amplification has aroused increasing interests because it does not require any external amplification medium, neither does it induce any noise. Motivated by this, the shape-changing collision of solitons in a generalized coupled higher-order nonlinear Schrodinger model is investigated through asymptotic analysis on the various features underlying the desirable collision between two solitons.
⑤It is of great significance to study localized coherent structures and their interaction behavior in (2+1)-dimensional integrable nonlinear systems. With the aid of symbolic computation, under investigation is the (2+1)-dimensional dispersive long wave equations. Based on the analytic multi-soliton solutions derived by the extended bilinear method, some novel soliton interaction behaviors for this model are revealed. At the same time, the main propagation properties are studied through the graphical illustration. Our analysis shows that the novel interactions include the fission phenomenon, which is a kind of non-elastic behavior. Compared with the completely elastic interaction of solitons, a simple answer to why the fission phenomenon exists is given from the viewpoint of mathematics.
In conclusion, combined with the technique of symbolic computation, the bilinear method is extended to investigate some complicated higher-order or higher-dimensional models. These models are of practical importance in various branches of physics like fluid dynamics, fiber communications, superconductors, Bose-Einstein condensates, plasma physics, atmosphere and oceans. The study of the models includes deriving soliton solutions, B(a|¨)cklund transformations and Lax pairs, where symbolic computation becomes a necessary and helpful implement. Furthermore, through computer simulation in software Mathematica, the stabilities, propagation characteristics, and the elastic and inelastic interaction properties of the solitonic waves are all graphically analyzed. Moreover, their underlying physical applications are discussed in detail. It is expected that the analytic results and relevant discussions in this dissertation will be observed in the future laboratory experiments, and will be helpful to the future studies.
引文
[1]Barnett M P,Capitani J F,Von Zur Gathen J,et al.Symbolic calculation in chemistry:selected examples.Int.J.Quantum Chem.,100,2004,80-104.
[2]Yan Z Y,Zhang.H Q.Symbolic computation and new families of exact soliton-like solutions to the integrable Broer-Kaup(BK) equations in(2+1)-dimensional spaces.J.Phys.A,34,2001,1785-1792.
[3]Duffy B R,Parks E J.Travelling solitary wave solutions to a seventh-order generalized KdV equation.Phys.Lett.A,214,1996,271-272.
[4]Hong W P.Comment on "Spherical Kadomtsev-Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computation".Phys.Lett.A,361,2007,520-522.
[5]Das G,Sarma J.Comment on "a new mathematical approach for finding the solitary waves in dusty plasma".Phys.Plasmas,6,1999,4394-4397.
[6]范恩贵.可积系统与计算机代数,科学出版社,2004.
[7]衷仁保,马建.计算机代数学,科学出版社,1996.
[8]张树功,刘停战,冯果忱.计算机代数基础,吉林大学出版社,1997.
[9]吴文俊.数学机械化,科学出版社,1999.
[10]周梦.计算代数与应用,武汉大学出版社,2002.
[11]陶庆生.计算机代数及其应用,浙江大学出版社,1991.
[12]赵东方.数学模型与计算,科学出版社,2007.
[13]张韵华.符号计算系统Mathematica教程,科学出版社,2001.
[14]田播.计算机符号计算在非线性研究中的若干应用[学位论文],北京航空航天大学,2003.
[15]李娟.基于计算机符号计算的若干变系数非线性模型可积性质的研究[学位沦文],北京邮电大学,2008.
[16]王爱银.符号计算对计算有理矩阵指数的实用方法[学位论文],新疆大学,2005.
[17]郭柏灵.非线性演化方程,上海科技教育出版社,1995.
[18]刘式达,刘式适.孤波和湍流,上海科技教育出版社,1995.
[19]刘式适,刘式达,谭本.非线性大气动力学,国防工业出版社,1996.
[20]郭柏灵,苏风秋.孤立子,辽宁教育出版社,1998.
[21]倪皖荪,魏容爵.水槽中的孤波,上海科技教育出版社,1998.
[22]谷超豪,胡和生,周子翔.孤立子中的Darboux变换及其几何应用,上海科技出版社,1999.
[23]闫振亚.复杂非线性波的构造性理论及其应用,科学出版社,2007.
[24]陆同兴.非线性物理概论,中国科学技术大学出版社,2002.
[25]刘式达,刘式适.物理学中的非线性方程,北京大学出版社,2002.
[26]潘祖梁.非线性问题的数学方法及其应用,浙江大学出版社,2002.
[27]庞小峰.非线性量子力学理论,重庆出版社,1994.
[28]罗德海.大气中大尺度包络孤立子理论与阻塞环流,气象出版社,1999.
[29]王德焴,吴德金,黄光力.空间等离子体中的孤波,上海科技教育出版社,2000.
[30]杨伯君,赵玉芳.高等数学物理方法,北京邮电大学出版社,2003.
[31]庞小峰.孤子物理学,四川科学技术出版社,2003.
[32]黄景宁,徐济仲,熊吟涛.孤子、概念、原理和应用,高等教育出版社,2004.
[33]陈登远.孤子引论,科学出版社,2006.
[34]楼森岳,唐晓艳.非线性数学物理方法,科学出版社,2006.
[35]Russell J S.Report on waves,14th meeting of the British association for advancement of science.London,John Murray,1844,311-390.
[36]Boussinesq M J.Th(?)orie de I'intumescencs liquide appel(?)e onde solitaire ou de translation se propageant dans un canal rectangulaire.C.R.Acad.Sci.Paris S(?)r,72,1871,755-759.
[37]Rayleigh L.On waves.Phil.Mag.,1,1876,257-279.
[38]Korteweg D J,de Vriss G.On the change of form of long waves advancing in a rectangular canal,and on a new type of long stationary waves.Philos.Mag.Ser.,39,1895,422-443.
[39]Fermi E,Pasta J,Ulam S.Studies of nonlinear problems I.Los Alamos Report LA-1940,1955.
[40]Toda M.Wave propagating in anharmonic lattices.J.Phys.Soc.Jpn.,23,1967,501-506.
[41]Perring K J and Skyrme T H R.A model unified field equation.Nucl.Phys.,31,1962,550-555.
[42]Zabusky N J,Kruskal M D.Interaction of"solitons" in a collisionless plasma and the recurrence of initial states.Phys.Rev.Lett.,15,1965,240-243.
[43]Li Y S,Ma W X,Zhang J E.Darboux transformations of classical Boussinesq system and its new solutions.Phys.Lett.A,275,2000,60-66.
[44]谷内俊弥,西原功修.非线性波动(徐福元,张家泰,丁启明译),原子能出版社,1981.
[45]邓瑞基,陈自力,唐小弟.基于Mathematica的符号计算方法.中南林学院学报,26,2006,90-93.
[46]吴剑,胡波.掌握和精通Mathematica4.0,机械工业出版社,2001.
[47]徐安农.Mathematica数学实验,电子工业出版社,2004.
[48]阳明盛,林建华.Mathematica基础及数学软件,大连理工大学出版社,2006.
[49]张宝善.Mathematica符号运算与数学实验,南京大学出版社,2007.
[50]沃尔夫雷姆.Mathematica全书(赫孝良,周义仓译),两安交通大学出版社,2002.
[51]陈杰.Matlab宝典,电子工业出版社,2007.
[52]张德丰.Matlab数值分析与应用,国防工业出版社,2007.
[53]钟益林,彭乐群,刘炳文.常微分方程及其Maple,Matlab求解,清华大学出版社,2007.
[54]刘辉.Maple符号处理及应用,国防工业出版社,2001.
[55]何青,王丽芬.Maple教程,科学出版社,2006.
[56]Buchberger B,Groebner B.An algrorithmic method in polynomial ideal theory.Recent trends in multidimensional systems theory.Reidel Publishing Company,1985.
[57]Ibrahim R S.Tanh-travelling wave solutions,truncated Painlev(?) expansion and reduction of Bullough-Dodd equation to a quadrature in magnetohy drodynamic equilibrium.Chaos Solitons and Fractals,16,2003,675-683.
[58]Hong W P,Park S H.Extended tanh-methods and solitonic solutions of the general shallow water wave models.Int.J.Mod.Phys.C,15,2004,363-370.
[59]Parks E J,Duffy B R.An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations.Comput.Phys.Commun.,98,1996,288-300.
[60]Li Z B,Wang M L.Travelling wave solutions to the two-dimensional KdV-Burgers equation.J.Phys.A,26,1993,6027-6031.
[61]Yang X D,Ruan H Y,Lou S Y.A Maple package on symbolic computation of conserved densities for(1+1)-dimensional nonlinear evolution systems.Commun.Theor.Phys.,47,2007,961-968.
[62]柳银萍.基于吴方法的孤波自动求解软件包及其应用[学位论文],华东师范大学,2001.
[63]Liu Y P,L I Z B.A Maple package for finding exact solitary wave solutions of coupled nonlinear revolution equations.Comput.Phys.Comet.,155,2002,65-70.
[64]Gardner C S,Greene J G,Kruskal G D,et al.Method for solving the Korteweg-de Vries equation.Phys.Rev.Lett.,19,1967,1095-1097.
[65]Ablowitz M J,Clarkson P A.Solitons,nonlinear evolution equations and inverse scattering.Cambridge University Press,1991.
[66]Wahlquist H D,Estabrook F B.B(a|¨)cklund transformation for solutions of the Korteweg-de Vries equation.Phys.Rev.Lett.,31,1973,1386-1390.
[67]Zait R A.B(a|¨)cklund transformations and solutions for some evolution equations.Phys.Ser.,57,1998,545-548.
[68]Tian B,Gao Y T.(2+1) dimensional Haragus-Courcelle-Ilichev model for the liquid surface waves in the presence of sea ice or surface tension,B(a|¨)cklund transformation,exact solutions and possibly observable effects.Eur.Phys.J.B,42,2004,441-450.
[69]Tian L X,Yin J L.Stability of multi-compaction solutions and B(a|¨)cklund transformation in K(m,n,1).Chaos Solitons and Fractals,23,2005,159-169.
[70]Li B,Chen Y,Zhang H Q.Auto-B(a|¨)cklund transformation and exact solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order.Phys.Lett.A,305,2002,377-382.
[71]Khater A H,Callebaut D K,Malfliet W,et al.B(a|¨)cklund transformations and exact solutions for some nonlinear evolution equations in solar magnetostatic models.J.Comput.Appl.Math.,140,2002,435-467.
[72]Konno K,Wadati M.Simple derivation of B(a|¨)cklund transformation from Riccati form of a inverse method.Prog.Theor.Phys.,53,1975,1652-1656.
