基于计算机符号计算的若干非线性模型的可积性研究
|
摘要
随着信息及科学技术的迅速发展,作为人工智能新分支的计算机符号计算被广泛的应用于科学研究的各类分支中,它已经成为现代科学研究的重要辅助工具。非线性科学作为当前的前沿科学研究领域,引起了一定的关注。而孤子理论是非线性科学中的三大主要分支之一。本文主要是运用计算机符号计算及Darboux变换等方法研究某些具有物理背景的非线性发展方程的可积性质、解析解及物理应用,并得到一些代数算法使其可以在计算机系统上实现。研究的方程类型主要有变系数、耦合、高维及高阶的非线性发展方程。
本论文的主要工作为：
(一)在各种实际的物理背景下,由于变化的系数能够较好的反映出介质的非均匀性和边界的不一致性,所以,变系数方程往往比对应的常系数方程更具有实用性。为了能够描述真实环境中各种复杂的非线性波包现象,一些带有时间和空间参数的变系数方程在不同的物理背景下被提出。本文研究的是变系数Gross-Pitaevskii (GP)方程。由于与物理及科技的某些基础方面相关联,Bose-Einstein凝聚是当前的热点问题之一。Bose-Einstein凝聚的某些机制,比如说被界定在圆柱式对称抛物线形状的雪茄状凝聚,是由GP方程来描述。本文通过Painleve分析方法和计算机符号计算,得到了变系数GP方程的可积条件；该方程的Lax对由Ablowitz-Kaup-Newell-Segur (AKNS)方法直接求得；基于所得到的Lax对给出方程的自-Backlund变换,并由此得出单孤子解；一些解析解由ansatz方法给出；并得到了无穷守恒率。以上这些结果对低维凝聚的研究会有一定的指导作用。
(二)最近,人们对(2+1)维的非线性发展方程在孤子理论和数学物理等方面的应用产生了研究兴趣。由于维数的限制,单纯的(1+[)维方程很难描述人们所观察到的一些现象,而(2+1)维的非线性发展方程,作为空间维数得以推广的方程,能够更加真实地反映某些非线性现象的物理机制,并且这些方程具有比一维方程更加复杂的性质。(?)adomtsev-Petviashvili(KP)方程是一个(2+1)维的非线性发展方程,它是一个可积模型,其应用涉及Bose-Einstein凝聚、等离子体物理和非线性光学等等。在本文中我们研究的是耦合的(2+1)维KP方程,已经有文献揭示了这个方程可分解为耦合的Korteweg-de Vries方程和标准的KP方程,并且可以将该耦合(2+1)维KP方程分解成AKNS系统的前两个方程。我们在这个分解的基础上,求出了约化方程的三种Darboux变换；通过求出的Darboux变换的迭代过程求出约化方程的单、双和三孤子解,并通过约化方程和原方程的联系式求出原方程的多孤子解；最后画图分析孤子解的物理机制。在随机矩阵理论中产生的耦合的(2+1)维KP方程是当前的研究热点,以上这些结果对随机矩阵理论的研究具有一定的指导借鉴价值。
(三)本文通过Darboux变换方法和计算机符号计算研究Kundu-Eckhaus (KE)方程及变系数KE方程。KE方程可以描述在光纤通信中的超短飞秒脉冲的传播,并且在量子场论、弱非线性色散物质波和非线性光学中都有应用。变系数KE方程是KE方程的推广,与KE方程相比具有更多的自由参数,因此可以期待其在现实生活中有相对广泛的应用。我们利用Darboux变换方法研究了KE方程的多孤子解。分析了孤子解的振幅、宽度、初始相位及能量等物理量。利用渐进分析方法对双孤子碰撞前后的机制进行分析,得出弹性碰撞的结论,并得到了三孤子碰撞特点。并且我们利用画图分析孤子碰撞机制：(ⅰ)迎面和追赶的弹性双孤子碰撞；(ⅱ)周期的吸引和排斥的束缚的双孤子碰撞；(ⅲ)两种类型的三孤子动态性质：三个孤子碰撞在一点,碰撞之后除了轻微的相移之外保持形状不变：两个孤子保持束缚状态与另外一个孤子碰撞,碰撞之后保持形状不变。通过对KE方程的推广给出了变系数KE方程的Lax对及Darboux变换,并得出单、双和三孤子解,并将渐进分析方法应用到双孤子分析。为了讨论孤子解的传播特征,画出了变系数KE方程的孤子图：(ⅰ)对于单孤子解来说,通过选取不同的非线性色散项前的系数,我们得到不同的孤子结构：当取非线性色散项前面系数为常数时,孤子以不变的能量、振幅和速度传播；当取非线性色散项前面系数为非常数时,我们给出了五种带有变化的速度的孤子结构；(ⅱ)分析了双孤子中的迎面孤子碰撞、追赶孤子碰撞、抛物线型孤子、周期振荡孤子和束缚孤子；(ⅲ)呈现了三孤子的四种不同的动态特征：三个孤子碰撞在一点,除了在碰撞前后的轻微的相移外保持它们的形状不变；两个孤子保持束缚状态而与另外一个孤子碰撞之后仍保持形状不变；三抛物线型孤子；一个抛物线型孤子和两个束缚状态孤子的碰撞。
(四)本文通过Darboux变换和计算机符号计算研究耦合高阶Schrodinger方程,该方程来源于双核非线性光学纤维和波导,描述五阶非线性项对超短光脉冲在非-Kerr介质中传播的影响。我们得到了方程的Lax对,并构建了相应的Darboux变换。在所得的Darboux变换的基础上方程的单、双和三孤子解以模的形式给出。孤子特征通过图像讨论：(ⅰ)双孤子的迎面和追赶弹性碰撞；(ⅱ)双孤子的周期性的吸引和排斥的束缚状态；(ⅲ)三孤子的能量交换碰撞。最后给出这个方程的拓展方程的无穷守恒律。
Accompanied with the rapid development of the information science technology, symbolic computation, which is the new branch of the artificial intelligence, has been extensively applied in certain branches of scientific researches, and it has been an assis-tant tool for the study of science and technology. Certain interest has been focused on the nonlinear science, which is one of the fields of current advanced scientific researches, and the soliton theory is one of the three main branches of the nonlinear science. Based on symbolic computation and Darboux transformation (DT), the dissertation is to in-vestigate the integrable properties, analytic solutions and physical applications of some nonlinear evolution equations which have physical backgrounds, some algebraic algo-rithms that can be performed with symbolic computation are also proposed. Types of equations investigated in the dissertation are mainly the variable-coefficient, coupled, higher-dimensional and higher-order nonlinear evolution equations.
The work of this dissertation includes the following aspects:
(1) Comparing with the equations with constant coefficients, the corresponding ones with variable-coefficients can describe the physical mechanism of nonlinear phenomena in physical situations with inhomogeneities media or nonuniformities of boundaries, and be considered to be more practical. To describe various complex nonlinear phenomena in real situation, some equations with time and space variable-coefficients from certain physical settings have been proposed. Due to its relevance to fundamental aspects of physics and technology, coherent atom optics is the subject of much current interest. For that purpose, the lower dimensional condensates have been the subject of active studies. Certain dynamics of the Bose-Einstein condensates (BECs) have been claimed to be governed by the one-dimensional Gross-Pitaevskii (GP) equation for a time-dependent trap. In this chapter, by virtue of the Painleve analysis and symbolic computation, we will derive the integrable condition for the GP equation with the time-dependent scat-tering length in the presence ot a confining or expulsive time-dependent trap. Lax pair for the equation will be directly obtained via the Ablowitz-Kaup-Newell-Segur (AKNS) scheme under the integrable condition. Bright one-soliton-like solution of the GP equa-tion will be presented via the auto-Backlund transformation and some analytic solutions with variable amplitudes will be obtained by the ansatz method. In addition, an infinite number of conservation laws will be derived. Those results could be of some values for the studies on the BECs.
(2) Nowadays,(2+1)-dimensional nonlinear evolution equations in soliton theory and mathematical physics are the subject of active studies. Due to the limitation of the dimension, the pure (1+1)-dimensional systems can not account for some ob-served features. In realistic situations, the higher dimensional systems may provide more complex models. As a typical example in (2+1)-dimensional systems, the Kadomtsev-Petviashvili (KP) system has been derived from many physical applications in the Bose-Einstein condensation, plasma physics and nonlinear optics, etc. In this chapter, a coupled KP system is investigated with symbolic computation, some researches have proposed that the system could be decomposed into the first two members of the (1+1)-dimensional AKNS hierarchy, and based on that decomposition, we will obtain three kinds of DTs of its reduced equations. Moreover, the multi-soliton-like solutions of the coupled KP system will be derived. Finally some figures will be plotted to discuss the propagation features of the soliton solutions. Those solutions could be of some values for the studies in the context of random matrix theory.
(3) In this chapter, by virtue of the DT and symbolic computation, the Kundu-Eckhaus (KE) equation with the cubic and quintic nonlinear terms and KE equation with variable coefficients will be investigated. The KE equation appears in the nonlin-ear optics, quantum field theory and weakly nonlinear dispersive matter waves. The KE equation with variable coefficients, which is an extension of the KE one, possesses the soliton solutions with more parametric freedom, therefore its solutions can be expected to model more complex situations in reality than the KE counterpart. Through DT, one-, two-and three-soli ton solutions of the KE equation will be presented in the form of modulus; Some physical quantities such as the amplitude, width, initial phase and energy of soliton solutions will be obtained; The asymptotic behavior of the two-soli ton solution will be analyzed, and shows that the collision is elastic; Figures will be plotted to discuss the propagation features of the soliton solutions:(ⅰ) Elastic collisions of the two solitons, the head-on and overtaking;(ii) Periodic attraction and repulsion of the bounded states of the two solitons;(iii) Two types of the dynamic characters of the three solitons:Three solitons intersect at a point at the moment of the collision and the soli-tons keep their shapes except for the phase shifts; Two solitons keep the bounded state and one another soliton retains its own shape invariant before and after the collision. Through the extension of the KE equation, the KE equation with variable coefficients will also be investigated in this chapter by virtue of the DT and symbolic computation. Lax pair of the equation will be obtained, and the corresponding DT will be constructed. One-, two-and three-soliton solutions will be presented, and the asymptotic behaviors of the two-soliton solution will be analyzed. Figures will be plotted to discuss the propaga-tion features of the soliton solutions:(i) As the one-soli ton solution, for different choices of the nonlinear dispersion, we obtain different soliton structures:When the nonlinear dispersion is constant, the soliton propagates with an invariant energy, amplitude and uniform velocity; When the nonlinear dispersion is variable, we present five different types of soliton structures with the varying velocities;(ii) The head-on collision, over-taking collision, parabolic solitons, periodic oscillation and the bounded states of the two solitons are analyzed;(iii) Four different dynamic characters of the three solitons are presented:Three solitons collide at a point, keeping their shapes unchanged except for the phase shifts before and after the collision; Two solitons keep the bounded state and the third one retains its own shape invariant before and after the collision; Three parabolic solitons; Collisions between one parabolic soliton and a two-soliton-bounded structure.
(4) In this chapter, the quintic generalization of the coupled cubic nonlinear Schrodinger equations from twin-core nonlinear optical fibers and waveguides will be studied, which describe the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. Lax pair of the equations will be obtained and the corresponding DT will be constructed. Moreover, one-, two-and three-soliton solutions will be presented in the form of modulus. Dynamic features of solitons will be graphically discussed:(i) Head-on and overtaking elastic collisions of the two solitons;(ii) Periodic attraction and repulsion of the bounded states of two solitons;(iii) Energy-exchanging collisions of the three solitons. Finally an infinite number of conservation laws of the extension equation will also be presented.