[73]Khater A H,Ibrahim R S,El-Kalaawy O H,et al.B(a|¨)cklund transformation and exact soliton solutions for some nonlinear evolution equations of the ZS/AKNS system.Chaos Solitons and Fractals,9,1998,1847-1855.
[74]Clarkson P A.Painlev(?) analysis and the complete integrability of a generalized variable-coefficient Kadomtsev-Petviashvili equation.IMA J.Appl.Math.,44,1990,27-53.
[75]Yana Z Y.Painlev(?) analysis,auto-B(a|¨)cklund transformations and exact solutions for a simplified model for reacting mixtures.Physica A,2003,326,344-359.
[76]Weiss J,Tabor M,Camevale G.The Painlev(?) property of partial differential equations.J.Math.Phys.,24,1983,522-526.
[77]Hlavat(?) L.Painlev(?) analysis ofnonautonomous evolution equations.Phys.Lett.A,128,1988,335-338.
[78]Lou S Y,Hu X B.Broer-Kaup system from Darboux transformation related symmetry constrants of Kadomtsev-Petviashvili equation.Commun.Theor.Phys.,29,1998,145-148.
[79]Li Y S,Zhang J E.Darboux transformations of classical Boussinesq system and its multi-soliton solutions.Phys.Lett.A,284,2001,253-258.
[80]Zhou L J.Darboux transformation for the nonisospectral AKNS system.Phys.Lett.A,345,2005,314-322.
[81]Hu H C,Liu Q P.New Darboux transformation for Hirota-Satsuma coupled KdV system.Chaos Solitons and Fractals,17,2003,921-928.
[82]Deng S F.Darboux transformations for the isospectral and nonisospectral mKP equation.Physica A,382,2007,487-493.
[83]Est(?)vez P G.Darboux transformation and solutions for an equation in 2+1dimensions.J.Math.Phys.,40,1999,1406-1419.
[84]Hu H C,Lou S Y.New interaction property of(2+1)-dimensional localized excitations from Darboux transformation.Chaos Solitons and Fractals,24,2005,1207-1216.
[85]Hirota R.Exact solution of the KdV equation for multiple collisions of solitons. Phys.Rev.Lett.27,1971,1192-1994.
[86]Hirota R.A new form of B(a|¨)cklund transformations and its relation to the inverse scattering problem.Progr.Theor.Phys.Suppl.52,1974,1498-1512.
[87]Hirota R.The direct method in soliton theory,(translated from the Jpn.ese and edited by Nagai A,Nimmo J,Gilson C),Cambridge University Press,2004.
[88]Li L L,Tian B,Zhang C Y,et al.On a generalized Kadomtsev-Petviashvili equation with variable coefficients via symbolic computation.Phys.Scr.,76,2007,411-417.
[89]Li L L,Tian B,Zhang C Y,et al.N-soliton solutions,auto-B(a|¨)cklund transformation and Lax pair for a nonisospectral and variable-coeffcient Korteweg-de Vries equation via symbolic computation.Int.J.Mod.Phys.B,Accepted.
[90]Li L L,Tian B,Zhang H Q,et al.Soliton solutions,pairwise collisions and partially coherent interactions for a generalized(1+1)-dimensional coupled nonlinear Schr(o|¨)dinger system via symbolic computation.Z.Naturforsch.,63a,2008,679-687.
[91]Hirota R,Satsuma J.Nsoliton solution of the KdV equation with loss and nonuniformity terms.J.Phys.Soc.Jpn.,41,1976,2141-2142.
[92]Matsuno Y.Bilinear Transformation Methods,Academic Press,1984.
[93]Hirota R,Hu X B,Tang X Y.A vector potential KdV equation and vector Ito equation:soliton solutions,bilinear B(a|¨)cklund transformations and Lax pairs J.Math.Anal.Appl.,288,2003,326-348.
[94]Hirota R,Satsuma J.A simple structure of superposition formula of the B(a|¨)cklund transformation.J.Phys.Soc.Jpn.,45,1978,1741-1750.
[95]Tian B,Gao Y T.Variable-coefficient balancing-act method and variable-coefficient KdV equation from fluid dynamics and plasma physics.Euro.Phys.J.B,22,2001,351-360.
[96]Zhang J F.Homogeneous balance method and chaotic and fractal solutions for the Nizhnik-Novikov-Veselov equation.Phys.Lett.A,313,2003,401-407.
[97]Zhang C Y,Yao Z Z,Zhu H W,et al.Exact analytic N-soliton-like solution in Wronskian form for a generalized variable-coefficient Korteweg-de Vries model from plasmas and fluid dynamics.Chin.Phys.Lett.,24,2007,1173-1176.
[98]Zhang C,Tian B,Li L L,et al.Analytical analysis on a generalized (2+1)-dimensional variable-coefficient Korteweg-de Vries equation using symbolic computation.Int.J.Mod.Phys.B,Accepted.
[99]Clarkson P A,Kruskal M D.New similarity solutions of Boussinesq equation.J.Math.Phys.,30,1989,2201-2213.
[100]Lou S Y.A note on the new similarity reductions of the Boussinesq equation.Phys.Lett.A.151,1990,133-135.
[101]Clarkson P A.Nonclassical symmetry reductions for the Boussinesq equation.Chaos Soliton and Fractals,5,1995,2261-2301.
[102]Saccomandi G.Potential symmetries and direct reduction methods of order two.J.Phys.A.30,1997,2211-2217.
[103]Olver P J.Application of Lie groups to differential equation 2nd ed.,Graduate Texts in Mathematics,Springer,New York,1993.
[104]Bluman G W,Kumei S.Symmetries and differential equation.Appl.Math.Sci.,Springer,Berlin,1989.
[105]Bluman G W,Cole J D.The general similarity solution of the heat equation.J.Math.Mech.,18,1969,1025-1042.
[106]Morris H C.A prolongation structure for the AKNS system and its generalization.J.Math.Phys.,18,1977,533-536.
[107]Ma W X,Strampp W.An explicit symmetry constraint for the Lax pairs and adjoint Lax pairs of AKNS systems.Phys.Lett.A,185,1994,277-286.
[108]Li Y S,Zhang J E.Bidirectional soliton solutions of the classical Boussinesq system and AKNS system.Chaos Soliton and Fractals,16,2003,271-277.
[109]Xu T,Li J,Zhang H Q,et al.New extension of the tanh-function method and application to the Whitham-Broer-Kaup shallow water model with symbolic computation.Physics Letters A,369,2007,458-463.
[110]Yomba E.The extended Fan's sub-equation method and its application to KdV-MKdV,BKK and variant Boussinesq equations.Phys.Lett.A,336,2005,463-476.
[111]郭玉翠.非线性偏微分方程引论,清华大学出版社,2008.
[112] Khater A H, Abdallah A A, El-Kalaawy O H. B(?)cklund transformations, a simple transformation and exact solutions for dust-acoustic solitary waves in dusty plasma consisting of cold dust particles and two-temperature isothermal ions. Phys.plasmas, 6,1999,4542-4547.
[113] Musette M, Verhoeven C. Nonlinear superposition formula for the Kaup-Kuper-shmidt partial differential equation. Physica D, 144,2000, 211-220.
[114] Boiti M, Leon J J P, Martina L, et al. Scattering of localized solitons in the plane.Phys. Lett. A, 132,1988,432- 439.
[115] Ruan H Y, Chen Y X. Dromion interactions of (2+1 )-dimensional nonlinear evolution equations. Phys. Rev. E, 62,2000, 5738-5344.
[116] Tang X Y, Li J M, Lou S Y. Reflection and reconnection interactions of resonant dromions. Phys. Scr., 75,2007, 201-205.
[117] Ruan H Y, Lou S Y. Dromion structure of (2+1)-dimensional Sine-Gordon system.Commun. Theor. Phys., 32,1999,109-114.
[118] Alagesan T, Uthayakumar A, Porsezian K. The generalisation of integrable (2+1)-dimensional dispersive long-wave equations. J. Phys. Soc. Jpn., 66,1997,1288-1290.
[119] Isojima S, Willox R, Satsuma J. On various solution of the coupled KP equation. J.Phys. A, 35,2002,6893-6909.
[120] Lou S Y. Symmtry analysis and exact solutions of the 2+1 dimensional Sine-Gordon system. J. Math. Phys., 41,2000,6509-6524.
[121] Tang X Y, Lou S Y, Zhang Y. Localized exicitations in. (2+1)-dimensional systems. Phys. Rev. E, 66,2002, 046601-046605.
[122] Lou S Y. Dromions, dromions lattice, breathers and instantons of the Davey-Stewartson equation. Phys. Scr., 65, 2002, 7-12.
[123] Ying J P, Lou S Y. Abundant coherent structures of the (2+1)-dimensional Broer-Kaup-Hupershmidt equation, Z. Naturforsch., 56a, 2001,619-625.
[124] Camassa R, Holm D D. An integrable shallow water equation with peaked solitons.Phys. Rev. Lett., 71,1993,1661-1664.
[125] Rosenau P, Hyman J. M. Compactons: solitons with finite wavelength. Phys. Rev.Lett., 70,1993, 564-567.
[126]Lu Z M,Tian E M,Grimshaw R.Interaction of two lump solitons described by the Kadomtsev-Petviashvili I equation.Wave Motion,40,2004,123-135.
[127]Fokas A S,Pelinovsky D E,Sulem C.Interaction of lumps with a line soliton for the DSII equation.Physica D,152,2001,189-198.
[128]Loginov E K.Multi-instantons in higher dimensions and superstring solitons.SIGMA 1,2005,002-005.
[129]Rajaraman R.Solitons and instantons:an introduction to solitons and instantons in quantum field theory,Elsevier,North-Holland,1982.
[130]Zhang J F,Meng J P,Zheng C L,et al.Folded solitary waves and foldons in the (2+1)-dimensional breaking soliton equation.Chaos,Solitons and Fractals,20,2004,523-527.
[131]Dai C Q,Ni Y Z.Novel interactions between semi-foldons of the (2+1)-dimensional Boiti-Leon-Pempinelli equation.Phys.Scr.,74 584-590.
[132]沈守枫.高维微分.差分模型的Virasoro对称子代数,多线性变量分离解和局域激发模式.物理学报,55,2006,5606-5610.
[133]Wu J R,Keolian R,Rudnick I,et al.Observation of a non-propagating hydrodynamic soliton.Phys.Rev.Lett.,47,1984,1421-1424.
[134]Fokas A S,Santini P M.On the simplest integrable equation in 2+1.Phyisica D,44,1990,99-104.
[135]Hietarinta J,Hirota R.Multidromion solutions to the Davey-Stewartson equation.Phys.Lett.A,145,1990,237-244.
[136]Zhang J F,Xu G D.Generalized dromion solutions of a new higher dimensional soliton equation.Commun.Nonlinear Sci.Numer.Simul.,6,2001,97-101.