本论文的主要工作为：
(一)在各种实际的物理背景下,由于变化的系数能够较好的反映出介质的非均匀性和边界的不一致性,所以,变系数方程往往比对应的常系数方程更具有实用性。为了能够描述真实环境中各种复杂的非线性波包现象,一些带有时间和空间参数的变系数方程在不同的物理背景下被提出。本文研究的是变系数Gross-Pitaevskii (GP)方程。由于与物理及科技的某些基础方面相关联,Bose-Einstein凝聚是当前的热点问题之一。Bose-Einstein凝聚的某些机制,比如说被界定在圆柱式对称抛物线形状的雪茄状凝聚,是由GP方程来描述。本文通过Painleve分析方法和计算机符号计算,得到了变系数GP方程的可积条件；该方程的Lax对由Ablowitz-Kaup-Newell-Segur (AKNS)方法直接求得；基于所得到的Lax对给出方程的自-Backlund变换,并由此得出单孤子解；一些解析解由ansatz方法给出；并得到了无穷守恒率。以上这些结果对低维凝聚的研究会有一定的指导作用。
(二)最近,人们对(2+1)维的非线性发展方程在孤子理论和数学物理等方面的应用产生了研究兴趣。由于维数的限制,单纯的(1+[)维方程很难描述人们所观察到的一些现象,而(2+1)维的非线性发展方程,作为空间维数得以推广的方程,能够更加真实地反映某些非线性现象的物理机制,并且这些方程具有比一维方程更加复杂的性质。(?)adomtsev-Petviashvili(KP)方程是一个(2+1)维的非线性发展方程,它是一个可积模型,其应用涉及Bose-Einstein凝聚、等离子体物理和非线性光学等等。在本文中我们研究的是耦合的(2+1)维KP方程,已经有文献揭示了这个方程可分解为耦合的Korteweg-de Vries方程和标准的KP方程,并且可以将该耦合(2+1)维KP方程分解成AKNS系统的前两个方程。我们在这个分解的基础上,求出了约化方程的三种Darboux变换；通过求出的Darboux变换的迭代过程求出约化方程的单、双和三孤子解,并通过约化方程和原方程的联系式求出原方程的多孤子解；最后画图分析孤子解的物理机制。在随机矩阵理论中产生的耦合的(2+1)维KP方程是当前的研究热点,以上这些结果对随机矩阵理论的研究具有一定的指导借鉴价值。
(三)本文通过Darboux变换方法和计算机符号计算研究Kundu-Eckhaus (KE)方程及变系数KE方程。KE方程可以描述在光纤通信中的超短飞秒脉冲的传播,并且在量子场论、弱非线性色散物质波和非线性光学中都有应用。变系数KE方程是KE方程的推广,与KE方程相比具有更多的自由参数,因此可以期待其在现实生活中有相对广泛的应用。我们利用Darboux变换方法研究了KE方程的多孤子解。分析了孤子解的振幅、宽度、初始相位及能量等物理量。利用渐进分析方法对双孤子碰撞前后的机制进行分析,得出弹性碰撞的结论,并得到了三孤子碰撞特点。并且我们利用画图分析孤子碰撞机制：(ⅰ)迎面和追赶的弹性双孤子碰撞；(ⅱ)周期的吸引和排斥的束缚的双孤子碰撞；(ⅲ)两种类型的三孤子动态性质：三个孤子碰撞在一点,碰撞之后除了轻微的相移之外保持形状不变：两个孤子保持束缚状态与另外一个孤子碰撞,碰撞之后保持形状不变。通过对KE方程的推广给出了变系数KE方程的Lax对及Darboux变换,并得出单、双和三孤子解,并将渐进分析方法应用到双孤子分析。为了讨论孤子解的传播特征,画出了变系数KE方程的孤子图：(ⅰ)对于单孤子解来说,通过选取不同的非线性色散项前的系数,我们得到不同的孤子结构：当取非线性色散项前面系数为常数时,孤子以不变的能量、振幅和速度传播；当取非线性色散项前面系数为非常数时,我们给出了五种带有变化的速度的孤子结构；(ⅱ)分析了双孤子中的迎面孤子碰撞、追赶孤子碰撞、抛物线型孤子、周期振荡孤子和束缚孤子；(ⅲ)呈现了三孤子的四种不同的动态特征：三个孤子碰撞在一点,除了在碰撞前后的轻微的相移外保持它们的形状不变；两个孤子保持束缚状态而与另外一个孤子碰撞之后仍保持形状不变；三抛物线型孤子；一个抛物线型孤子和两个束缚状态孤子的碰撞。
(四)本文通过Darboux变换和计算机符号计算研究耦合高阶Schrodinger方程,该方程来源于双核非线性光学纤维和波导,描述五阶非线性项对超短光脉冲在非-Kerr介质中传播的影响。我们得到了方程的Lax对,并构建了相应的Darboux变换。在所得的Darboux变换的基础上方程的单、双和三孤子解以模的形式给出。孤子特征通过图像讨论：(ⅰ)双孤子的迎面和追赶弹性碰撞；(ⅱ)双孤子的周期性的吸引和排斥的束缚状态；(ⅲ)三孤子的能量交换碰撞。最后给出这个方程的拓展方程的无穷守恒律。
Accompanied with the rapid development of the information science technology, symbolic computation, which is the new branch of the artificial intelligence, has been extensively applied in certain branches of scientific researches, and it has been an assis-tant tool for the study of science and technology. Certain interest has been focused on the nonlinear science, which is one of the fields of current advanced scientific researches, and the soliton theory is one of the three main branches of the nonlinear science. Based on symbolic computation and Darboux transformation (DT), the dissertation is to in-vestigate the integrable properties, analytic solutions and physical applications of some nonlinear evolution equations which have physical backgrounds, some algebraic algo-rithms that can be performed with symbolic computation are also proposed. Types of equations investigated in the dissertation are mainly the variable-coefficient, coupled, higher-dimensional and higher-order nonlinear evolution equations.
The work of this dissertation includes the following aspects:
(1) Comparing with the equations with constant coefficients, the corresponding ones with variable-coefficients can describe the physical mechanism of nonlinear phenomena in physical situations with inhomogeneities media or nonuniformities of boundaries, and be considered to be more practical. To describe various complex nonlinear phenomena in real situation, some equations with time and space variable-coefficients from certain physical settings have been proposed. Due to its relevance to fundamental aspects of physics and technology, coherent atom optics is the subject of much current interest. For that purpose, the lower dimensional condensates have been the subject of active studies. Certain dynamics of the Bose-Einstein condensates (BECs) have been claimed to be governed by the one-dimensional Gross-Pitaevskii (GP) equation for a time-dependent trap. In this chapter, by virtue of the Painleve analysis and symbolic computation, we will derive the integrable condition for the GP equation with the time-dependent scat-tering length in the presence ot a confining or expulsive time-dependent trap. Lax pair for the equation will be directly obtained via the Ablowitz-Kaup-Newell-Segur (AKNS) scheme under the integrable condition. Bright one-soliton-like solution of the GP equa-tion will be presented via the auto-Backlund transformation and some analytic solutions with variable amplitudes will be obtained by the ansatz method. In addition, an infinite number of conservation laws will be derived. Those results could be of some values for the studies on the BECs.
(2) Nowadays,(2+1)-dimensional nonlinear evolution equations in soliton theory and mathematical physics are the subject of active studies. Due to the limitation of the dimension, the pure (1+1)-dimensional systems can not account for some ob-served features. In realistic situations, the higher dimensional systems may provide more complex models. As a typical example in (2+1)-dimensional systems, the Kadomtsev-Petviashvili (KP) system has been derived from many physical applications in the Bose-Einstein condensation, plasma physics and nonlinear optics, etc. In this chapter, a coupled KP system is investigated with symbolic computation, some researches have proposed that the system could be decomposed into the first two members of the (1+1)-dimensional AKNS hierarchy, and based on that decomposition, we will obtain three kinds of DTs of its reduced equations. Moreover, the multi-soliton-like solutions of the coupled KP system will be derived. Finally some figures will be plotted to discuss the propagation features of the soliton solutions. Those solutions could be of some values for the studies in the context of random matrix theory.
(3) In this chapter, by virtue of the DT and symbolic computation, the Kundu-Eckhaus (KE) equation with the cubic and quintic nonlinear terms and KE equation with variable coefficients will be investigated. The KE equation appears in the nonlin-ear optics, quantum field theory and weakly nonlinear dispersive matter waves. The KE equation with variable coefficients, which is an extension of the KE one, possesses the soliton solutions with more parametric freedom, therefore its solutions can be expected to model more complex situations in reality than the KE counterpart. Through DT, one-, two-and three-soli ton solutions of the KE equation will be presented in the form of modulus; Some physical quantities such as the amplitude, width, initial phase and energy of soliton solutions will be obtained; The asymptotic behavior of the two-soli ton solution will be analyzed, and shows that the collision is elastic; Figures will be plotted to discuss the propagation features of the soliton solutions:(ⅰ) Elastic collisions of the two solitons, the head-on and overtaking;(ii) Periodic attraction and repulsion of the bounded states of the two solitons;(iii) Two types of the dynamic characters of the three solitons:Three solitons intersect at a point at the moment of the collision and the soli-tons keep their shapes except for the phase shifts; Two solitons keep the bounded state and one another soliton retains its own shape invariant before and after the collision. Through the extension of the KE equation, the KE equation with variable coefficients will also be investigated in this chapter by virtue of the DT and symbolic computation. Lax pair of the equation will be obtained, and the corresponding DT will be constructed. One-, two-and three-soliton solutions will be presented, and the asymptotic behaviors of the two-soliton solution will be analyzed. Figures will be plotted to discuss the propaga-tion features of the soliton solutions:(i) As the one-soli ton solution, for different choices of the nonlinear dispersion, we obtain different soliton structures:When the nonlinear dispersion is constant, the soliton propagates with an invariant energy, amplitude and uniform velocity; When the nonlinear dispersion is variable, we present five different types of soliton structures with the varying velocities;(ii) The head-on collision, over-taking collision, parabolic solitons, periodic oscillation and the bounded states of the two solitons are analyzed;(iii) Four different dynamic characters of the three solitons are presented:Three solitons collide at a point, keeping their shapes unchanged except for the phase shifts before and after the collision; Two solitons keep the bounded state and the third one retains its own shape invariant before and after the collision; Three parabolic solitons; Collisions between one parabolic soliton and a two-soliton-bounded structure.
(4) In this chapter, the quintic generalization of the coupled cubic nonlinear Schrodinger equations from twin-core nonlinear optical fibers and waveguides will be studied, which describe the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. Lax pair of the equations will be obtained and the corresponding DT will be constructed. Moreover, one-, two-and three-soliton solutions will be presented in the form of modulus. Dynamic features of solitons will be graphically discussed:(i) Head-on and overtaking elastic collisions of the two solitons;(ii) Periodic attraction and repulsion of the bounded states of two solitons;(iii) Energy-exchanging collisions of the three solitons. Finally an infinite number of conservation laws of the extension equation will also be presented.
引文
[1]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.
[2]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[博士学位论文].北京邮电大学,2008.
[3]郝柏林.混沌与分形.上海科学技术出版社,2004.
[4]李丽莉.基于计算机符号计算的非线性模型孤子解研究[博士学位论文].北京邮电大学,2009.
[5]陈登远.孤子引论.科学出版社,2006.
[6]田播.计算机符号计算在非线性研究中的若干应用[博士学位论文].北京航空航天大学,2003.
[7]李翊神.孤子与可积系统.上海科技教育出版社,1999.
[8]范恩贵.可积系统与计算机代数.科学出版社,2004.
[9]吴文俊.数学机械化.科学出版社,1999.
[10]吴文俊.吴文俊论数学机械化,山东教育出版社,1986.
[11]郝孝良,周义仓译.Mathematica全书.西安交通人学出版社,2002.
[12]王东明.符号计算选讲.清华大学出版社,2003.
[13]王东明.计算机代数.清华大学出版社,2004.
[14]张宝善Mathematica符号运算与数学实验.南京大学出版社,2007.
[15]Yan Z Y. Advances in nonlinear waves and symbolic computation. Nova Science Publishers,2009.
[16]魏诺.非线性科学基础与应用.科学出版社.2004.
[17]Russell J S. Report on waves. Report of the 14th Meeting of the British Association for Advancement of Science, John Murray Press,1844,311-390.
[18]许韬.符号计算与若干非线性发展方程的解析研究[博士学位论文].北京邮电大学,2011.
[19]Korteweg D J, de Vries G. On the change of form of long waves advancing in a rectangular canal and a new type of long solitary waves. Phil. Mag.,39,1895,422-443.
[20]王明亮.非线性发展方程与孤立子.兰州大学出版社,1990.
[21]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.
[22]庞小峰.孤子物理学.四川科学技术出版社,2003.
[23]刘式适,刘式达.非线性大气动力学.国防工业出版社,1996.
[24]郭柏灵.非线性演化方程.上海科技教育出版社,1995.
[25]庞小峰.非线性量子力学理论.重庆出版社,1994.
[26]Ablowitz M J, Clarkson P A. Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press,1992.
[27]Agrawal G P. Nonlinear fiber optics. Academic Press,2002.
[28]Drazin P G, Johnson R S. Solitons:an Introduction. Cambridge University Press,1989.
[29]谷超豪,郭柏灵,李翊神等.孤立子理论与应用.浙江科学出版社,1990.
[30]Ablowitz M J, Segur H. Solitons and the inverse scattering transform. SIAM,1981.
[31]Dickey L A. Soliton equations and Hamiltonian systems. World Scientific Pre Singapore,2003.
[32]Steeb W H, Euler N. Nonlinear evolution equations and Painleve test. World Sci-entific Press,1989.
[33]Ablowitz M J, Kaup D J, Newell A C, et al. Nonlinear-evolution equations of physical significance. Phys. Rev. Lett.,31,1973,125-127.
[34]Kaup D J, Newell A C. An exact solution for a derivative nonlinear Schrodinger equation. J. Math. Phys.,19,1978,798-801.
[35]Wadati M, Konno K, Ichikawa Y H. A generalization of inverse scattering method. J. Phys. Soc. Jpn.,46,1979,1965-1966.
[36]Gao Y T, Tian B. On the non-planar dust-ion-acoustic waves in cosmic dusty plas-mas with transverse perturbations. Europhys. Lett.,77,2007,15001:1-5.
[37]杨伯君,赵玉芳.高等数学物理方法.北京邮电大学出版社,2003.
[38]Matveev V B, Salle M A. Darboux Transformations and Solitons. Springer,1991.
[39]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.
[40]Miura R M, Gardner C S, Kruskal M D. Korteweg-de Vries equation and general-izations. Ⅱ. Existence of conservation laws and constants of motion. J. Math. Phys., 9,1968,1204-1209.
[41]Lax P D. Integrals of nonlinear equations of evolution and solitray waves. Commun. Pure Appl. Math.,21,1968,467-490.
[42]Zakharov V E, Shabat A B. Interaction between solitons in a stable medium. Sov. Phys. JETP,37,1973,823-828.
[43]Hasegawa A. Theory and computer experiment on self-trapping instability of plasma cyclotron waves. Phys. Fluids,15,1972,870-881.
[44]Darboux G. Sure une proposition relative aux equations lineaires. Compt. Rend. Acad. Sci. Paris,94,1882.1456-1459.
[45]Cao C W, Wu Y T, Geng X G. Relation between the Kadometsev-Petviashvili equation and the confocal involutive system. J. Math. Phys..40,1999,3948-3970.
[46]Cao C W, Geng X G, Wang H G. Algebro—geometric solution of the 2+1 dimen-sional Burgers equation with a discrete variable. J. Math. Phys.,43,2002,621-643.
[47]Cheng Y. Li Y S. The constraint of the Kadomtsev-Petviashvili equation and its special solutions. Phys. Lett. A,157.1991,22-26.
[48]Li B, Chen Y, Zhang H Q. Auto-Backlund 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.
[49]Elwakil S A, El-labany S K, Zahran M A, et al. Exact travelling wave solutions for the generalized shallow water wave equation. Chaos Solitons and Fractals,17,2003, 121-126.
[50]Yana Z Y. Painleve analysis,auto-Bacldund transformations and exact solutions for a simplified model for reacting mixtures. Physica A,326,2003,344-359.
[51]Gao Y T, Tian B. On the non-planar dust-ion-acoustic waves in cosmic dusty plas-mas with transverse perturbations. Europhys. Lett.,77,2007,15001:1-5.
[52]Kumar V R, Radha R, Panigrahi P K. Dynamics of Bose-Einstein condensates in a time-dependent trap. Phys. Rev. A,77,2008,023611-023616.
[53]Bao W Z, Jaksch D, Markowich P A. Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys.,187,2003,318-342.
[54]Pitaevskii L P, Stringari S. Bose-Einstein Condensation. Oxford University Press, 2003.
[55]Zhang J F. Generalized noether theorem of high-order nonholonomic nonconserva-tive dynamical systems. Opt. Commun.,258,2006,35-42.
[56]Hirota R, Ohta Y. Hierarchies of coupled soli ton equations. J. Phys. Soc. Jpn.,60, 1991,798-809.
[57]Kakei S. Dressing method and the coupled KP hierarchy. Phys. Lett. A,246,2002, 449-458.
[58]Isojima S, Willox R, Satsuma J. Spider-web solutions of the coupled KP equation. J. Phys. A,36,2003,9533-9552.
[59]Geng X G. Algebraic-geometrical solutions of some multidimensional nonlinear evo-lution equations. J. Phys. A.36,2003,2289-2303.
[60]Adler M, Horozov E, Moerbeke P. The Pfaff Lattice and Skew-orthogonal polyno-mials. Int. Math. Res. Notices,11,1999,569-588.
[61]Kundu A. Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrodinger type equation. J. Math. Phys.,25,1984,3433-3438.
[62]Calogero F, Eckhaus W. Nonlinear evolution equations, rescalings, model PDES and their integrability:I. Inv. Prob.3,1987,229-262.
[63]Levi D, Scimiterna C. The Kundu-Eckhaus equation and its discretizations. J. Phys. A,42,2009,465203-465210.
[64]Radhakrishnan R, Kundu A, Lakshmanan M. Coupled nonlinear Schrodinger equa-tions with cubic-quintic nonlinearity:Integrability and soliton interaction in non-Kerr media. Phys. Rev. E,60,1999,3314-3323.
[65]Zong F D, Dai C Q, Zhang J F. Optical Solitary waves in fourth-order dispersive nonlinear Schrodinger equation with self-steepening and self-frequency shift. Com-mun. Theor. Phys.,45,2006,721-726.
[66]Wang M L, Zhang J L, Li X Z. Solitary wave solutions of a generalized derivative nonlinear Schrodinger equation. Commun. Theor. Phys.,50,2008,39-42.
[67]Clarkson P A, Tuszynski J A. Exact solutions of the multidimensional derivative nonlinear Schrodinger equation for many-body systems of criticality. J. Phys. A,23, 1990,4269-4288.
[68]Kodama Y. Optical solitons in a monomode fiber. J. Stat. Phys.,39,1985,597-614.
[69]Johnson R S. On the modulation of water waves in the neighbourhood of kh≈ 1.363. Proc. Roy. Soc. London A,357,1977,131-141.
[70]Albuch L, Malomed B A. Transitions between symmetric and asymmetric solitons in dual-core systems with cubie-quintic nonlinearity. Math. Commun. Simul.,74, 2007,312-322.
[1]Winiecki T, Jackson B, McCann J F, et al. Vortex shedding and drag in dilute Bose-Einstein condensates, J. Phys. B,33,2000,4069-4078.
[2]Svidzinsky A A, Fetter A L. Stability of a vortex in a trapped Bose-Einstein con-densate. Phys. Rev. Lett.,84,2000,5919-5923.
[3]Fetter A L, Svidzinsky A A. Vortices in a trapped dilute Bose-Einstein condensate. J. Phys. Condens. Matter,13,2001,135-194.
[4]Demokritov S O, Demidov V E, Dzyapko O, et al. Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping. Nature,443,2006, 430-433.
[5]Mewes M O, Andrews M R, Druten N J van, et al. Bose-Einstein condensation in a tightly confining de magnetic trap. Phys. Rev. Lett.,77,1996,416-419.
[6]Wu L, Zhang J F, Li L. Modulational instability and bright solitary wave solution for Bose-Einstein condensates with time-dependent scattering length and harmonic potential. New J. Phys.,9,2007,69:1-13.
[7]Davis K B, Mewes M O, Andrews M R, et al. Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett.,75,1995,3969-3974.
[8]Parkins A S, Walls D F. The physics of trapped dilute-gas Bose-Einstein condensates. Phys. Rep.,303,1998,1-80.
[9]Lu X, Tian B, Xu T, et al. Analytical study of the nonlinear Schrodinger equation with an arbitrary linear time-dependent potential in quasi-one-dimensional Bose-Einstein condensates. Ann. Phys. (NY),323.2008,2554-2565.
[10]Radha R. Kumar V R. Bright matter wave solitons and their collision in Bose-Einstein condensates. Phys. Lett. A,370,2008,46-50.
[11]Konotop V V, Pacciani P. Collapse of solutions of the nonlinear schrodinger equation with a time-dependent nonlinearity:application to Bose-Einstein condensates. Phys. Rev. Lett,,94.2005,240405-240408.
[12]Slasnich L, Parola A, Reatto L. Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates. Phys. Rev. A,65,2002,043614-043619.
[13]Liang Z X, Zhang Z D, Liu W M. Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic-potential. Phys. Rev. Lett.,94,2005,050402-050405.
[14]Gross E P. Hydrodynamics of a superfluid condensate. J. Math. Phys.,4,1963, 195-207.
[15]Gross E P. Structure of a quantized vortex in boson systems. Nuov. Cim.,20,1961, 454-477.
[16]Bao W Z, Jaksch D, Markowich P A. Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation. J. Comput. Phys.,187,2003,318-342.
[17]Pitaevskii L P, Stringari S. Bose-Einstein condensation. Oxford University Press, 2003.
[18]Yang Q, Zhang J F. Bose-Einstein soli tons in time-dependent linear potential. Opt. Commun.,258,2006,35-42.
[19]Kumar V R, Radha R, Panigrahi P K. Dynamics of Bose-Einstein condensates in a time-dependent trap. Phys. Rev. A,77,2008,023611-023616.
[20]Pelinovsky D E, Kevrekidis P G, Frantzeskakis D J. Averaging for solitons with nonlinearity management. Phys. Rev. Lett.,91,2003,240201-240206.
[21]Brazhnyi V A, Konotop V V. Theory of nonlinear matter waves in optical lattices. Mod. Phys. Lett. B,18,2004,627-551.
[22]Sakaguehi H, Malomed B A. Resonant nonlinearity management for nonlinear Sehrodinger solitons. Phys. Rev. E,70,2004,066613-066618.
[23]Liu S K, Gao B, Fu Z T, et al. Envelope periodie solutions to one-dimensional Gross-Pitaevskii equation in Bose-Einstein condensation. Commun. Theor. Phys., 51,2009,1069-1072.
[24]Tian B, Gao Y T, Zhu H W. Variable-coefficient higher-order nonlinear Schrodinger model in optical fibers:Variable-coefficient bilinear form, Backlund transformation, brightons and symbolic computation. Phys. Lett. A,366,2007,223-229.
[25]Tian B, Wei G M, Zhang C Y, et al. Transformations for a generalized variable-coefficient Korteweg-de Vries model from blood vessels, Bose-Einstein condensates, rods and positons with symbolic computation. Phys. Lett. A,356,2006,8-16.