[137]Hietarinta J.One-dromion solutions for generic classes of equations.Phys.Lett.A,149,1990,113-118.
[138]Radha R,Lakshmanan M.Singularity analysis and localized coherent structures in (2+1)-dimensional generalized Korteweg-de Vries equations.J.Math.Phys.,35,1994,4746-4756.
[139]Radha R,Lakshmanan M.Dromion like structures in the(2+1)-dimensional breaking soliton equation.Phys.Lett.A,1995,197,7-12.
[140]Radha R,Lakshmanan M.Exotic coherent Structures in the(2+1)-dimensional long dispersive wave equation.J.Math.Phys.,38,1997,292-299.
[141]Radha R and Lakshmanan M.A New Class of induced Localized Coherent Structures in the(2+1)-dimensional scalar nonlinear Schr(o|¨)dinger equations.J.Phys.A,30,1997,3229-3232.
[142]Radha R and Lakshmanan M.Localized coherent structures and integrability in a generalized(2+1)-dimensional nonlinear Schrodinger equation.Chaos,Solitons and Fractals,8,1997,17-25.
[143]Lou S Y.Dromion-like structures in a(3+1)-dimensional KdV type equation.J.Phys.A,29,1996,5989-6001.
[144]Ruan H Y,Lou S Y.Higher dimensional dromion structures:Jimbo-Miwa-Kadomtsev-Petviashvili system.J.Math.Phys.,38,1997,3123-3136.
[145]Lou S Y.Generalized dromion solutions of the(2+1) -dimensional KdV equation.J.Phys.A,28,1995,7227-7232.
[146]Lou S Y.On the dromion solution of a(2+1) -dimensional KdV-type equation.Commun.Theor.,26,1996,487-490.
[147]Radha R,Lakshmanan M.Generalized dromions in the(2+1)-dimensional long dispersive wave(2LDW) and scalar nonlinear Schrodinger(NLS) equations.Chaos,Solitons and Fractals,10,1999,1-4.
[148]Zhang J F.Generalized dromions of the(2+1)-dimensional nonlinear Schrodinger equations.Commun.Nonlinear Sci.Numer.Simul.,6,2001,50-53.
[149]Lou S Y.On the coherent structures of Nizhnik-Novikov-Veselov equation.Phys.Lett.A,277,2000,94-100.
[150]Lou S Y,Ruan H Y.Revisitation of the localized excitations of the 2+1 dimensional KdV equarion.J.Phys.A,34,2001,305-316.
[151]Ruan H Y,Chert Y X.Ring solitons,dromions,breathers and instantons of the NLS equation.Acta Phys.Sin.50,2001,586-591.
[152]Kadomtsov B B,Petviashvili V I.On the stability of solitary waves in weakly dispersive media.Soy.Phys.Dokl.,15,1970,539-541.
[153]Davey A,Stewartson K.On three-dimensional packets of surface waves.Proc.R.Soc.A,338,1974,101-110.
[154]Zhang J F,Huang W H,Zheng C L.Coherent soliton structures of a new (2+1)-dimensional evolution equation.Acta Phys.Sin.,51,2002,2676-2682.
[155]Matveev V B,Salle M A.Darboux transformations and solitons,Springer,Berlin,1991.
[156]Kako F,Yajima N.Interaction of ion-acoustic solitons in two-dimensional space.J.Phys.Soc.Jpn.,49,1980,2063-2071.
[157]Hasegawa A,Tappert F.Transmission of stationary nonlinear optical pulses in dispersion dielectric fibers.App.Phys.Lett.,23,1973,142-144.
[158]贾东方,余震虹,谈斌等译.非线性光纤光学原理及应用,电子工业出版社,2002.
[159]Law C T,Swartzlander G A.Optical vortex solitons and the stability of dark soliton stripes.Optics Letts.18,1993,586-588.
[160]L.D.Carr.Solitons in Bose-Einstein Condensates,American Physical Society,34th Meeting of the Division of Atomic,Molecular and Optical Physics,2003.
[161]Denschlag J,Simsarian J E,Feder D L,et al.Science,287,2000,97-101.
[162]Chan W L,Li K S.Nonpropagating solitons of the variable coefficient and nonisospeetral KdV equation.J.Math.Phys.30,1989,2521-2526.
[163]Chan W L,Zhang X.Symmetries,conservation laws and Hamiltonian structures of the nonisospectral and variable coefficient KdV and MKdV equations.J.Phys.A 28,1995,407-419.
[164]Brugarino T.Painlev(?) property,auto-B(a|¨)cklund transformation,and reduction to the standard form for the Korteweg-de Vries equation.J.Math.Phys.30,1989,1013-1015.
[165]Zhang J F,Chen F Y.Truncated expansion method and new exact soliton -like solution of the general variable coefficient KdV equation.Phys.Soc.,50,2001,1648-1650.
[166]Zhao X Q,Tang D B,Wang M L.New soliton-like solutions for KdV equation with variable coefficient.Phys.Lett.A,346,2005,288-291.
[167]Calogero F,Degasperis A.Extension of the spectral transform method for solving nonlinear evolution equations.Lett.Nuovo Cimento 22,1978,131-137.
[168]Wang Y L,Zhou Z X,Jiang X Q,et al.Cylindrical Kadomtsev-Petviashvili equation for relativistically magnetosonic solitary wave in the collisionless plasma. Phys. Lett. A, 355,2006,386-389.
[169] Wang Y Y, Zhang J F. Variable-coefficient KP equation and solitonic solution for two-temperature ions in dusty plasma. Phys. Lett. A, 352,2006,155-162.
[170] Gao Y T, Tian B. Phys. Cylindrical Kadomtsev-Petviashvili model, nebulons and symbolic computation for cosmic dust ion-acoustic waves. Lett. A, 349,2006,314-319.
[171] Tian B, Gao Y T. Symbolic computation on cylindrical-modified dust-ion-acoustic nebulons in dusty plasmas. Phys. Lett. A, 362,2007,283-288.
[172] Johnson R S. Water waves and Korteweg-de Vries equations. J. Fluid Mech. 97,1980,701-719.
[173] Milewski P. Long wave interaction over varying topography. Physica D, 123, 1998,36-47.
[174] Anders I, Monvel B D A. Asymptotic Solitons of the Johnson Equation. J.Nonlinear Math. Phys., 7,2000,284-302.
[ 175] Gao Y T, Tian B. Some two-dimensional and non-travelling-wave observable effects of the shallow-water waves. Phys. Lett. A, 301, 2002, 74-84.
[176] Shen S F, Zhang J, Ye C E, et al. New exact solutions of the (2+1)-dimensional KdV equation with variable coefficients. Phys. Lett. A, 337, 2005,101-106.
[177] Oevel W, Steeb W H. Painlev(?) analysis for a time-dependent KP equation. Phys.Lett. A, 103,1984,239-242.
[ 178] Emmanuel Y. Abundant families of Jacobi elliptic function-like solutions for a generalized variable coefficients 2D KdV equation via the extended mapping method. Phys. Lett. A, 349,2006, 212-219.
[179] Ablowitz M J and Clarkson P A. Solitons, Nonlinear evolution equations and inverse scattering, Cambridge University Press, 1991.
[180] Xie Y C. Exact solutions of the Wick-type stochastic Kadomtsev-Petviashvili equations. Chaos Solitons Fractals, 21,2004,473-480.
[181] Malik H K. Effect of electron inertia on KP solitons in a relativistic plasma. Physica D, 125,1999,295-301.
[ 182] Duan W S. Acoustic mode in a two-ion-temperature warm dusty plasma under the weakly transverse perturbations. Chaos Solitons Fractals, 14,2002,1315-1318.
[183] Lin M M, Duan W S. The Kadomtsev-Petviashvili (KP), MKP, and coupled KP equations for two-ion-temperature dusty plasmas. Chaos Solitons Fractals, 23,2005, 929-937.
[184] Noubissi(?) S, Woafo P. Dynamics of solitary waves over an erodible surface.Physica A,345,2005,9-16.
[185] A. Hasegawa, Optical Solitons in Fibers, Springer, Berlin, 1989.
[186] K. Porsezian and V. C. Kuriakose, Optical Solitons: Theoretical and Experimental Challenges, Springer, Berlin, 2003.
[187] Erbay H A, Erbay S, Dost S. Nonlinear wave modulation in micropolar elastic media-II. Transverse waves. Int. J. Eng. Sci., 29,1991, 859-868.
[188] Kajinaga Y, Tsuchida T, M Wadati. Coupled Nonlinear Schrodinger Equations for Two-component Wave Systems. J. Phys. Soc. Jpn., 67, 1998,1565-1568.
[189] Tsang S C, Chow K W. The evolution of periodic waves of the coupled nonlinear Schr(?)dinger equations. Math. Comput. Simul., 66, 2004, 551-564.
[190] Hacinliyan I, Erbay S. A higher-order model for transverse waves in a generalized elastic solid. Chaos, Solitons and Fractals, 14,2002,1127-1135.
[191] Kanna T, Lakshmanan M, Dinda P T, et al. Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schr(?)dinger equations. Phys.Rev. E, 73, 2006, 026604-026618.
[192] Manakov S V. On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Phys. JETP, 38,1974, 248-253;
[193] Rothenberg J E. Observation of the buildup of modulational instability from wave breaking. Opt. Lett., 16, 1991,18-20.
[194] Wai P K A, Menyuk C R, Chen H H. Stability of solitons in randomly varying birefringent fibers. Opt. Lett., 16, 1991, 1231-1233.
[195] Angelis C D, Wabnitz S, Haelterman M. Bandwidths limits due to polarization multiplexed soliton interactions. Electron. Lett., 29, 1993, 1568-1570.
[196] Zhang H Q, Meng X H, Xu T, et al. Interactions of bright solitons for the (2+1)-dimensional coupled nonlinear Schrodinger equations from optical fibres with symbolic computation. Physica Scripta. Phys. Scr., 75, 2007, 537-542.
[197] Menyuk C R. Nonlinear puke propagation in birefringent optical fibers. IEEE J. Quant.Elec.,23,1987,174-176.
[198]Wadati M,Izuka T,Hisakado M.A coupled nonlinear Schrodinger equation and optical solitons.J.Phys.Soc.Jpn.61,1992,2241-2245.
[199]Lakshmanan M,Karma T,Radhakrishnan R.Shape-changing collisions of coupled bright solitons in birifringent optical fibers.Rep.Math.Phys.46,2000,143-156.
[200]Newboult G K,Parker D F,Faulkner T R.Coupled nonlinear Schr(o|¨)linger equations arising in the study of monomode step-index optical fibers.J.Math.Phys.30,1989,930-936.
[201]Nakkeeran K.Optical Solitons in a new type of coupled nonlinear Schr(o|¨)dinger equations.J.Mod.Opt.48,2001,1863-1867.
[202]Park Q H,Shin H J.Painlev(?) analysis of the coupled nonlinear Schr(o|¨)dinger equation for polarized optical waves in an isotropic medium.Phys.Rev.E,59,1999,2373-2379.
[203]Bindu S G,Mahalingam A,Porsezian K.Phys.Lett.A,286,2001,321-.
[204]Gilson C,Hietarinta J,Nimmo J,et al.Sasa-Satsuma higher-order nonlinear Schr(o|¨)dinger equation and its bilinearization and multisoliton solutions.Phys.Rev.E,68,2003,016614-016623.
[205]Chow K W.Periodic waves for a system of coupled,higher order nonlinear Schr(o|¨)dinger equations with third order dispersion.Phys.Lett.A,308,2003,426-431.
[206]Hirota R.Exact envelope-soliton solutions of a nonlinear wave equation.J.Math.Phys.,14,1973,805-809.
[207]Tasgal R S,Potasek M J.Soliton solutions to coupled higher-order nonlinear Schr(o|¨)dinger equations.J.Math.Phys.,33,1992,1208-1215.
[208]Radhakrishnan R,Lakshmanan M,Daniel M.Bright and dark optical solitons in coupled higher-order nonlinear Schr(o|¨)dinger equations through singularity structure analysis.J.Phys.A,28,1995,7299-7314.
[209]Boiti M,Leon J,Pempinelli F.Spectral transform for a two spatial dimension extension of the dispersive long wave equation.Inverse Problems,3,1987,371-387.
[210]Tang X Y,Lou S Y.Abundant coherent structures of the dispersive long-wave equation in (2+1)-dimensional spaces. Chaos, Solitons and Fractals, 14, 2002,1451-1456.
[211] Tang X Y, Chen C L, Lou S Y. Localized solutions with chaotic and fractal behaviours in a (2+1)-dimensional dispersive long-wave system. J. Phys. A, 35,2002.L293-L301.
[212] Ma W X. Diversity of exact solutions to a restricted Boiti-Leon-Pempinelli dispersive long-wave system. Phys. Lett. A, 319,2003, 325-333.
[213] Xue Y S, Tian B, Li L L, et al. Solitons and localized excitations for the (2+1)-dimensional dispersive long wave system via symbolic computation. Int. J.Mod. Phys. B, Accepted.
[214] Ankiewicz Z, Krolikowski W, Akhmediev N N. Partially coherent solitons of variable shape in a slow Kerr-like medium: Exact solutions. Phys. Rev. E, 59, 1999,6079-6087.
[215] Christodoulides D N, Joseph R I. Vector solitons in birefringent nonlinear dispersive media. Opt. Lett., 13,1988, 53-55.
[216] Tratnik M V, Sipe J E. Bound solitary waves in a birefringent optical fiber. Phys.Rev. A,38, 1988,2011-2017.
[217] Akhmediev N, Krolikowski W, Snyder A W. Partially coherent solitons of variable shape. Phys. Rev. Lett., 81, 1998,4632-4635.
[218] Christodoulides D N, Coskun T H, Joseph R I. Incoherent spatial solitons in saturable nonlinear media. Opt. Lett., 22, 1997, 1080-1082.
[219] Christodoulides D N, Coskun T H, Mitchell M, et al. Theory of incoherent self-focusing in biased photorefractive media. Phys. Rev. Lett., 78, 1997, 646-649.
[220] Krolikowski W, Akhmediev N, Luther-Davies B. Collision-induced shape transformations of partially coherent solitons. Phys. Rev. E, 59,1999,4654-4658.
[221] Kanna T, Lakshmanan M. Exact soliton solutions of coupled nonlinear SchrSdinger equations: Shape-changing collisions, logic gates, and partially coherent solitons.Phys. Rev. E, 67, 2003, 046617-046641.
[222] Radhakrishnan R, Lakshmanan M, Hietarinta J. Inelastic collision and switching of coupled bright solitons in optical fibers. Phys. Rev. E, 56, 1997, 2213-2216.
[223] Kanna T, Lakshmanan M. Exact soliton solutions, shape shanging sollisions, and partially coherent solitons in coupled nonlinear Schr(o|¨)dinger equations.Phys.Rev.Lett.,86,2001,5043-5046.
[224]Zhang H Q,Xu T,Li J,et al.Integrability of an N-coupled nonlinear Schr(o|¨)dinger system for polarized optical waves in an isotropic medium via symbolic computation.Phys.Rev.E,77,2008,026605-026614.
[225]Wang S,Tang X Y,Lou S Y.Fission and fusion:Burgers equation and Sharma-Tasso-Olver equation.Chaos,Solitons and Fractals,21,2004,231-239.
[226]Lin J,Wu F M.Fission and fusion of localized coherent structures for a (2+1)-dimensional KdV equation.Chaos,Solitons and Fractals,19,2004,189-193.
[227]Zhang C Y,Gao Y T,Xu T,et al.Various methods for constructing auto-B(a|¨)cklund transformations for a generalized variable-coefficient Korteweg-de Vries model from plasmas and fluid dynamics.Commun.Theor.Phys.,49,2008,673-678.
[228]Wei G M,Gao Y T,Hu W.Painlev(?) analysis,auto-B(a|¨)cklund transformation and new analytic solutions for a generalized variable-coefficient Korteweg-de Vries (KdV) equation.Eur.Phys.J.B,53,2006,343-350.
[229]Zhang C Y,Gao Y T,Meng X H,et al.Integrable properties of a variable-coefficient Korteweg-de Vries model from Bose-Einstein condensates and fluid dynamics.J.Phys.A,39,2006,14353-14362.
[230]Kalita B C,Das R.A comparative study of modified Korteweg-de Vries(MKdV)and Korteweg-de Vries(KdV) solitons in plasmas with negative ions under the influence of electrons drift motion.Phys.Plasmas,5,1998,3588-3594.
[231]Mohamad M N B.Exact solutions to the combined KdV and mKdV equation.Math.Meth.Appl.Sci.,15,1992,73-78.
[232]Liu H L,Slemrod M.KdV dynamics in the plasma-sheath transition.Appl.Math.Lett.,17,2004,401-410.
[233]Elwakil S A,El-labany S K,Zahran M A,et al.New exact solutions for a generalized variable coefficients 2D KdV equation.Chaos Solitons and Fractals,19,2004,1083-1086.
[234]Ma W X,You Y C.Solving the Korteweg-de Vries equation by its bilinear form:Wronskian solutions.Trans.Amer.Math.Sot.,357,2005,1753-1778.
[235]陆振球,经典和现代数学物理方程,上海科技出版社,1991.
[236] Vvedensky D. Partial Differential equations with Mathematica, Addison-Wesley Publishing Co., 1993.
[237] Coffey M. Nonlinear dynamics of vortices in ultraclean type-II superconductors: Integrable wave equations in cylindrical geometry. Phys. Rev. B, 54,1996,1279-1285.
[238] Tian B, Gao Y T. Spherical Kadomtsev-Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computation. Phys. Lett. A, 340, 2005,243-250.
[239] Tian B, Gao Y T. Spherical nebulons and B(?)cklund transformation for a space or laboratory un-magnetized dusty plasma with symbolic computation. Eur. Phys. J. D,33, 2005, 59-65.
[240] Tian B, Gao Y T. Cylindrical nebulons, symbolic computation and B(?)cklund transformation for the cosmic dust acoustic waves. Phys. Plasmas Lett., 12, 2005,070703-070704.
[241] Gao Y T, Tian B. (3+1)-dimensional generalized Johnson model for cosmic dust-ion-acoustic nebulons with symbolic computation. Phys. Plasmas Lett., 13,2006,120703-120706.
[242] Miura R M. BScklund transformations, Springer, Berlin, 1976.
[243] Rogers C, Shadwick W F. B(?)cklund transformations and their applications,Academic, 1982.
[244] Lou S Y, Ma H C. Finite symmetry transformation groups and exact solutions of Lax integrable systems. Chaos Solitons and Fractrals, 302, 2006, 804-821.
[245] Li J, Xu T, Meng X H, Yang Z C, et al. Symbolic computation on integrable properties of a variable-coefficient Korteweg-de Vries equation from arterial mechanics and Bose-Einstein condensates. Phys. Scr., 75,2007, 278-284
[246] Xu T, Zhang C Y, Wei G M, et al. Symbolic-computation construction of transformations for a more generalized nonlinear Schr(?)dinger equation with applications in inhomogeneous plasmas, optical fibers, viscous fluids and Bose-Einstein condensates. Eur. Phys. J. B, 55,2007, 323-332.
[247] Ohkuma K, Wadati M. The Kadomtsev-Petviashvili equation: the trace method and the soliton resonances. J. Phys. Soc. Jpn., 52,1983, 749-760.
[248] Xu T, Zhang C Y, Li J, et al. Inelastic interaction and non-traveling-wave effects for two multi-dimensional Burgers models from fluid dynamics and astrophysics with symbolic computation. Z. Naturforsch., 61a, 2006, 652-660.
[249] Ma W X. An approach for constructing nonisospectral hierarchies of evolution equations. J. Phys. A, 25,1992, L719-L726.
[250] Ma W X. A simple scheme for generating nonisospectral flows from the zero curvature representation. Phys. Lett. A, 179,1993,179-185.
[251] Ma W X, Wu H Y, He J S. Partial differential equations possessing Frobenius integrable decompositions. Phys. Lett. A, 364,2007,29-32.
[252] Agrawal G P. Nonlinear Fiber Optics, Academic Press, 1995.
[253] Sahadevan R, Tamizhmani K M, Lakshmanan M. Painleve analysis and integrability of coupled non-linear Schrodinger equations. J. Phys. A 19,1986,1783-1791.
[254] H.Q. Zhang, J. Li, T. Xu, et al. Optical soliton solutions for two coupled nonlinear Schr(?)dinger systems via Darboux transformation. Phys. Scr., 76,2007,452-460.
[255] Menyuk C R. Pulse propagation in an elliptically birefringent Kerr medium. IEEE J.Quant. Elec., 25,1989,2674-2682.
[256] Liu W J, Tian B, Zhang H Q, et al. Soliton interaction in the higher-order nonlinear Schr(?)dinger equation investigated with Hirota's bilinear method. Phys. Rev. E, 77,2008,066605-066611.
[257] Jakubowski M H, Steiglitz K, Squier R. State transformations of colliding optical solitons and possible application to computation in bulk media. Phys. Rev. E, 581998,6752-6758.