[26]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.
[27]Gao Y T, Tian B. Cylindrical Kadomtsev-Petviashvili model, nebulons and sym-bolic computation for cosmic dust ion-acoustic waves. Phys. Lett. A,349,2006, 314-319.
[28]Gao Y T, Tian B. Reply to:"Comment on:'Spherical Kadomtsev-Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computa-tion'" [Phys. Lett. A 361 (2007) 520]. Phys. Lett. A,361,2007,523-528.
[29]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, 120703:1-4.
[30]Das G, Sarma J. Response to "Comment on'A new mathematical approach for finding the solitary waves in dusty plasma'" [Phys. Plasmas 6,4392 (1999)]. Phys Plasmas,6,1999,4394:1-4.
[31]Hong W P. Comment on:"Spherical Kadomtsev-Petviashvili equation and neb-ulons for dust ion-acoustic waves with symbolic computation" [Phys. Lett. A 340 (2005) 243]. Phys. Lett. A,361,2007,520-522.
[32]Atre R, Panigrahi P K, Agarwal G S. Class of solitary wave solutions of the one-dimensional Gross-Pitaevskii equation. Phys. Rev. E,73,2006,056611-05665.
[33]Weiss J, Tabor M, Carnevale G. The Painleve property for partial differential equa-tions. J. Math. Phys.,24,1983.522-526.
[34]Ma W X. Darboux transformations for a Lax integrable system in 2n-dimensions. Lett. Math. Phys.,39,1997,33-49.
[35]Ma W X, Guo F K. Lax representations and zero-curvature representations by the Kronecker product. Int. J. Theor. Phys.,36,1997,697-704.
[36]Wright O C, Forest M G. On the Backlund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrodinger system. Phys. D,141,2000,104-106.
[37]Ablowitz M J, Clarkson P A. Nonlinear evolution equations and inverse scattering. Cambridge University Press,1991.
[38]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.
[39]李志斌.非线性数学物理方程的行波解.科学出版社,2007.
[40]Conte R, Musette M. Link between solitary waves and projective Riccati equations. J. Phys. A,25,1992,5609-5623.
[41]Lu B Q, Xiu B Z, Pang Z L, et al. Exact traveling wave solution of one class of nonlinear diffusion equations. Phys. Lett. A,175,1993,113-115.
[42]Parkes E J, Duffy B R, Abbott P C. The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations. Phys. Lett. A,295,2002, 280-286.
[43]Ma W X, Chen M. Direct search for exact solutions to the nonlinear Schrodinger equation. Appl. Math. Comput.,215,2009,2835-2842.
[44]Parkes E J, Duffy B R. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. Phys. Commun.,98,1996, 288-300.
[45]Parkes E J, Duffy B R. Travelling solitary wave solutions to a compound KdVBurg-ers equation. Phys. Lett. A,229,1997,217-220.
[46]Li Z B, Liu Y P. RAEEM:A Maple package for finding a series of exact traveling wave solutions for nonlinear evolution equations. Comput. Phys. Commun.,163, 2004,191-201.
[47]杨伯君,赵玉芳.高等数学物理方法.北京邮电大学山版社,2003.
[48]倪皖荪,魏荣爵.水槽中的孤波.上海科技教育出版社,1997.
[49]Zhang D J, Chen D Y. The conservation laws of some discrete soliton systems. Chaos Solitons Frac.,14,2002,573-579.
[1]Xue J K. The nonlinear evolution of ring dark solitons in Bose-Einstein condensates. J. Phys. A,37,2004,11223-11228.
[2]Hong W P. Comment on:" Spherical Kadomtsev-Petviashvili equation and neb-ulons for dust ion-acoustic waves with symbolic computation" [Phys. Lett. A 340 (2005) 243]. Phys. Lett. A,361,2007,520-522.
[3]David D, Levi D, Winternitz P. Integrable nonlinear equations for water waves in straits of varying width and depth. Stud. Appl. Math.,76,1987,133-168.
[4]Santini P M. On the evolution of two-dimensional packets of water waves over an uneven bottom. Lett. Nuov. Cim.,30,1983,236-240.
[5]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.
[6]Kakei S. Dressing method and the coupled KP hierarchy. Phys. Lett. A,246,2002, 449-458.
[7]Isojima S, Willox R, Satsuma J. Spider-web solutions of the coupled KP equation. J. Phys. A,36,2003,9533-9552.
[8]Isojima S, Willox R, Satsuma J. On various solutions of the coupled KP equation. J. Phys. A,35,2002,6893-6909.
[9]Adler M, Horozov E, Moerbeke P. The Pfaff Lattice and Skew-orthogonal polynomi-als. Int. Math. Res. Notices,11,1999,569-588.
[10]Jimbo M, Miwa T. Solitons and infinite dimensional Lie algebras. Publ. Res. Inst. Math. Sci.,19.1983,943-1001.
[11]Wang J, Li B. Symmetry and general symmetry groups of the coupled Kadomtsev-Petviashvili equation. Chin. Phys. Soc.,18,2009,2109-2114.
[12]Hietarinta J. Partially integrable evolution equations in physics. Springer,1990.
[13]Hirota R, Ohta Y. Hierarchies of coupled soliton equations. I. J. Phys. Soc. Jpn. 60,1991,798-809.
[14]Tian B, Gao Y T, Zhu H W. Variable-coefficient higher-order nonlinear Schrodinger model in optical fibers:Variable-coefficient bilinear form, Backlund transformation, brightons and symbolic computation. Phys. Lett. A,366,2007,223-229.
[15]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.
[16]许韬.符号计算与若干非线性发展方程的解析研究[博士学位论文].北京邮电大学,2011.
[17]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.
[18]Gao Y T, Tian B. Cylindrical Kadomtsev-Petviashvili model, nebulons and sym-bolic computation for cosmic dust ion-acoustic waves. Phys. Lett. A,349,2006, 314-319.
[19]Gao Y T, Tian B. Reply to:"Comment on:'Spherical Kadomtsev-Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computa-tion'" [Phys. Lett. A 361 (2007) 520]. Phys. Lett. A,361,2007,523-528.
[20]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[博士学位论文].北京邮电大学,2008.
[21]李丽莉.基于计算机符号计算的非线性模型孤子解研究[博士学位论文].北京邮电大学,2009.
[22]Das G, Sarma J. Response to "Comment on'A new mathematical approach for finding the solitary waves in dusty plasma'" [Phys. Plasmas 6,4392 (1999)]. Phys. Plasmas,6,1999,4394:1-4.
[23]Geng X G. Algebraic-geometrical solutions of some multidimensional nonlinear evo-lution equations. J. Phys. A,36.2003,2289-2303.
[24]Van de Leur J. Backlund-Darboux transformations for the coupled KP hierarchy. J. Phys. A.37,2004.4395-4405.
[25]Yang G Y, Liu Q P. A Darboux transformation for the coupled Kadomtsev-Petviashvili equation. Chin. Phys. Lett.,25,2008,1-3.
[26]Ablowitz M J, Kaup D J, Newell A C, et al. Nonlinear-evolution equations of physical significance. Phys. Rev. Lett.,31,1973,125-127.
[27]Li J, Zhang H Q, Xu T, et al. Soliton-like solutions of a generalized variable-coefficient higher order nonlinear Schrodinger equation from inhomogeneous optical fibers with symbolic computation. J. Phys. A,40,2007,13299-13309.
[28]Porsezian K, Nakkeeran K. Optical soli tons in presence of Kerr dispersion and self-frequency shift. Phys. Rev. Lett.,76,1996,3955-3958.
[29]Mahalingam A, Porsezian K. Propagation of dark solitons with higher-order effects in optical fibers. Phys. Rev. E,64,2001,046608-046616.
[30]Wright O C, Forest M G. On the Backlund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrodinger system. Phys. D,141,2000,104-116.
[31]Li Y S. Soliton and Integrable System. Shanghai Scientific and Technological Edu-cation Publishing,1999.
[32]Maxworhy T. On the formation of nonlinear internal waves from the gravitational collapse of mixed regions in two and three dimensions. J. Fluid. Mech.,96,1980, 47-64.
[33]Biondini G, Chakravarty S. Soliton solutions of the Kadomtsev-Petviashvili II equa-tion. J. Math. Phys.,47,2006,033514-033539.
[34]Lonngren K E. Ion acoustic soliton experiments in a plasma. Opt. Quantum Elec-tron.,30,1998,615-630.
[35]Folkes P A, Ikezi H, Davis R. Two-dimensional interaction of ion-acoustic solitons. J. Phys. Rev. Lett.,45,1980,902-904.
[36]Tsukabayashi I, Nakamura Y. Resonant interaction of cylindrical ion-acoustic soli-tons. Phy. Lett. A,85,1981,151-154.
[37]Kakei S. Orthogonal and symplectic matrix integrals and coupled KP hierarchy. J. Phys. Soc. Jpn.,68,1999,2875-2877.
[1]Hasegawa A, Tappert F. Transmission of stationary nonlinear optical pulses in dis-persive dielectric fibers. II. Normal dispersion. Appl. Phys. Lett.,23,1973,171-172.
[2]Mollenauer L F, Stolen R H, Gordon J P. Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys. Rev. Lett.,45,1980,1095-1098.
[3]Zhang H Q, Xu T, Li J, et al. Integrability of an N-coupled nonlinear Schrodinger system for polarized optical waves in an isotropic medium via symbolic computation. Phys. Rev. E,77,2008,026605-026614.
[4]Liu W J, Tian B, Zhang H Q, et al. Soliton interaction in the higher-order nonlinear Schrodinger equation investigated with Hirota's bilinear method, Phys. Rev. E,77, 2008,066605-066611.
[5]Gedalin M, Scott T C, Band Y B. Optical solitons in the higher order nonlinear Schrodinger equation. Phys. Rev. Lett.,78,1997,448-451.
[6]Mihalache D, Truta N, Crasovan L C. Painleve analysis and bright solitary waves of the higher-order nonlinear Schrodinger equation containing third-order dispersion and self-steepening term. Phys. Rev. E,56,1997,1064-1070.
[7]Yan Z Y. Optical solitary wave solutions to nonlinear schrodinger equation with cubic-quintic nonlinearity in non-Kerr media. J. Phys. Soc. Jpn.,73,2004,2397-2401.
[8]Newell A C, Moloney J V. Nonlinear Optics. Addison-Wesley.1992.
[9]Hong W P. Optical solitary wave solutions for the higher order nonlinear Schrodinger equation with cubic-quintic non-Kerr terms. Opt. Commun.,194,2001,217-223.
[10]Levi D, Scimiterna C. The Kundu-Eckhaus equation and its discretizations. J. Phys. A,42,2009,465203-465210.
[11]Radhakrishnan R. Kundu A, Lakshmanan M. Coupled nonlinear Schrodinger equa-tions with cubic-quintic nonlinearity:Integrability and soliton interaction in non-Kerr media. Phys. Rev. E,60,1999,3314-3323.
[12]Calogero F, Eckhaus W. Nonlinear evolution equations, rescalings, model PDES and their integrability:I. Inv. Prob.,3,1987,229-262.
[13]Kundu A. Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrodinger type equation. J. Math. Phys.,25,1984,3433-3438.
[14]Zong F D, Dai C Q, Zhang J F. Optical solitary waves in fourth-order dispersive nonlinear Schrodinger equation with self-steepening and self-frequency shift. Com-mun. Theor. Phys.,45,2006,721-726.
[15]Clarkson P A, Tuszynski J A. Exact solutions of the multidimensional derivative nonlinear Schrodinger equation for many-body systems of criticality. J. Phys. A,23, 1990,4269-4288.
[16]Kodama Y. Optical solitons in a monomode fiber. J. Stat. Phys.,39,1985,597-614.
[17]Wang M L, Zhang J L, Li X Z. Solitary wave solutions of a generalized derivative nonlinear Schrodinger equation. Commun. Theor. Phys.,50,2008,39-42.