[258] Steiglitz K. Time-gated Manakov spatial solitons are computationally universal.Phys. Rev. E, 63, 2000,016608-016614.
[259] Chan K C, Liu H F. Effects of optical fiber characteristics. IEEE J. Quantum Electron., 31, 1995,2226-2235.
[260] Gordoa P R, Conde J M. A linear algebraic nonlinear superposition formula. Phys.Lett. A, 295,2002, 287-298.
[261] Yomba E, Peng Y Z. Fission, fusion and annihilation in the interaction of localized structures for the (2 +1)-dimensional generalized Broer-Kaup system. Chaos, Solitons and Fractals,28,2006,650-667.
[262]Zheng C L,Zhang J F,Huang W H,et al.Peakon and foldon excitations in a (2+1)-dimensional breaking soliton system.Chin.Phys.Lett.20,2003,783-788.
[263]Lou S Y.Painlev(?) test for the integrable dispersive long wave equation.Phys.Lett.A,176,1993,96-100.
[2]Yan Z Y,Zhang.H Q.Symbolic computation and new families of exact soliton-like solutions to the integrable Broer-Kaup(BK) equations in(2+1)-dimensional spaces.J.Phys.A,34,2001,1785-1792.
[3]Duffy B R,Parks E J.Travelling solitary wave solutions to a seventh-order generalized KdV equation.Phys.Lett.A,214,1996,271-272.
[4]Hong W P.Comment on "Spherical Kadomtsev-Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computation".Phys.Lett.A,361,2007,520-522.
[5]Das G,Sarma J.Comment on "a new mathematical approach for finding the solitary waves in dusty plasma".Phys.Plasmas,6,1999,4394-4397.
[6]范恩贵.可积系统与计算机代数,科学出版社,2004.
[7]衷仁保,马建.计算机代数学,科学出版社,1996.
[8]张树功,刘停战,冯果忱.计算机代数基础,吉林大学出版社,1997.
[9]吴文俊.数学机械化,科学出版社,1999.
[10]周梦.计算代数与应用,武汉大学出版社,2002.
[11]陶庆生.计算机代数及其应用,浙江大学出版社,1991.
[12]赵东方.数学模型与计算,科学出版社,2007.
[13]张韵华.符号计算系统Mathematica教程,科学出版社,2001.
[14]田播.计算机符号计算在非线性研究中的若干应用[学位论文],北京航空航天大学,2003.
[15]李娟.基于计算机符号计算的若干变系数非线性模型可积性质的研究[学位沦文],北京邮电大学,2008.
[16]王爱银.符号计算对计算有理矩阵指数的实用方法[学位论文],新疆大学,2005.
[17]郭柏灵.非线性演化方程,上海科技教育出版社,1995.
[18]刘式达,刘式适.孤波和湍流,上海科技教育出版社,1995.
[19]刘式适,刘式达,谭本.非线性大气动力学,国防工业出版社,1996.
[20]郭柏灵,苏风秋.孤立子,辽宁教育出版社,1998.
[21]倪皖荪,魏容爵.水槽中的孤波,上海科技教育出版社,1998.
[22]谷超豪,胡和生,周子翔.孤立子中的Darboux变换及其几何应用,上海科技出版社,1999.
[23]闫振亚.复杂非线性波的构造性理论及其应用,科学出版社,2007.
[24]陆同兴.非线性物理概论,中国科学技术大学出版社,2002.
[25]刘式达,刘式适.物理学中的非线性方程,北京大学出版社,2002.
[26]潘祖梁.非线性问题的数学方法及其应用,浙江大学出版社,2002.
[27]庞小峰.非线性量子力学理论,重庆出版社,1994.
[28]罗德海.大气中大尺度包络孤立子理论与阻塞环流,气象出版社,1999.
[29]王德焴,吴德金,黄光力.空间等离子体中的孤波,上海科技教育出版社,2000.
[30]杨伯君,赵玉芳.高等数学物理方法,北京邮电大学出版社,2003.
[31]庞小峰.孤子物理学,四川科学技术出版社,2003.
[32]黄景宁,徐济仲,熊吟涛.孤子、概念、原理和应用,高等教育出版社,2004.
[33]陈登远.孤子引论,科学出版社,2006.
[34]楼森岳,唐晓艳.非线性数学物理方法,科学出版社,2006.
[35]Russell J S.Report on waves,14th meeting of the British association for advancement of science.London,John Murray,1844,311-390.
[36]Boussinesq M J.Th(?)orie de I'intumescencs liquide appel(?)e onde solitaire ou de translation se propageant dans un canal rectangulaire.C.R.Acad.Sci.Paris S(?)r,72,1871,755-759.
[37]Rayleigh L.On waves.Phil.Mag.,1,1876,257-279.
[38]Korteweg D J,de Vriss G.On the change of form of long waves advancing in a rectangular canal,and on a new type of long stationary waves.Philos.Mag.Ser.,39,1895,422-443.
[39]Fermi E,Pasta J,Ulam S.Studies of nonlinear problems I.Los Alamos Report LA-1940,1955.
[40]Toda M.Wave propagating in anharmonic lattices.J.Phys.Soc.Jpn.,23,1967,501-506.
[41]Perring K J and Skyrme T H R.A model unified field equation.Nucl.Phys.,31,1962,550-555.
[42]Zabusky N J,Kruskal M D.Interaction of"solitons" in a collisionless plasma and the recurrence of initial states.Phys.Rev.Lett.,15,1965,240-243.
[43]Li Y S,Ma W X,Zhang J E.Darboux transformations of classical Boussinesq system and its new solutions.Phys.Lett.A,275,2000,60-66.
[44]谷内俊弥,西原功修.非线性波动(徐福元,张家泰,丁启明译),原子能出版社,1981.
[45]邓瑞基,陈自力,唐小弟.基于Mathematica的符号计算方法.中南林学院学报,26,2006,90-93.
[46]吴剑,胡波.掌握和精通Mathematica4.0,机械工业出版社,2001.
[47]徐安农.Mathematica数学实验,电子工业出版社,2004.
[48]阳明盛,林建华.Mathematica基础及数学软件,大连理工大学出版社,2006.
[49]张宝善.Mathematica符号运算与数学实验,南京大学出版社,2007.
[50]沃尔夫雷姆.Mathematica全书(赫孝良,周义仓译),两安交通大学出版社,2002.
[51]陈杰.Matlab宝典,电子工业出版社,2007.
[52]张德丰.Matlab数值分析与应用,国防工业出版社,2007.
[53]钟益林,彭乐群,刘炳文.常微分方程及其Maple,Matlab求解,清华大学出版社,2007.
[54]刘辉.Maple符号处理及应用,国防工业出版社,2001.
[55]何青,王丽芬.Maple教程,科学出版社,2006.
[56]Buchberger B,Groebner B.An algrorithmic method in polynomial ideal theory.Recent trends in multidimensional systems theory.Reidel Publishing Company,1985.
[57]Ibrahim R S.Tanh-travelling wave solutions,truncated Painlev(?) expansion and reduction of Bullough-Dodd equation to a quadrature in magnetohy drodynamic equilibrium.Chaos Solitons and Fractals,16,2003,675-683.
[58]Hong W P,Park S H.Extended tanh-methods and solitonic solutions of the general shallow water wave models.Int.J.Mod.Phys.C,15,2004,363-370.
[59]Parks E J,Duffy B R.An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations.Comput.Phys.Commun.,98,1996,288-300.
[60]Li Z B,Wang M L.Travelling wave solutions to the two-dimensional KdV-Burgers equation.J.Phys.A,26,1993,6027-6031.
[61]Yang X D,Ruan H Y,Lou S Y.A Maple package on symbolic computation of conserved densities for(1+1)-dimensional nonlinear evolution systems.Commun.Theor.Phys.,47,2007,961-968.
[62]柳银萍.基于吴方法的孤波自动求解软件包及其应用[学位论文],华东师范大学,2001.
[63]Liu Y P,L I Z B.A Maple package for finding exact solitary wave solutions of coupled nonlinear revolution equations.Comput.Phys.Comet.,155,2002,65-70.
[64]Gardner C S,Greene J G,Kruskal G D,et al.Method for solving the Korteweg-de Vries equation.Phys.Rev.Lett.,19,1967,1095-1097.
[65]Ablowitz M J,Clarkson P A.Solitons,nonlinear evolution equations and inverse scattering.Cambridge University Press,1991.
[66]Wahlquist H D,Estabrook F B.B(a|¨)cklund transformation for solutions of the Korteweg-de Vries equation.Phys.Rev.Lett.,31,1973,1386-1390.
[67]Zait R A.B(a|¨)cklund transformations and solutions for some evolution equations.Phys.Ser.,57,1998,545-548.
[68]Tian B,Gao Y T.(2+1) dimensional Haragus-Courcelle-Ilichev model for the liquid surface waves in the presence of sea ice or surface tension,B(a|¨)cklund transformation,exact solutions and possibly observable effects.Eur.Phys.J.B,42,2004,441-450.
[69]Tian L X,Yin J L.Stability of multi-compaction solutions and B(a|¨)cklund transformation in K(m,n,1).Chaos Solitons and Fractals,23,2005,159-169.
[70]Li B,Chen Y,Zhang H Q.Auto-B(a|¨)cklund transformation and exact solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order.Phys.Lett.A,305,2002,377-382.
[71]Khater A H,Callebaut D K,Malfliet W,et al.B(a|¨)cklund transformations and exact solutions for some nonlinear evolution equations in solar magnetostatic models.J.Comput.Appl.Math.,140,2002,435-467.
[72]Konno K,Wadati M.Simple derivation of B(a|¨)cklund transformation from Riccati form of a inverse method.Prog.Theor.Phys.,53,1975,1652-1656.
[73]Khater A H,Ibrahim R S,El-Kalaawy O H,et al.B(a|¨)cklund transformation and exact soliton solutions for some nonlinear evolution equations of the ZS/AKNS system.Chaos Solitons and Fractals,9,1998,1847-1855.
[74]Clarkson P A.Painlev(?) analysis and the complete integrability of a generalized variable-coefficient Kadomtsev-Petviashvili equation.IMA J.Appl.Math.,44,1990,27-53.
[75]Yana Z Y.Painlev(?) analysis,auto-B(a|¨)cklund transformations and exact solutions for a simplified model for reacting mixtures.Physica A,2003,326,344-359.
[76]Weiss J,Tabor M,Camevale G.The Painlev(?) property of partial differential equations.J.Math.Phys.,24,1983,522-526.
[77]Hlavat(?) L.Painlev(?) analysis ofnonautonomous evolution equations.Phys.Lett.A,128,1988,335-338.
[78]Lou S Y,Hu X B.Broer-Kaup system from Darboux transformation related symmetry constrants of Kadomtsev-Petviashvili equation.Commun.Theor.Phys.,29,1998,145-148.
[79]Li Y S,Zhang J E.Darboux transformations of classical Boussinesq system and its multi-soliton solutions.Phys.Lett.A,284,2001,253-258.
[80]Zhou L J.Darboux transformation for the nonisospectral AKNS system.Phys.Lett.A,345,2005,314-322.
[81]Hu H C,Liu Q P.New Darboux transformation for Hirota-Satsuma coupled KdV system.Chaos Solitons and Fractals,17,2003,921-928.
[82]Deng S F.Darboux transformations for the isospectral and nonisospectral mKP equation.Physica A,382,2007,487-493.
[83]Est(?)vez P G.Darboux transformation and solutions for an equation in 2+1dimensions.J.Math.Phys.,40,1999,1406-1419.
[84]Hu H C,Lou S Y.New interaction property of(2+1)-dimensional localized excitations from Darboux transformation.Chaos Solitons and Fractals,24,2005,1207-1216.
[85]Hirota R.Exact solution of the KdV equation for multiple collisions of solitons. Phys.Rev.Lett.27,1971,1192-1994.
[86]Hirota R.A new form of B(a|¨)cklund transformations and its relation to the inverse scattering problem.Progr.Theor.Phys.Suppl.52,1974,1498-1512.
[87]Hirota R.The direct method in soliton theory,(translated from the Jpn.ese and edited by Nagai A,Nimmo J,Gilson C),Cambridge University Press,2004.
[88]Li L L,Tian B,Zhang C Y,et al.On a generalized Kadomtsev-Petviashvili equation with variable coefficients via symbolic computation.Phys.Scr.,76,2007,411-417.
[89]Li L L,Tian B,Zhang C Y,et al.N-soliton solutions,auto-B(a|¨)cklund transformation and Lax pair for a nonisospectral and variable-coeffcient Korteweg-de Vries equation via symbolic computation.Int.J.Mod.Phys.B,Accepted.
[90]Li L L,Tian B,Zhang H Q,et al.Soliton solutions,pairwise collisions and partially coherent interactions for a generalized(1+1)-dimensional coupled nonlinear Schr(o|¨)dinger system via symbolic computation.Z.Naturforsch.,63a,2008,679-687.
[91]Hirota R,Satsuma J.Nsoliton solution of the KdV equation with loss and nonuniformity terms.J.Phys.Soc.Jpn.,41,1976,2141-2142.
[92]Matsuno Y.Bilinear Transformation Methods,Academic Press,1984.
[93]Hirota R,Hu X B,Tang X Y.A vector potential KdV equation and vector Ito equation:soliton solutions,bilinear B(a|¨)cklund transformations and Lax pairs J.Math.Anal.Appl.,288,2003,326-348.
[94]Hirota R,Satsuma J.A simple structure of superposition formula of the B(a|¨)cklund transformation.J.Phys.Soc.Jpn.,45,1978,1741-1750.
[95]Tian B,Gao Y T.Variable-coefficient balancing-act method and variable-coefficient KdV equation from fluid dynamics and plasma physics.Euro.Phys.J.B,22,2001,351-360.
[96]Zhang J F.Homogeneous balance method and chaotic and fractal solutions for the Nizhnik-Novikov-Veselov equation.Phys.Lett.A,313,2003,401-407.
[97]Zhang C Y,Yao Z Z,Zhu H W,et al.Exact analytic N-soliton-like solution in Wronskian form for a generalized variable-coefficient Korteweg-de Vries model from plasmas and fluid dynamics.Chin.Phys.Lett.,24,2007,1173-1176.
[98]Zhang C,Tian B,Li L L,et al.Analytical analysis on a generalized (2+1)-dimensional variable-coefficient Korteweg-de Vries equation using symbolic computation.Int.J.Mod.Phys.B,Accepted.
[99]Clarkson P A,Kruskal M D.New similarity solutions of Boussinesq equation.J.Math.Phys.,30,1989,2201-2213.
[100]Lou S Y.A note on the new similarity reductions of the Boussinesq equation.Phys.Lett.A.151,1990,133-135.
[101]Clarkson P A.Nonclassical symmetry reductions for the Boussinesq equation.Chaos Soliton and Fractals,5,1995,2261-2301.
[102]Saccomandi G.Potential symmetries and direct reduction methods of order two.J.Phys.A.30,1997,2211-2217.
[103]Olver P J.Application of Lie groups to differential equation 2nd ed.,Graduate Texts in Mathematics,Springer,New York,1993.
[104]Bluman G W,Kumei S.Symmetries and differential equation.Appl.Math.Sci.,Springer,Berlin,1989.
[105]Bluman G W,Cole J D.The general similarity solution of the heat equation.J.Math.Mech.,18,1969,1025-1042.
[106]Morris H C.A prolongation structure for the AKNS system and its generalization.J.Math.Phys.,18,1977,533-536.
[107]Ma W X,Strampp W.An explicit symmetry constraint for the Lax pairs and adjoint Lax pairs of AKNS systems.Phys.Lett.A,185,1994,277-286.
[108]Li Y S,Zhang J E.Bidirectional soliton solutions of the classical Boussinesq system and AKNS system.Chaos Soliton and Fractals,16,2003,271-277.
[109]Xu T,Li J,Zhang H Q,et al.New extension of the tanh-function method and application to the Whitham-Broer-Kaup shallow water model with symbolic computation.Physics Letters A,369,2007,458-463.
[110]Yomba E.The extended Fan's sub-equation method and its application to KdV-MKdV,BKK and variant Boussinesq equations.Phys.Lett.A,336,2005,463-476.
[111]郭玉翠.非线性偏微分方程引论,清华大学出版社,2008.
[112] Khater A H, Abdallah A A, El-Kalaawy O H. B(?)cklund transformations, a simple transformation and exact solutions for dust-acoustic solitary waves in dusty plasma consisting of cold dust particles and two-temperature isothermal ions. Phys.plasmas, 6,1999,4542-4547.
[113] Musette M, Verhoeven C. Nonlinear superposition formula for the Kaup-Kuper-shmidt partial differential equation. Physica D, 144,2000, 211-220.
[114] Boiti M, Leon J J P, Martina L, et al. Scattering of localized solitons in the plane.Phys. Lett. A, 132,1988,432- 439.
[115] Ruan H Y, Chen Y X. Dromion interactions of (2+1 )-dimensional nonlinear evolution equations. Phys. Rev. E, 62,2000, 5738-5344.
[116] Tang X Y, Li J M, Lou S Y. Reflection and reconnection interactions of resonant dromions. Phys. Scr., 75,2007, 201-205.
[117] Ruan H Y, Lou S Y. Dromion structure of (2+1)-dimensional Sine-Gordon system.Commun. Theor. Phys., 32,1999,109-114.
[118] Alagesan T, Uthayakumar A, Porsezian K. The generalisation of integrable (2+1)-dimensional dispersive long-wave equations. J. Phys. Soc. Jpn., 66,1997,1288-1290.
[119] Isojima S, Willox R, Satsuma J. On various solution of the coupled KP equation. J.Phys. A, 35,2002,6893-6909.
[120] Lou S Y. Symmtry analysis and exact solutions of the 2+1 dimensional Sine-Gordon system. J. Math. Phys., 41,2000,6509-6524.
[121] Tang X Y, Lou S Y, Zhang Y. Localized exicitations in. (2+1)-dimensional systems. Phys. Rev. E, 66,2002, 046601-046605.
[122] Lou S Y. Dromions, dromions lattice, breathers and instantons of the Davey-Stewartson equation. Phys. Scr., 65, 2002, 7-12.
[123] Ying J P, Lou S Y. Abundant coherent structures of the (2+1)-dimensional Broer-Kaup-Hupershmidt equation, Z. Naturforsch., 56a, 2001,619-625.
[124] Camassa R, Holm D D. An integrable shallow water equation with peaked solitons.Phys. Rev. Lett., 71,1993,1661-1664.
[125] Rosenau P, Hyman J. M. Compactons: solitons with finite wavelength. Phys. Rev.Lett., 70,1993, 564-567.
[126]Lu Z M,Tian E M,Grimshaw R.Interaction of two lump solitons described by the Kadomtsev-Petviashvili I equation.Wave Motion,40,2004,123-135.
[127]Fokas A S,Pelinovsky D E,Sulem C.Interaction of lumps with a line soliton for the DSII equation.Physica D,152,2001,189-198.
[128]Loginov E K.Multi-instantons in higher dimensions and superstring solitons.SIGMA 1,2005,002-005.
[129]Rajaraman R.Solitons and instantons:an introduction to solitons and instantons in quantum field theory,Elsevier,North-Holland,1982.
[130]Zhang J F,Meng J P,Zheng C L,et al.Folded solitary waves and foldons in the (2+1)-dimensional breaking soliton equation.Chaos,Solitons and Fractals,20,2004,523-527.
[131]Dai C Q,Ni Y Z.Novel interactions between semi-foldons of the (2+1)-dimensional Boiti-Leon-Pempinelli equation.Phys.Scr.,74 584-590.
[132]沈守枫.高维微分.差分模型的Virasoro对称子代数,多线性变量分离解和局域激发模式.物理学报,55,2006,5606-5610.
[133]Wu J R,Keolian R,Rudnick I,et al.Observation of a non-propagating hydrodynamic soliton.Phys.Rev.Lett.,47,1984,1421-1424.
[134]Fokas A S,Santini P M.On the simplest integrable equation in 2+1.Phyisica D,44,1990,99-104.
[135]Hietarinta J,Hirota R.Multidromion solutions to the Davey-Stewartson equation.Phys.Lett.A,145,1990,237-244.
[136]Zhang J F,Xu G D.Generalized dromion solutions of a new higher dimensional soliton equation.Commun.Nonlinear Sci.Numer.Simul.,6,2001,97-101.
[137]Hietarinta J.One-dromion solutions for generic classes of equations.Phys.Lett.A,149,1990,113-118.
[138]Radha R,Lakshmanan M.Singularity analysis and localized coherent structures in (2+1)-dimensional generalized Korteweg-de Vries equations.J.Math.Phys.,35,1994,4746-4756.
[139]Radha R,Lakshmanan M.Dromion like structures in the(2+1)-dimensional breaking soliton equation.Phys.Lett.A,1995,197,7-12.
[140]Radha R,Lakshmanan M.Exotic coherent Structures in the(2+1)-dimensional long dispersive wave equation.J.Math.Phys.,38,1997,292-299.
[141]Radha R and Lakshmanan M.A New Class of induced Localized Coherent Structures in the(2+1)-dimensional scalar nonlinear Schr(o|¨)dinger equations.J.Phys.A,30,1997,3229-3232.