[18]Johnson R S. On the modulation of water waves in the neighbourhood of kh≈ 1.363. Proc. Roy. Soc. London A,357,1977,131-141.
[19]Levko D, Volkov A. Modeling of Kundu-Eckhaus equation. ArXiv:nlin.PS/0702050, 2007.
[20]Geng X G, Tam H W. Darboux transformation and soli ton solutions for generalized nonlinear Schrodinger equations. J. Phys. Soc. Jpn.,68,1999.1508-1512.
[21]Hao R Y, Li L, Li Z H, et al. A new way to exact quasi-soliton solutions and soliton interaction for the cubic-quintic nonlinear Schrodinger equation with variable coefficients. Opt. Commun.,245,2005,383-390.
[22]Tian B. Gao Y T, Zhu H W. Variable-coefficient higher-order nonlinear Schrodinger model in optical fibers:Variable-coefficient bilinear form, Backlund transformation. brightons and symbolic computation. Phys. Lett. A,366,2007,223-229.
[23]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.
[24]李丽莉.基于计算机符号计算的非线性模型孤子解研究[博士学位论文].北京邮电大学,2009.
[25]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[博士学位论文].北京邮电大学,2008.
[26]Tian B, Wei G M, Zhang C Y, et al. Transformations for a generalized variable-coefficient Korteweg-de Vries model from blood vessels, Bose-Einstein condensates, rods and positons with symbolic computation. Phys. Lett. A,356,2006,8-16.
[27]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.
[28]Gao Y T, Tian B. Cylindrical Kadomtsev-Petviashvili model, nebulons and sym-bolic computation for cosmic dust ion-acoustic waves. Phys. Lett. A,349,2006, 314-319.
[29]Gao Y T, Tian B. Reply to:"Comment on:'Spherical Kadomtsev-Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computa-tion'" [Phys. Lett. A 361 (2007) 520]. Phys. Lett. A,361,2007,523-528.
[30]许韬.符号计算与若干非线性发展方程的解析研究[博士学位论文].北京邮电大学,2011.
[31]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, 120703:1-4.
[32]Gao Y T, Tian B. On the non-planar dust-ion-acoustic waves in cosmic dusty plas-mas with transverse perturbations. Europhys. Lett.,77,2007,15001:1-6.
[33]Das G, Sarma J. Response to "Comment on 'A new mathematical approach for finding the solitary waves in dusty plasma'" [Phys. Plasmas 6,4392 (1999)]. Phys. Plasmas.6,1999.4394:1-4.
[34]Hong W P. Comment on:"Spherical Kadomtsev-Petviashvili equation and neb-ulons for dust ion-acoustic waves with symbolic computation" [Phys. Lett. A 340 (2005) 243]. Phys. Lett. A,361,2007,520-522.
[35]田播.计算机符号计算在非线性研究中的若干应用[博士学位论文].北京航空航天大学,2003.
[36]Ablowitz M J, Kaup D J, Newell A C, et al. Nonlinear-evolution equations of physical significance. Phys. Rev. Lett.,31,1973,125-127.
[37]Li J, Zhang H Q, Xu T, et al. Soliton-like solutions of a generalized variable-coefficient higher order nonlinear Schrodinger equation from inhomogeneous optical fibers with symbolic computation. J. Phys. A,40,2007,13299-13309.
[38]Porsezian K, Nakkeeran K. Optical solitons in presence of Kerr dispersion and self-frequency shift. Phys. Rev. Lett.,76,1996,3955-3958.
[39]Mahalingam A, Porsezian K. Propagation of dark solitons with higher-order effects in optical fibers. Phys. Rev. E,64,2001,046608-046616.
[40]Ablowitz M J, Kaup D J, Newell A C, et al. Method for solving the Sine-Gordon equation. Phys. Rev. Lett.,30,1973,1262-1264.
[41]Wright O C, Forest M G. On the Backlund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrodinger system. Phys. D,141,2000,104-116.
[42]Li J, Zhang H Q, Xu T, et al. Symbolic computation on the multi-soliton-like solutions of the cylindrical Kadomtsev-Petviashvili equation from dusty plasmas. J. Phys. A,40,2007,7643-7657.
[43]Li Y S. Soliton and Integrable System, Shanghai Scientific and Technological Edu-cation Publishing,1999.
[44]Zhang H Q, Tian B, Lu X. et al. Soliton interaction in the coupled mixed derivative nonlinear Schrodinger equations. Phys. Lett. A,373,2009,4315-4321.
[45]Seong N H. Kim D Y. Experimental observation of stable bound solitons in a figure-eight fiber laser. Opt. Lett.,27,2002,1321-1323.
[46]Kim W S, Moon H T. Soliton-kink interactions in a generalized nonlinear Schrodinger system. Phys. Lett. A,266,2000,364-369.
[47]Malomed B A. Bound solitons in coupled nonlinear Schrodinger equations. Phys. Rev. A,45,1991,8321-8323.
[48]Haelterman M, Shepppard A. Bifurcation phenomena and multiple soliton-bound states in isotropic Kerr media. Phys. Rev. E,49,1994,3376-3381.
[49]Sun Z Y, Gao Y T, Yu X, et al. Bound vector solitons and soliton complexes for the coupled nonlinear Schrodinger equations. Phys. Rev. E,80,2009,066608-066618.
[1]Radhakrishnan R, Kundu A, Lakshmanan M. Coupled nonlinear Schrodinger equa-tions with cubic-quintic nonlinearity:Integrability and soliton interaction in non-Kerr media. Phys. Rev. E,60,1999,3314-3323.
[2]Zhang H Q, Xu T, Li J, et al. Integrability of an N-coupled nonlinear Schrodinger system for polarized optical waves in an isotropic medium via symbolic computation. Phys. Rev. E,77,2008,026605-026614.
[3]Mollenauer L F, Stolen R H, Gordon J P. Experimental observation of picosecond pulse narrowing and solitons in Optical fibers. Phys. Rev. Lett.,45,1980,1095-1098.
[4]Liu W J, Tian B, Zhang H Q, et al. Soliton interaction in the higher-order nonlinear Schrodinger equation investigated with Hirota's bilinear method. Phys. Rev. E,77, 2008,066605-066611.
[5]Porsezian K, Kuriakose V C. Optical Solitons:Theoretical and experimental chal-lenges. Springer,2003.
[6]Hasegawa A, Kodama Y. Solitons in Optical Communications. Oxford University, 1995.
[7]Islam M N. Ultrafast Fiber Switching Devices and Systems. Cambridge University, 1992.
[8]Newell A C, Moloney J V. Nonlinear Optics. Addison-Wesley,1992.
[9]Abdullaev F, Darmanyan S, Khabibullaev P. Optical Solitons. Springer-Verlag,1993.
[10]Hong W P. Optical solitary wave solutions for the higher order nonlinear Schrodinger equation with cubic-quintic non-Kerr terms. Opt. Commun.,194,2001, 217-223.
[11]Skarka V, Berezhiani V I, Miklaszewski R. Spatiotemporal soliton propagation in saturating nonlinear optical media. Phys. Rev. E,56,1997,1080-1087.
[12]Afanasjev V V, Chu P L, Kivshar Y S. Breathing spatial solitons in non-Kerr media. Opt. Lett.,22.1997,1388-1390.
[13]Dattoli G, Orsitto F P, Torre A. Evidence for multistability of light solitons in SF6 absorption measurements. Opt. Lett.,14,1989,456-458.
[14]Gedalin M, Scott T C, Band Y B. Optical solitons in the higher order nonlinear Schrodinger equation. Phys. Rev. Lett.,78,1997,448-451.
[15]Mihalache D, Truta N, Crasovan L C. Painleve analysis and bright solitary waves of the higher-order nonlinear Schrodinger equation containing third-order dispersion and self-steepening term. Phys. Rev. E,56,1997,1064-1070.
[16]Yan Z Y. Optical solitary wave solutions to nonlinear schrodinger equation with cubic-quintic nonlinearity in non-Kerr media. J. Phys. Soc. Jpn.,73,2004,2397-2401.
[17]Zong F D, Dai C Q, Zhang J F. Optical Solitary waves in fourth-order dispersive nonlinear Schrodinger equation with self-steepening and self-frequency shift. Com-mun. Theor. Phys.,45,2006,721-726.
[18]Kundu A. Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrodinger type equation. J. Math. Phys.,25,1984,3433-3438.
[19]Calogero F, Eckhaus W. Nonlinear evolution equations, rescalings, model PDES and their integrability:Ⅰ, Inv. Prob.,3,1987,229-262.
[20]Levi D, Scimiterna C. The Kundu-Eckhaus equation and its discretizations. J. Phys. A,42,2009,465203-465210.
[21]Wang M L, Zhang J L, Li X Z. Solitary wave solutions of a generalized derivative nonlinear Schrodinger equation,Commun. Theor. Phys.,50,2008,39-42.
[22]Clarkson P A, Tuszynski J A. Exact solutions of the multidimensional derivative nonlinear Schrodinger equation for many-body systems of criticality. J. Phys. A,23, 1990,4269-4288.
[23]Kodama Y. Optical solitons in a monomode fiber. J. Stat. Phys.,39,1985,597-614.
[24]Johnson R S. On the modulation of water waves in the neighbourhood of kh≈ 1.363. Proc. Roy. Soc. London A,357,1977,131-141.
[25]Albuch L, Malomed B A. Transitions between symmetric and asymmetric solitons in dual-core systems with cubic-quintic nonlinearity. Math. Commun. Simul.,74, 2007,312-322.
[26]Hisakado M, Wadati M. J. Gauge transformations among generalised nonlinear Schrodinger equations. Phys. Soc. Jpn.,63,1994,3962-3966.
[27]Hisakado M, Wadati M J. Integrable multi-component hybrid nonlinear Schrodinger equations. Phys. Soc. Jpn.,64,1995,408-413.
[28]Tian B, Gao Y T, Zhu H W. Variable-coefficient higher-order nonlinear Schrodinger model in optical fibers:Variable-coefficient bilinear form, Backlund transformation, brightons and symbolic computation. Phys. Lett. A,366,2007,223-229.
[29]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.
[30]Tian B, Gao Y T. Variable-coefficient higher-order nonlinear Schrodinger model in optical fibers:New transformation with burstons, brightons and symbolic computa-tion. Phys. Lett. A,359,2006,241-248.
[31]Tian B, Wei G M, Zhang C Y, et al. Transformations for a generalized variable-coefficient Korteweg-de Vries model from blood vessels, Bose-Einstein condensates, rods and positons with symbolic computation. Phys. Lett. A,356,2006,8-16.
[32]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.
[33]Gao Y T, Tian B. Cylindrical Kadomtsev-Petviashvili model, nebulous and sym-bolic computation for cosmic dust ion-acoustic waves. Phys. Lett. A,349,2006, 314-319.
[34]许韬.符号计算与若干非线性发展方程的解析研究[博士学位论文].北京邮电大学,2011.
[35]Gao Y T, Tian B. Cosmic dust-ion-acoustic waves, spherical modified Kadomtsev-Petviashvili model, and symbolic computation. Phys. Plasmas,13,2006,112901: 1-6.
[36]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, 120703:1-4.
[37]李丽莉.基于计算机符号计算的非线性模型孤子解研究[博士学位论文].北京邮电大学,2009.
[38]Das G, Sarma J. Response to "Comment on'A new mathematical approach for finding the solitary waves in dusty plasma'" [Phys. Plasmas 6.4392 (1999)]. Phys. Plasmas,6,1999,4394:1-4.
[39]Hong W P. Comment on:"Spherical Kadomtsev-Petviashvili equation and neb-ulons for dust ion-acoustic waves with symbolic computation" [Phys. Lett. A 340 (2005) 243]. Phys. Lett. A,361,2007,520-522.