[142]Radha R and Lakshmanan M.Localized coherent structures and integrability in a generalized(2+1)-dimensional nonlinear Schrodinger equation.Chaos,Solitons and Fractals,8,1997,17-25.
[143]Lou S Y.Dromion-like structures in a(3+1)-dimensional KdV type equation.J.Phys.A,29,1996,5989-6001.
[144]Ruan H Y,Lou S Y.Higher dimensional dromion structures:Jimbo-Miwa-Kadomtsev-Petviashvili system.J.Math.Phys.,38,1997,3123-3136.
[145]Lou S Y.Generalized dromion solutions of the(2+1) -dimensional KdV equation.J.Phys.A,28,1995,7227-7232.
[146]Lou S Y.On the dromion solution of a(2+1) -dimensional KdV-type equation.Commun.Theor.,26,1996,487-490.
[147]Radha R,Lakshmanan M.Generalized dromions in the(2+1)-dimensional long dispersive wave(2LDW) and scalar nonlinear Schrodinger(NLS) equations.Chaos,Solitons and Fractals,10,1999,1-4.
[148]Zhang J F.Generalized dromions of the(2+1)-dimensional nonlinear Schrodinger equations.Commun.Nonlinear Sci.Numer.Simul.,6,2001,50-53.
[149]Lou S Y.On the coherent structures of Nizhnik-Novikov-Veselov equation.Phys.Lett.A,277,2000,94-100.
[150]Lou S Y,Ruan H Y.Revisitation of the localized excitations of the 2+1 dimensional KdV equarion.J.Phys.A,34,2001,305-316.
[151]Ruan H Y,Chert Y X.Ring solitons,dromions,breathers and instantons of the NLS equation.Acta Phys.Sin.50,2001,586-591.
[152]Kadomtsov B B,Petviashvili V I.On the stability of solitary waves in weakly dispersive media.Soy.Phys.Dokl.,15,1970,539-541.
[153]Davey A,Stewartson K.On three-dimensional packets of surface waves.Proc.R.Soc.A,338,1974,101-110.
[154]Zhang J F,Huang W H,Zheng C L.Coherent soliton structures of a new (2+1)-dimensional evolution equation.Acta Phys.Sin.,51,2002,2676-2682.
[155]Matveev V B,Salle M A.Darboux transformations and solitons,Springer,Berlin,1991.
[156]Kako F,Yajima N.Interaction of ion-acoustic solitons in two-dimensional space.J.Phys.Soc.Jpn.,49,1980,2063-2071.
[157]Hasegawa A,Tappert F.Transmission of stationary nonlinear optical pulses in dispersion dielectric fibers.App.Phys.Lett.,23,1973,142-144.
[158]贾东方,余震虹,谈斌等译.非线性光纤光学原理及应用,电子工业出版社,2002.
[159]Law C T,Swartzlander G A.Optical vortex solitons and the stability of dark soliton stripes.Optics Letts.18,1993,586-588.
[160]L.D.Carr.Solitons in Bose-Einstein Condensates,American Physical Society,34th Meeting of the Division of Atomic,Molecular and Optical Physics,2003.
[161]Denschlag J,Simsarian J E,Feder D L,et al.Science,287,2000,97-101.
[162]Chan W L,Li K S.Nonpropagating solitons of the variable coefficient and nonisospeetral KdV equation.J.Math.Phys.30,1989,2521-2526.
[163]Chan W L,Zhang X.Symmetries,conservation laws and Hamiltonian structures of the nonisospectral and variable coefficient KdV and MKdV equations.J.Phys.A 28,1995,407-419.
[164]Brugarino T.Painlev(?) property,auto-B(a|¨)cklund transformation,and reduction to the standard form for the Korteweg-de Vries equation.J.Math.Phys.30,1989,1013-1015.
[165]Zhang J F,Chen F Y.Truncated expansion method and new exact soliton -like solution of the general variable coefficient KdV equation.Phys.Soc.,50,2001,1648-1650.
[166]Zhao X Q,Tang D B,Wang M L.New soliton-like solutions for KdV equation with variable coefficient.Phys.Lett.A,346,2005,288-291.
[167]Calogero F,Degasperis A.Extension of the spectral transform method for solving nonlinear evolution equations.Lett.Nuovo Cimento 22,1978,131-137.
[168]Wang Y L,Zhou Z X,Jiang X Q,et al.Cylindrical Kadomtsev-Petviashvili equation for relativistically magnetosonic solitary wave in the collisionless plasma. Phys. Lett. A, 355,2006,386-389.
[169] Wang Y Y, Zhang J F. Variable-coefficient KP equation and solitonic solution for two-temperature ions in dusty plasma. Phys. Lett. A, 352,2006,155-162.
[170] Gao Y T, Tian B. Phys. Cylindrical Kadomtsev-Petviashvili model, nebulons and symbolic computation for cosmic dust ion-acoustic waves. Lett. A, 349,2006,314-319.
[171] Tian B, Gao Y T. Symbolic computation on cylindrical-modified dust-ion-acoustic nebulons in dusty plasmas. Phys. Lett. A, 362,2007,283-288.
[172] Johnson R S. Water waves and Korteweg-de Vries equations. J. Fluid Mech. 97,1980,701-719.
[173] Milewski P. Long wave interaction over varying topography. Physica D, 123, 1998,36-47.
[174] Anders I, Monvel B D A. Asymptotic Solitons of the Johnson Equation. J.Nonlinear Math. Phys., 7,2000,284-302.
[ 175] Gao Y T, Tian B. Some two-dimensional and non-travelling-wave observable effects of the shallow-water waves. Phys. Lett. A, 301, 2002, 74-84.
[176] Shen S F, Zhang J, Ye C E, et al. New exact solutions of the (2+1)-dimensional KdV equation with variable coefficients. Phys. Lett. A, 337, 2005,101-106.
[177] Oevel W, Steeb W H. Painlev(?) analysis for a time-dependent KP equation. Phys.Lett. A, 103,1984,239-242.
[ 178] Emmanuel Y. Abundant families of Jacobi elliptic function-like solutions for a generalized variable coefficients 2D KdV equation via the extended mapping method. Phys. Lett. A, 349,2006, 212-219.
[179] Ablowitz M J and Clarkson P A. Solitons, Nonlinear evolution equations and inverse scattering, Cambridge University Press, 1991.
[180] Xie Y C. Exact solutions of the Wick-type stochastic Kadomtsev-Petviashvili equations. Chaos Solitons Fractals, 21,2004,473-480.
[181] Malik H K. Effect of electron inertia on KP solitons in a relativistic plasma. Physica D, 125,1999,295-301.
[ 182] Duan W S. Acoustic mode in a two-ion-temperature warm dusty plasma under the weakly transverse perturbations. Chaos Solitons Fractals, 14,2002,1315-1318.
[183] Lin M M, Duan W S. The Kadomtsev-Petviashvili (KP), MKP, and coupled KP equations for two-ion-temperature dusty plasmas. Chaos Solitons Fractals, 23,2005, 929-937.
[184] Noubissi(?) S, Woafo P. Dynamics of solitary waves over an erodible surface.Physica A,345,2005,9-16.
[185] A. Hasegawa, Optical Solitons in Fibers, Springer, Berlin, 1989.
[186] K. Porsezian and V. C. Kuriakose, Optical Solitons: Theoretical and Experimental Challenges, Springer, Berlin, 2003.
[187] Erbay H A, Erbay S, Dost S. Nonlinear wave modulation in micropolar elastic media-II. Transverse waves. Int. J. Eng. Sci., 29,1991, 859-868.
[188] Kajinaga Y, Tsuchida T, M Wadati. Coupled Nonlinear Schrodinger Equations for Two-component Wave Systems. J. Phys. Soc. Jpn., 67, 1998,1565-1568.
[189] Tsang S C, Chow K W. The evolution of periodic waves of the coupled nonlinear Schr(?)dinger equations. Math. Comput. Simul., 66, 2004, 551-564.
[190] Hacinliyan I, Erbay S. A higher-order model for transverse waves in a generalized elastic solid. Chaos, Solitons and Fractals, 14,2002,1127-1135.
[191] Kanna T, Lakshmanan M, Dinda P T, et al. Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schr(?)dinger equations. Phys.Rev. E, 73, 2006, 026604-026618.
[192] Manakov S V. On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Phys. JETP, 38,1974, 248-253;
[193] Rothenberg J E. Observation of the buildup of modulational instability from wave breaking. Opt. Lett., 16, 1991,18-20.
[194] Wai P K A, Menyuk C R, Chen H H. Stability of solitons in randomly varying birefringent fibers. Opt. Lett., 16, 1991, 1231-1233.
[195] Angelis C D, Wabnitz S, Haelterman M. Bandwidths limits due to polarization multiplexed soliton interactions. Electron. Lett., 29, 1993, 1568-1570.
[196] Zhang H Q, Meng X H, Xu T, et al. Interactions of bright solitons for the (2+1)-dimensional coupled nonlinear Schrodinger equations from optical fibres with symbolic computation. Physica Scripta. Phys. Scr., 75, 2007, 537-542.
[197] Menyuk C R. Nonlinear puke propagation in birefringent optical fibers. IEEE J. Quant.Elec.,23,1987,174-176.
[198]Wadati M,Izuka T,Hisakado M.A coupled nonlinear Schrodinger equation and optical solitons.J.Phys.Soc.Jpn.61,1992,2241-2245.
[199]Lakshmanan M,Karma T,Radhakrishnan R.Shape-changing collisions of coupled bright solitons in birifringent optical fibers.Rep.Math.Phys.46,2000,143-156.
[200]Newboult G K,Parker D F,Faulkner T R.Coupled nonlinear Schr(o|¨)linger equations arising in the study of monomode step-index optical fibers.J.Math.Phys.30,1989,930-936.
[201]Nakkeeran K.Optical Solitons in a new type of coupled nonlinear Schr(o|¨)dinger equations.J.Mod.Opt.48,2001,1863-1867.
[202]Park Q H,Shin H J.Painlev(?) analysis of the coupled nonlinear Schr(o|¨)dinger equation for polarized optical waves in an isotropic medium.Phys.Rev.E,59,1999,2373-2379.
[203]Bindu S G,Mahalingam A,Porsezian K.Phys.Lett.A,286,2001,321-.
[204]Gilson C,Hietarinta J,Nimmo J,et al.Sasa-Satsuma higher-order nonlinear Schr(o|¨)dinger equation and its bilinearization and multisoliton solutions.Phys.Rev.E,68,2003,016614-016623.
[205]Chow K W.Periodic waves for a system of coupled,higher order nonlinear Schr(o|¨)dinger equations with third order dispersion.Phys.Lett.A,308,2003,426-431.