[40]Tian B, Gao Y T. On the solitonic structures of the cylindrical dust-acoustic dust-ion-acoustic waves with symbolic computation. Phys. Lett. A,340,2005,449-455.
[41]Tian B, Gao Y T. Spherical nebulons and Backlund transformation for a space or laboratory un-magnetized dusty plasma with symbolic computation. Eur. Phys. J. D,33,2005,59-65.
[42]Tian B, Gao Y T. Comment on "Exact solutions of cylindrical and spherical dust ion acoustic waves" [Phys. Plasmas 10,4162 (2003)]. Phys. Plasmas,12,2005, 054701:1-4.
[43]Tian B, Gao Y T. Cylindrical nebulous, symbolic computation and Backlund trans-formation for the cosmic dust acoustic waves. Phys. Plasmas (Lett.),12,2005, 070703:1-4.
[44]Tian B, Gao Y T. Symbolic computation on cylindrical-modified dust-ion-acoustic nebulons in dusty plasmas. Phys. Lett. A.362,2007,283-288.
[45]Ablowitz M J, Kaup D J, Newell A C, et al. Nonlinear-evolution equations of physical significance. Phys. Rev. Lett.,31,1973,125-127.
[46]Li J, Zhang H Q, Xu T, et al. Soliton-like solutions of a generalized variable-coefficient higher order nonlinear Schrodinger equation from inhomogeneous optical fibers with symbolic computation. J. Phys. A,40,2007,13299-13309.
[47]Porsezian K, Nakkeeran K. Optical Solitons in Presence of Kerr Dispersion and Self-Frequency Shift. Phys. Rev. Lett.,76,1996,3955-3958.
[48]Mahalingam A, Porsezian K. Propagation of dark solitons with higher-order effects in optical fibers. Phys. Rev. E,64,2001,046608-046616.
[49]Ablowitz M J, Kaup D J, Newell A C, et al. Method for solving the Sine-Gordon equation. Phys. Rev. Lett.,30,1973,1262-1264.
[50]Wright O C, Forest M G. On the Backlund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrodinger system. Phys. D,141,2000,104-116.
[51]Li J, Zhang H Q, Xu T, et al. Symbolic computation on the multi-soliton-like solutions of the cylindrical Kadomtsev-Petviashvili equation from dusty plasmas. J. Phys. A,40,2007,7643-7657.
[52]Li Y S. Soliton and Integrable System. Shanghai Scientific and Technological Edu-cation Publishing,1999.
[53]Seong N H, Kim D Y. Experimental observation of stable bound solitons in a figure-eight fiber laser. Opt, Lett.,27,2002,1321-1323.
[54]Kim W S, Moon H T. Soliton-kink interactions in a generalized nonlinear Schrodinger system. Phys. Lett. A,266,2000,364-369.
[55]Malomed B A. Bound solitons in coupled nonlinear Schrodinger equations. Phys. Rev. A,45,1991,8321-8323.
[56]Haelterman M, Shepppard A. Bifurcation phenomena and multiple soliton-bound states in isotropie Kerr media. Phys. Rev. E,49,1994,3376-3381.
[57]Sun Z Y, Gao Y T, Yu X, et al. Bound vector solitons and soliton complexes for the coupled nonlinear Schrodinger equations. Phys. Rev. E,80,2009,066608-066618.
[58]Shan W R, Qi F H, Guo R, et al. Conservation laws and solitons for the coupled cubic-quintic nonlinear Schrodinger equations in nonlinear optics. Phys. Scr.,85, 2012,015002:1-9.
[59]Wadati M, Sanuki H, Konno K. Relationships among inverse method, Backlund transformation and an infinite number of conservation Laws. Prog. Theor. Phys.,53, 1975,419-436.
[60]Zhang D J, Chen D Y. The conservation laws of some discrete soliton systems. Chaos Solitons Frac., 14,2002,573-579.
[2]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[博士学位论文].北京邮电大学,2008.
[3]郝柏林.混沌与分形.上海科学技术出版社,2004.
[4]李丽莉.基于计算机符号计算的非线性模型孤子解研究[博士学位论文].北京邮电大学,2009.
[5]陈登远.孤子引论.科学出版社,2006.
[6]田播.计算机符号计算在非线性研究中的若干应用[博士学位论文].北京航空航天大学,2003.
[7]李翊神.孤子与可积系统.上海科技教育出版社,1999.
[8]范恩贵.可积系统与计算机代数.科学出版社,2004.
[9]吴文俊.数学机械化.科学出版社,1999.
[10]吴文俊.吴文俊论数学机械化,山东教育出版社,1986.
[11]郝孝良,周义仓译.Mathematica全书.西安交通人学出版社,2002.
[12]王东明.符号计算选讲.清华大学出版社,2003.
[13]王东明.计算机代数.清华大学出版社,2004.
[14]张宝善Mathematica符号运算与数学实验.南京大学出版社,2007.
[15]Yan Z Y. Advances in nonlinear waves and symbolic computation. Nova Science Publishers,2009.
[16]魏诺.非线性科学基础与应用.科学出版社.2004.
[17]Russell J S. Report on waves. Report of the 14th Meeting of the British Association for Advancement of Science, John Murray Press,1844,311-390.
[18]许韬.符号计算与若干非线性发展方程的解析研究[博士学位论文].北京邮电大学,2011.
[19]Korteweg D J, de Vries G. On the change of form of long waves advancing in a rectangular canal and a new type of long solitary waves. Phil. Mag.,39,1895,422-443.
[20]王明亮.非线性发展方程与孤立子.兰州大学出版社,1990.
[21]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.
[22]庞小峰.孤子物理学.四川科学技术出版社,2003.
[23]刘式适,刘式达.非线性大气动力学.国防工业出版社,1996.
[24]郭柏灵.非线性演化方程.上海科技教育出版社,1995.
[25]庞小峰.非线性量子力学理论.重庆出版社,1994.
[26]Ablowitz M J, Clarkson P A. Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press,1992.
[27]Agrawal G P. Nonlinear fiber optics. Academic Press,2002.
[28]Drazin P G, Johnson R S. Solitons:an Introduction. Cambridge University Press,1989.
[29]谷超豪,郭柏灵,李翊神等.孤立子理论与应用.浙江科学出版社,1990.
[30]Ablowitz M J, Segur H. Solitons and the inverse scattering transform. SIAM,1981.
[31]Dickey L A. Soliton equations and Hamiltonian systems. World Scientific Pre Singapore,2003.
[32]Steeb W H, Euler N. Nonlinear evolution equations and Painleve test. World Sci-entific Press,1989.
[33]Ablowitz M J, Kaup D J, Newell A C, et al. Nonlinear-evolution equations of physical significance. Phys. Rev. Lett.,31,1973,125-127.
[34]Kaup D J, Newell A C. An exact solution for a derivative nonlinear Schrodinger equation. J. Math. Phys.,19,1978,798-801.
[35]Wadati M, Konno K, Ichikawa Y H. A generalization of inverse scattering method. J. Phys. Soc. Jpn.,46,1979,1965-1966.
[36]Gao Y T, Tian B. On the non-planar dust-ion-acoustic waves in cosmic dusty plas-mas with transverse perturbations. Europhys. Lett.,77,2007,15001:1-5.
[37]杨伯君,赵玉芳.高等数学物理方法.北京邮电大学出版社,2003.
[38]Matveev V B, Salle M A. Darboux Transformations and Solitons. Springer,1991.
[39]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.
[40]Miura R M, Gardner C S, Kruskal M D. Korteweg-de Vries equation and general-izations. Ⅱ. Existence of conservation laws and constants of motion. J. Math. Phys., 9,1968,1204-1209.
[41]Lax P D. Integrals of nonlinear equations of evolution and solitray waves. Commun. Pure Appl. Math.,21,1968,467-490.
[42]Zakharov V E, Shabat A B. Interaction between solitons in a stable medium. Sov. Phys. JETP,37,1973,823-828.
[43]Hasegawa A. Theory and computer experiment on self-trapping instability of plasma cyclotron waves. Phys. Fluids,15,1972,870-881.
[44]Darboux G. Sure une proposition relative aux equations lineaires. Compt. Rend. Acad. Sci. Paris,94,1882.1456-1459.
[45]Cao C W, Wu Y T, Geng X G. Relation between the Kadometsev-Petviashvili equation and the confocal involutive system. J. Math. Phys..40,1999,3948-3970.
[46]Cao C W, Geng X G, Wang H G. Algebro—geometric solution of the 2+1 dimen-sional Burgers equation with a discrete variable. J. Math. Phys.,43,2002,621-643.
[47]Cheng Y. Li Y S. The constraint of the Kadomtsev-Petviashvili equation and its special solutions. Phys. Lett. A,157.1991,22-26.
[48]Li B, Chen Y, Zhang H Q. Auto-Backlund 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.
[49]Elwakil S A, El-labany S K, Zahran M A, et al. Exact travelling wave solutions for the generalized shallow water wave equation. Chaos Solitons and Fractals,17,2003, 121-126.
[50]Yana Z Y. Painleve analysis,auto-Bacldund transformations and exact solutions for a simplified model for reacting mixtures. Physica A,326,2003,344-359.
[51]Gao Y T, Tian B. On the non-planar dust-ion-acoustic waves in cosmic dusty plas-mas with transverse perturbations. Europhys. Lett.,77,2007,15001:1-5.
[52]Kumar V R, Radha R, Panigrahi P K. Dynamics of Bose-Einstein condensates in a time-dependent trap. Phys. Rev. A,77,2008,023611-023616.
[53]Bao W Z, Jaksch D, Markowich P A. Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys.,187,2003,318-342.
[54]Pitaevskii L P, Stringari S. Bose-Einstein Condensation. Oxford University Press, 2003.
[55]Zhang J F. Generalized noether theorem of high-order nonholonomic nonconserva-tive dynamical systems. Opt. Commun.,258,2006,35-42.
[56]Hirota R, Ohta Y. Hierarchies of coupled soli ton equations. J. Phys. Soc. Jpn.,60, 1991,798-809.
[57]Kakei S. Dressing method and the coupled KP hierarchy. Phys. Lett. A,246,2002, 449-458.
[58]Isojima S, Willox R, Satsuma J. Spider-web solutions of the coupled KP equation. J. Phys. A,36,2003,9533-9552.
[59]Geng X G. Algebraic-geometrical solutions of some multidimensional nonlinear evo-lution equations. J. Phys. A.36,2003,2289-2303.
[60]Adler M, Horozov E, Moerbeke P. The Pfaff Lattice and Skew-orthogonal polyno-mials. Int. Math. Res. Notices,11,1999,569-588.
[61]Kundu A. Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrodinger type equation. J. Math. Phys.,25,1984,3433-3438.
[62]Calogero F, Eckhaus W. Nonlinear evolution equations, rescalings, model PDES and their integrability:I. Inv. Prob.3,1987,229-262.
[63]Levi D, Scimiterna C. The Kundu-Eckhaus equation and its discretizations. J. Phys. A,42,2009,465203-465210.
[64]Radhakrishnan R, Kundu A, Lakshmanan M. Coupled nonlinear Schrodinger equa-tions with cubic-quintic nonlinearity:Integrability and soliton interaction in non-Kerr media. Phys. Rev. E,60,1999,3314-3323.
[65]Zong F D, Dai C Q, Zhang J F. Optical Solitary waves in fourth-order dispersive nonlinear Schrodinger equation with self-steepening and self-frequency shift. Com-mun. Theor. Phys.,45,2006,721-726.
[66]Wang M L, Zhang J L, Li X Z. Solitary wave solutions of a generalized derivative nonlinear Schrodinger equation. Commun. Theor. Phys.,50,2008,39-42.
[67]Clarkson P A, Tuszynski J A. Exact solutions of the multidimensional derivative nonlinear Schrodinger equation for many-body systems of criticality. J. Phys. A,23, 1990,4269-4288.