[206]Hirota R.Exact envelope-soliton solutions of a nonlinear wave equation.J.Math.Phys.,14,1973,805-809.
[207]Tasgal R S,Potasek M J.Soliton solutions to coupled higher-order nonlinear Schr(o|¨)dinger equations.J.Math.Phys.,33,1992,1208-1215.
[208]Radhakrishnan R,Lakshmanan M,Daniel M.Bright and dark optical solitons in coupled higher-order nonlinear Schr(o|¨)dinger equations through singularity structure analysis.J.Phys.A,28,1995,7299-7314.
[209]Boiti M,Leon J,Pempinelli F.Spectral transform for a two spatial dimension extension of the dispersive long wave equation.Inverse Problems,3,1987,371-387.
[210]Tang X Y,Lou S Y.Abundant coherent structures of the dispersive long-wave equation in (2+1)-dimensional spaces. Chaos, Solitons and Fractals, 14, 2002,1451-1456.
[211] Tang X Y, Chen C L, Lou S Y. Localized solutions with chaotic and fractal behaviours in a (2+1)-dimensional dispersive long-wave system. J. Phys. A, 35,2002.L293-L301.
[212] Ma W X. Diversity of exact solutions to a restricted Boiti-Leon-Pempinelli dispersive long-wave system. Phys. Lett. A, 319,2003, 325-333.
[213] Xue Y S, Tian B, Li L L, et al. Solitons and localized excitations for the (2+1)-dimensional dispersive long wave system via symbolic computation. Int. J.Mod. Phys. B, Accepted.
[214] Ankiewicz Z, Krolikowski W, Akhmediev N N. Partially coherent solitons of variable shape in a slow Kerr-like medium: Exact solutions. Phys. Rev. E, 59, 1999,6079-6087.
[215] Christodoulides D N, Joseph R I. Vector solitons in birefringent nonlinear dispersive media. Opt. Lett., 13,1988, 53-55.
[216] Tratnik M V, Sipe J E. Bound solitary waves in a birefringent optical fiber. Phys.Rev. A,38, 1988,2011-2017.
[217] Akhmediev N, Krolikowski W, Snyder A W. Partially coherent solitons of variable shape. Phys. Rev. Lett., 81, 1998,4632-4635.
[218] Christodoulides D N, Coskun T H, Joseph R I. Incoherent spatial solitons in saturable nonlinear media. Opt. Lett., 22, 1997, 1080-1082.
[219] Christodoulides D N, Coskun T H, Mitchell M, et al. Theory of incoherent self-focusing in biased photorefractive media. Phys. Rev. Lett., 78, 1997, 646-649.
[220] Krolikowski W, Akhmediev N, Luther-Davies B. Collision-induced shape transformations of partially coherent solitons. Phys. Rev. E, 59,1999,4654-4658.
[221] Kanna T, Lakshmanan M. Exact soliton solutions of coupled nonlinear SchrSdinger equations: Shape-changing collisions, logic gates, and partially coherent solitons.Phys. Rev. E, 67, 2003, 046617-046641.
[222] Radhakrishnan R, Lakshmanan M, Hietarinta J. Inelastic collision and switching of coupled bright solitons in optical fibers. Phys. Rev. E, 56, 1997, 2213-2216.
[223] Kanna T, Lakshmanan M. Exact soliton solutions, shape shanging sollisions, and partially coherent solitons in coupled nonlinear Schr(o|¨)dinger equations.Phys.Rev.Lett.,86,2001,5043-5046.
[224]Zhang H Q,Xu T,Li J,et al.Integrability of an N-coupled nonlinear Schr(o|¨)dinger system for polarized optical waves in an isotropic medium via symbolic computation.Phys.Rev.E,77,2008,026605-026614.
[225]Wang S,Tang X Y,Lou S Y.Fission and fusion:Burgers equation and Sharma-Tasso-Olver equation.Chaos,Solitons and Fractals,21,2004,231-239.
[226]Lin J,Wu F M.Fission and fusion of localized coherent structures for a (2+1)-dimensional KdV equation.Chaos,Solitons and Fractals,19,2004,189-193.
[227]Zhang C Y,Gao Y T,Xu T,et al.Various methods for constructing auto-B(a|¨)cklund transformations for a generalized variable-coefficient Korteweg-de Vries model from plasmas and fluid dynamics.Commun.Theor.Phys.,49,2008,673-678.
[228]Wei G M,Gao Y T,Hu W.Painlev(?) analysis,auto-B(a|¨)cklund transformation and new analytic solutions for a generalized variable-coefficient Korteweg-de Vries (KdV) equation.Eur.Phys.J.B,53,2006,343-350.
[229]Zhang C Y,Gao Y T,Meng X H,et al.Integrable properties of a variable-coefficient Korteweg-de Vries model from Bose-Einstein condensates and fluid dynamics.J.Phys.A,39,2006,14353-14362.
[230]Kalita B C,Das R.A comparative study of modified Korteweg-de Vries(MKdV)and Korteweg-de Vries(KdV) solitons in plasmas with negative ions under the influence of electrons drift motion.Phys.Plasmas,5,1998,3588-3594.
[231]Mohamad M N B.Exact solutions to the combined KdV and mKdV equation.Math.Meth.Appl.Sci.,15,1992,73-78.
[232]Liu H L,Slemrod M.KdV dynamics in the plasma-sheath transition.Appl.Math.Lett.,17,2004,401-410.
[233]Elwakil S A,El-labany S K,Zahran M A,et al.New exact solutions for a generalized variable coefficients 2D KdV equation.Chaos Solitons and Fractals,19,2004,1083-1086.
[234]Ma W X,You Y C.Solving the Korteweg-de Vries equation by its bilinear form:Wronskian solutions.Trans.Amer.Math.Sot.,357,2005,1753-1778.
[235]陆振球,经典和现代数学物理方程,上海科技出版社,1991.
[236] Vvedensky D. Partial Differential equations with Mathematica, Addison-Wesley Publishing Co., 1993.
[237] Coffey M. Nonlinear dynamics of vortices in ultraclean type-II superconductors: Integrable wave equations in cylindrical geometry. Phys. Rev. B, 54,1996,1279-1285.
[238] Tian B, Gao Y T. Spherical Kadomtsev-Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computation. Phys. Lett. A, 340, 2005,243-250.
[239] Tian B, Gao Y T. Spherical nebulons and B(?)cklund transformation for a space or laboratory un-magnetized dusty plasma with symbolic computation. Eur. Phys. J. D,33, 2005, 59-65.
[240] Tian B, Gao Y T. Cylindrical nebulons, symbolic computation and B(?)cklund transformation for the cosmic dust acoustic waves. Phys. Plasmas Lett., 12, 2005,070703-070704.
[241] Gao Y T, Tian B. (3+1)-dimensional generalized Johnson model for cosmic dust-ion-acoustic nebulons with symbolic computation. Phys. Plasmas Lett., 13,2006,120703-120706.
[242] Miura R M. BScklund transformations, Springer, Berlin, 1976.
[243] Rogers C, Shadwick W F. B(?)cklund transformations and their applications,Academic, 1982.
[244] Lou S Y, Ma H C. Finite symmetry transformation groups and exact solutions of Lax integrable systems. Chaos Solitons and Fractrals, 302, 2006, 804-821.
[245] Li J, Xu T, Meng X H, Yang Z C, et al. Symbolic computation on integrable properties of a variable-coefficient Korteweg-de Vries equation from arterial mechanics and Bose-Einstein condensates. Phys. Scr., 75,2007, 278-284
[246] Xu T, Zhang C Y, Wei G M, et al. Symbolic-computation construction of transformations for a more generalized nonlinear Schr(?)dinger equation with applications in inhomogeneous plasmas, optical fibers, viscous fluids and Bose-Einstein condensates. Eur. Phys. J. B, 55,2007, 323-332.
[247] Ohkuma K, Wadati M. The Kadomtsev-Petviashvili equation: the trace method and the soliton resonances. J. Phys. Soc. Jpn., 52,1983, 749-760.
[248] Xu T, Zhang C Y, Li J, et al. Inelastic interaction and non-traveling-wave effects for two multi-dimensional Burgers models from fluid dynamics and astrophysics with symbolic computation. Z. Naturforsch., 61a, 2006, 652-660.
[249] Ma W X. An approach for constructing nonisospectral hierarchies of evolution equations. J. Phys. A, 25,1992, L719-L726.
[250] Ma W X. A simple scheme for generating nonisospectral flows from the zero curvature representation. Phys. Lett. A, 179,1993,179-185.
[251] Ma W X, Wu H Y, He J S. Partial differential equations possessing Frobenius integrable decompositions. Phys. Lett. A, 364,2007,29-32.
[252] Agrawal G P. Nonlinear Fiber Optics, Academic Press, 1995.
[253] Sahadevan R, Tamizhmani K M, Lakshmanan M. Painleve analysis and integrability of coupled non-linear Schrodinger equations. J. Phys. A 19,1986,1783-1791.
[254] H.Q. Zhang, J. Li, T. Xu, et al. Optical soliton solutions for two coupled nonlinear Schr(?)dinger systems via Darboux transformation. Phys. Scr., 76,2007,452-460.
[255] Menyuk C R. Pulse propagation in an elliptically birefringent Kerr medium. IEEE J.Quant. Elec., 25,1989,2674-2682.
[256] Liu W J, Tian B, Zhang H Q, et al. Soliton interaction in the higher-order nonlinear Schr(?)dinger equation investigated with Hirota's bilinear method. Phys. Rev. E, 77,2008,066605-066611.
[257] Jakubowski M H, Steiglitz K, Squier R. State transformations of colliding optical solitons and possible application to computation in bulk media. Phys. Rev. E, 581998,6752-6758.
[258] Steiglitz K. Time-gated Manakov spatial solitons are computationally universal.Phys. Rev. E, 63, 2000,016608-016614.
[259] Chan K C, Liu H F. Effects of optical fiber characteristics. IEEE J. Quantum Electron., 31, 1995,2226-2235.
[260] Gordoa P R, Conde J M. A linear algebraic nonlinear superposition formula. Phys.Lett. A, 295,2002, 287-298.
[261] Yomba E, Peng Y Z. Fission, fusion and annihilation in the interaction of localized structures for the (2 +1)-dimensional generalized Broer-Kaup system. Chaos, Solitons and Fractals,28,2006,650-667.
[262]Zheng C L,Zhang J F,Huang W H,et al.Peakon and foldon excitations in a (2+1)-dimensional breaking soliton system.Chin.Phys.Lett.20,2003,783-788.
[263]Lou S Y.Painlev(?) test for the integrable dispersive long wave equation.Phys.Lett.A,176,1993,96-100.