[68]Kodama Y. Optical solitons in a monomode fiber. J. Stat. Phys.,39,1985,597-614.
[69]Johnson R S. On the modulation of water waves in the neighbourhood of kh≈ 1.363. Proc. Roy. Soc. London A,357,1977,131-141.
[70]Albuch L, Malomed B A. Transitions between symmetric and asymmetric solitons in dual-core systems with cubie-quintic nonlinearity. Math. Commun. Simul.,74, 2007,312-322.
[1]Winiecki T, Jackson B, McCann J F, et al. Vortex shedding and drag in dilute Bose-Einstein condensates, J. Phys. B,33,2000,4069-4078.
[2]Svidzinsky A A, Fetter A L. Stability of a vortex in a trapped Bose-Einstein con-densate. Phys. Rev. Lett.,84,2000,5919-5923.
[3]Fetter A L, Svidzinsky A A. Vortices in a trapped dilute Bose-Einstein condensate. J. Phys. Condens. Matter,13,2001,135-194.
[4]Demokritov S O, Demidov V E, Dzyapko O, et al. Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping. Nature,443,2006, 430-433.
[5]Mewes M O, Andrews M R, Druten N J van, et al. Bose-Einstein condensation in a tightly confining de magnetic trap. Phys. Rev. Lett.,77,1996,416-419.
[6]Wu L, Zhang J F, Li L. Modulational instability and bright solitary wave solution for Bose-Einstein condensates with time-dependent scattering length and harmonic potential. New J. Phys.,9,2007,69:1-13.
[7]Davis K B, Mewes M O, Andrews M R, et al. Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett.,75,1995,3969-3974.
[8]Parkins A S, Walls D F. The physics of trapped dilute-gas Bose-Einstein condensates. Phys. Rep.,303,1998,1-80.
[9]Lu X, Tian B, Xu T, et al. Analytical study of the nonlinear Schrodinger equation with an arbitrary linear time-dependent potential in quasi-one-dimensional Bose-Einstein condensates. Ann. Phys. (NY),323.2008,2554-2565.
[10]Radha R. Kumar V R. Bright matter wave solitons and their collision in Bose-Einstein condensates. Phys. Lett. A,370,2008,46-50.
[11]Konotop V V, Pacciani P. Collapse of solutions of the nonlinear schrodinger equation with a time-dependent nonlinearity:application to Bose-Einstein condensates. Phys. Rev. Lett,,94.2005,240405-240408.
[12]Slasnich L, Parola A, Reatto L. Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates. Phys. Rev. A,65,2002,043614-043619.
[13]Liang Z X, Zhang Z D, Liu W M. Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic-potential. Phys. Rev. Lett.,94,2005,050402-050405.
[14]Gross E P. Hydrodynamics of a superfluid condensate. J. Math. Phys.,4,1963, 195-207.
[15]Gross E P. Structure of a quantized vortex in boson systems. Nuov. Cim.,20,1961, 454-477.
[16]Bao W Z, Jaksch D, Markowich P A. Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation. J. Comput. Phys.,187,2003,318-342.
[17]Pitaevskii L P, Stringari S. Bose-Einstein condensation. Oxford University Press, 2003.
[18]Yang Q, Zhang J F. Bose-Einstein soli tons in time-dependent linear potential. Opt. Commun.,258,2006,35-42.
[19]Kumar V R, Radha R, Panigrahi P K. Dynamics of Bose-Einstein condensates in a time-dependent trap. Phys. Rev. A,77,2008,023611-023616.
[20]Pelinovsky D E, Kevrekidis P G, Frantzeskakis D J. Averaging for solitons with nonlinearity management. Phys. Rev. Lett.,91,2003,240201-240206.
[21]Brazhnyi V A, Konotop V V. Theory of nonlinear matter waves in optical lattices. Mod. Phys. Lett. B,18,2004,627-551.
[22]Sakaguehi H, Malomed B A. Resonant nonlinearity management for nonlinear Sehrodinger solitons. Phys. Rev. E,70,2004,066613-066618.
[23]Liu S K, Gao B, Fu Z T, et al. Envelope periodie solutions to one-dimensional Gross-Pitaevskii equation in Bose-Einstein condensation. Commun. Theor. Phys., 51,2009,1069-1072.
[24]Tian B, Gao Y T, Zhu H W. Variable-coefficient higher-order nonlinear Schrodinger model in optical fibers:Variable-coefficient bilinear form, Backlund transformation, brightons and symbolic computation. Phys. Lett. A,366,2007,223-229.
[25]Tian B, Wei G M, Zhang C Y, et al. Transformations for a generalized variable-coefficient Korteweg-de Vries model from blood vessels, Bose-Einstein condensates, rods and positons with symbolic computation. Phys. Lett. A,356,2006,8-16.
[26]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.
[27]Gao Y T, Tian B. Cylindrical Kadomtsev-Petviashvili model, nebulons and sym-bolic computation for cosmic dust ion-acoustic waves. Phys. Lett. A,349,2006, 314-319.
[28]Gao Y T, Tian B. Reply to:"Comment on:'Spherical Kadomtsev-Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computa-tion'" [Phys. Lett. A 361 (2007) 520]. Phys. Lett. A,361,2007,523-528.
[29]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, 120703:1-4.
[30]Das G, Sarma J. Response to "Comment on'A new mathematical approach for finding the solitary waves in dusty plasma'" [Phys. Plasmas 6,4392 (1999)]. Phys Plasmas,6,1999,4394:1-4.
[31]Hong W P. Comment on:"Spherical Kadomtsev-Petviashvili equation and neb-ulons for dust ion-acoustic waves with symbolic computation" [Phys. Lett. A 340 (2005) 243]. Phys. Lett. A,361,2007,520-522.
[32]Atre R, Panigrahi P K, Agarwal G S. Class of solitary wave solutions of the one-dimensional Gross-Pitaevskii equation. Phys. Rev. E,73,2006,056611-05665.
[33]Weiss J, Tabor M, Carnevale G. The Painleve property for partial differential equa-tions. J. Math. Phys.,24,1983.522-526.
[34]Ma W X. Darboux transformations for a Lax integrable system in 2n-dimensions. Lett. Math. Phys.,39,1997,33-49.
[35]Ma W X, Guo F K. Lax representations and zero-curvature representations by the Kronecker product. Int. J. Theor. Phys.,36,1997,697-704.
[36]Wright O C, Forest M G. On the Backlund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrodinger system. Phys. D,141,2000,104-106.
[37]Ablowitz M J, Clarkson P A. Nonlinear evolution equations and inverse scattering. Cambridge University Press,1991.
[38]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.
[39]李志斌.非线性数学物理方程的行波解.科学出版社,2007.
[40]Conte R, Musette M. Link between solitary waves and projective Riccati equations. J. Phys. A,25,1992,5609-5623.
[41]Lu B Q, Xiu B Z, Pang Z L, et al. Exact traveling wave solution of one class of nonlinear diffusion equations. Phys. Lett. A,175,1993,113-115.
[42]Parkes E J, Duffy B R, Abbott P C. The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations. Phys. Lett. A,295,2002, 280-286.
[43]Ma W X, Chen M. Direct search for exact solutions to the nonlinear Schrodinger equation. Appl. Math. Comput.,215,2009,2835-2842.
[44]Parkes E J, Duffy B R. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. Phys. Commun.,98,1996, 288-300.
[45]Parkes E J, Duffy B R. Travelling solitary wave solutions to a compound KdVBurg-ers equation. Phys. Lett. A,229,1997,217-220.
[46]Li Z B, Liu Y P. RAEEM:A Maple package for finding a series of exact traveling wave solutions for nonlinear evolution equations. Comput. Phys. Commun.,163, 2004,191-201.
[47]杨伯君,赵玉芳.高等数学物理方法.北京邮电大学山版社,2003.
[48]倪皖荪,魏荣爵.水槽中的孤波.上海科技教育出版社,1997.
[49]Zhang D J, Chen D Y. The conservation laws of some discrete soliton systems. Chaos Solitons Frac.,14,2002,573-579.
[1]Xue J K. The nonlinear evolution of ring dark solitons in Bose-Einstein condensates. J. Phys. A,37,2004,11223-11228.
[2]Hong W P. Comment on:" Spherical Kadomtsev-Petviashvili equation and neb-ulons for dust ion-acoustic waves with symbolic computation" [Phys. Lett. A 340 (2005) 243]. Phys. Lett. A,361,2007,520-522.
[3]David D, Levi D, Winternitz P. Integrable nonlinear equations for water waves in straits of varying width and depth. Stud. Appl. Math.,76,1987,133-168.
[4]Santini P M. On the evolution of two-dimensional packets of water waves over an uneven bottom. Lett. Nuov. Cim.,30,1983,236-240.
[5]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.
[6]Kakei S. Dressing method and the coupled KP hierarchy. Phys. Lett. A,246,2002, 449-458.
[7]Isojima S, Willox R, Satsuma J. Spider-web solutions of the coupled KP equation. J. Phys. A,36,2003,9533-9552.
[8]Isojima S, Willox R, Satsuma J. On various solutions of the coupled KP equation. J. Phys. A,35,2002,6893-6909.
[9]Adler M, Horozov E, Moerbeke P. The Pfaff Lattice and Skew-orthogonal polynomi-als. Int. Math. Res. Notices,11,1999,569-588.
[10]Jimbo M, Miwa T. Solitons and infinite dimensional Lie algebras. Publ. Res. Inst. Math. Sci.,19.1983,943-1001.
[11]Wang J, Li B. Symmetry and general symmetry groups of the coupled Kadomtsev-Petviashvili equation. Chin. Phys. Soc.,18,2009,2109-2114.
[12]Hietarinta J. Partially integrable evolution equations in physics. Springer,1990.
[13]Hirota R, Ohta Y. Hierarchies of coupled soliton equations. I. J. Phys. Soc. Jpn. 60,1991,798-809.
[14]Tian B, Gao Y T, Zhu H W. Variable-coefficient higher-order nonlinear Schrodinger model in optical fibers:Variable-coefficient bilinear form, Backlund transformation, brightons and symbolic computation. Phys. Lett. A,366,2007,223-229.
[15]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.
[16]许韬.符号计算与若干非线性发展方程的解析研究[博士学位论文].北京邮电大学,2011.
[17]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.
[18]Gao Y T, Tian B. Cylindrical Kadomtsev-Petviashvili model, nebulons and sym-bolic computation for cosmic dust ion-acoustic waves. Phys. Lett. A,349,2006, 314-319.
[19]Gao Y T, Tian B. Reply to:"Comment on:'Spherical Kadomtsev-Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computa-tion'" [Phys. Lett. A 361 (2007) 520]. Phys. Lett. A,361,2007,523-528.
[20]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[博士学位论文].北京邮电大学,2008.
[21]李丽莉.基于计算机符号计算的非线性模型孤子解研究[博士学位论文].北京邮电大学,2009.
[22]Das G, Sarma J. Response to "Comment on'A new mathematical approach for finding the solitary waves in dusty plasma'" [Phys. Plasmas 6,4392 (1999)]. Phys. Plasmas,6,1999,4394:1-4.
[23]Geng X G. Algebraic-geometrical solutions of some multidimensional nonlinear evo-lution equations. J. Phys. A,36.2003,2289-2303.
[24]Van de Leur J. Backlund-Darboux transformations for the coupled KP hierarchy. J. Phys. A.37,2004.4395-4405.
[25]Yang G Y, Liu Q P. A Darboux transformation for the coupled Kadomtsev-Petviashvili equation. Chin. Phys. Lett.,25,2008,1-3.
[26]Ablowitz M J, Kaup D J, Newell A C, et al. Nonlinear-evolution equations of physical significance. Phys. Rev. Lett.,31,1973,125-127.
[27]Li J, Zhang H Q, Xu T, et al. Soliton-like solutions of a generalized variable-coefficient higher order nonlinear Schrodinger equation from inhomogeneous optical fibers with symbolic computation. J. Phys. A,40,2007,13299-13309.
[28]Porsezian K, Nakkeeran K. Optical soli tons in presence of Kerr dispersion and self-frequency shift. Phys. Rev. Lett.,76,1996,3955-3958.
[29]Mahalingam A, Porsezian K. Propagation of dark solitons with higher-order effects in optical fibers. Phys. Rev. E,64,2001,046608-046616.
[30]Wright O C, Forest M G. On the Backlund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrodinger system. Phys. D,141,2000,104-116.
[31]Li Y S. Soliton and Integrable System. Shanghai Scientific and Technological Edu-cation Publishing,1999.
[32]Maxworhy T. On the formation of nonlinear internal waves from the gravitational collapse of mixed regions in two and three dimensions. J. Fluid. Mech.,96,1980, 47-64.
[33]Biondini G, Chakravarty S. Soliton solutions of the Kadomtsev-Petviashvili II equa-tion. J. Math. Phys.,47,2006,033514-033539.
[34]Lonngren K E. Ion acoustic soliton experiments in a plasma. Opt. Quantum Elec-tron.,30,1998,615-630.
[35]Folkes P A, Ikezi H, Davis R. Two-dimensional interaction of ion-acoustic solitons. J. Phys. Rev. Lett.,45,1980,902-904.
[36]Tsukabayashi I, Nakamura Y. Resonant interaction of cylindrical ion-acoustic soli-tons. Phy. Lett. A,85,1981,151-154.
[37]Kakei S. Orthogonal and symplectic matrix integrals and coupled KP hierarchy. J. Phys. Soc. Jpn.,68,1999,2875-2877.
[1]Hasegawa A, Tappert F. Transmission of stationary nonlinear optical pulses in dis-persive dielectric fibers. II. Normal dispersion. Appl. Phys. Lett.,23,1973,171-172.
[2]Mollenauer L F, Stolen R H, Gordon J P. Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys. Rev. Lett.,45,1980,1095-1098.
[3]Zhang H Q, Xu T, Li J, et al. Integrability of an N-coupled nonlinear Schrodinger system for polarized optical waves in an isotropic medium via symbolic computation. Phys. Rev. E,77,2008,026605-026614.
[4]Liu W J, Tian B, Zhang H Q, et al. Soliton interaction in the higher-order nonlinear Schrodinger equation investigated with Hirota's bilinear method, Phys. Rev. E,77, 2008,066605-066611.
[5]Gedalin M, Scott T C, Band Y B. Optical solitons in the higher order nonlinear Schrodinger equation. Phys. Rev. Lett.,78,1997,448-451.
[6]Mihalache D, Truta N, Crasovan L C. Painleve analysis and bright solitary waves of the higher-order nonlinear Schrodinger equation containing third-order dispersion and self-steepening term. Phys. Rev. E,56,1997,1064-1070.
[7]Yan Z Y. Optical solitary wave solutions to nonlinear schrodinger equation with cubic-quintic nonlinearity in non-Kerr media. J. Phys. Soc. Jpn.,73,2004,2397-2401.
[8]Newell A C, Moloney J V. Nonlinear Optics. Addison-Wesley.1992.
[9]Hong W P. Optical solitary wave solutions for the higher order nonlinear Schrodinger equation with cubic-quintic non-Kerr terms. Opt. Commun.,194,2001,217-223.
[10]Levi D, Scimiterna C. The Kundu-Eckhaus equation and its discretizations. J. Phys. A,42,2009,465203-465210.
[11]Radhakrishnan R. Kundu A, Lakshmanan M. Coupled nonlinear Schrodinger equa-tions with cubic-quintic nonlinearity:Integrability and soliton interaction in non-Kerr media. Phys. Rev. E,60,1999,3314-3323.
[12]Calogero F, Eckhaus W. Nonlinear evolution equations, rescalings, model PDES and their integrability:I. Inv. Prob.,3,1987,229-262.
[13]Kundu A. Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrodinger type equation. J. Math. Phys.,25,1984,3433-3438.
[14]Zong F D, Dai C Q, Zhang J F. Optical solitary waves in fourth-order dispersive nonlinear Schrodinger equation with self-steepening and self-frequency shift. Com-mun. Theor. Phys.,45,2006,721-726.
[15]Clarkson P A, Tuszynski J A. Exact solutions of the multidimensional derivative nonlinear Schrodinger equation for many-body systems of criticality. J. Phys. A,23, 1990,4269-4288.
[16]Kodama Y. Optical solitons in a monomode fiber. J. Stat. Phys.,39,1985,597-614.
[17]Wang M L, Zhang J L, Li X Z. Solitary wave solutions of a generalized derivative nonlinear Schrodinger equation. Commun. Theor. Phys.,50,2008,39-42.
[18]Johnson R S. On the modulation of water waves in the neighbourhood of kh≈ 1.363. Proc. Roy. Soc. London A,357,1977,131-141.
[19]Levko D, Volkov A. Modeling of Kundu-Eckhaus equation. ArXiv:nlin.PS/0702050, 2007.
[20]Geng X G, Tam H W. Darboux transformation and soli ton solutions for generalized nonlinear Schrodinger equations. J. Phys. Soc. Jpn.,68,1999.1508-1512.
[21]Hao R Y, Li L, Li Z H, et al. A new way to exact quasi-soliton solutions and soliton interaction for the cubic-quintic nonlinear Schrodinger equation with variable coefficients. Opt. Commun.,245,2005,383-390.
[22]Tian B. Gao Y T, Zhu H W. Variable-coefficient higher-order nonlinear Schrodinger model in optical fibers:Variable-coefficient bilinear form, Backlund transformation. brightons and symbolic computation. Phys. Lett. A,366,2007,223-229.
[23]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.
[24]李丽莉.基于计算机符号计算的非线性模型孤子解研究[博士学位论文].北京邮电大学,2009.
[25]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[博士学位论文].北京邮电大学,2008.
[26]Tian B, Wei G M, Zhang C Y, et al. Transformations for a generalized variable-coefficient Korteweg-de Vries model from blood vessels, Bose-Einstein condensates, rods and positons with symbolic computation. Phys. Lett. A,356,2006,8-16.
[27]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.
[28]Gao Y T, Tian B. Cylindrical Kadomtsev-Petviashvili model, nebulons and sym-bolic computation for cosmic dust ion-acoustic waves. Phys. Lett. A,349,2006, 314-319.
[29]Gao Y T, Tian B. Reply to:"Comment on:'Spherical Kadomtsev-Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computa-tion'" [Phys. Lett. A 361 (2007) 520]. Phys. Lett. A,361,2007,523-528.
[30]许韬.符号计算与若干非线性发展方程的解析研究[博士学位论文].北京邮电大学,2011.
[31]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, 120703:1-4.
[32]Gao Y T, Tian B. On the non-planar dust-ion-acoustic waves in cosmic dusty plas-mas with transverse perturbations. Europhys. Lett.,77,2007,15001:1-6.
[33]Das G, Sarma J. Response to "Comment on 'A new mathematical approach for finding the solitary waves in dusty plasma'" [Phys. Plasmas 6,4392 (1999)]. Phys. Plasmas.6,1999.4394:1-4.
[34]Hong W P. Comment on:"Spherical Kadomtsev-Petviashvili equation and neb-ulons for dust ion-acoustic waves with symbolic computation" [Phys. Lett. A 340 (2005) 243]. Phys. Lett. A,361,2007,520-522.
[35]田播.计算机符号计算在非线性研究中的若干应用[博士学位论文].北京航空航天大学,2003.
[36]Ablowitz M J, Kaup D J, Newell A C, et al. Nonlinear-evolution equations of physical significance. Phys. Rev. Lett.,31,1973,125-127.
[37]Li J, Zhang H Q, Xu T, et al. Soliton-like solutions of a generalized variable-coefficient higher order nonlinear Schrodinger equation from inhomogeneous optical fibers with symbolic computation. J. Phys. A,40,2007,13299-13309.
[38]Porsezian K, Nakkeeran K. Optical solitons in presence of Kerr dispersion and self-frequency shift. Phys. Rev. Lett.,76,1996,3955-3958.
[39]Mahalingam A, Porsezian K. Propagation of dark solitons with higher-order effects in optical fibers. Phys. Rev. E,64,2001,046608-046616.
[40]Ablowitz M J, Kaup D J, Newell A C, et al. Method for solving the Sine-Gordon equation. Phys. Rev. Lett.,30,1973,1262-1264.
[41]Wright O C, Forest M G. On the Backlund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrodinger system. Phys. D,141,2000,104-116.
[42]Li J, Zhang H Q, Xu T, et al. Symbolic computation on the multi-soliton-like solutions of the cylindrical Kadomtsev-Petviashvili equation from dusty plasmas. J. Phys. A,40,2007,7643-7657.
[43]Li Y S. Soliton and Integrable System, Shanghai Scientific and Technological Edu-cation Publishing,1999.
[44]Zhang H Q, Tian B, Lu X. et al. Soliton interaction in the coupled mixed derivative nonlinear Schrodinger equations. Phys. Lett. A,373,2009,4315-4321.
[45]Seong N H. Kim D Y. Experimental observation of stable bound solitons in a figure-eight fiber laser. Opt. Lett.,27,2002,1321-1323.
[46]Kim W S, Moon H T. Soliton-kink interactions in a generalized nonlinear Schrodinger system. Phys. Lett. A,266,2000,364-369.
[47]Malomed B A. Bound solitons in coupled nonlinear Schrodinger equations. Phys. Rev. A,45,1991,8321-8323.
[48]Haelterman M, Shepppard A. Bifurcation phenomena and multiple soliton-bound states in isotropic Kerr media. Phys. Rev. E,49,1994,3376-3381.
[49]Sun Z Y, Gao Y T, Yu X, et al. Bound vector solitons and soliton complexes for the coupled nonlinear Schrodinger equations. Phys. Rev. E,80,2009,066608-066618.
[1]Radhakrishnan R, Kundu A, Lakshmanan M. Coupled nonlinear Schrodinger equa-tions with cubic-quintic nonlinearity:Integrability and soliton interaction in non-Kerr media. Phys. Rev. E,60,1999,3314-3323.
[2]Zhang H Q, Xu T, Li J, et al. Integrability of an N-coupled nonlinear Schrodinger system for polarized optical waves in an isotropic medium via symbolic computation. Phys. Rev. E,77,2008,026605-026614.
[3]Mollenauer L F, Stolen R H, Gordon J P. Experimental observation of picosecond pulse narrowing and solitons in Optical fibers. Phys. Rev. Lett.,45,1980,1095-1098.
[4]Liu W J, Tian B, Zhang H Q, et al. Soliton interaction in the higher-order nonlinear Schrodinger equation investigated with Hirota's bilinear method. Phys. Rev. E,77, 2008,066605-066611.
[5]Porsezian K, Kuriakose V C. Optical Solitons:Theoretical and experimental chal-lenges. Springer,2003.
[6]Hasegawa A, Kodama Y. Solitons in Optical Communications. Oxford University, 1995.
[7]Islam M N. Ultrafast Fiber Switching Devices and Systems. Cambridge University, 1992.
[8]Newell A C, Moloney J V. Nonlinear Optics. Addison-Wesley,1992.
[9]Abdullaev F, Darmanyan S, Khabibullaev P. Optical Solitons. Springer-Verlag,1993.
[10]Hong W P. Optical solitary wave solutions for the higher order nonlinear Schrodinger equation with cubic-quintic non-Kerr terms. Opt. Commun.,194,2001, 217-223.
[11]Skarka V, Berezhiani V I, Miklaszewski R. Spatiotemporal soliton propagation in saturating nonlinear optical media. Phys. Rev. E,56,1997,1080-1087.
[12]Afanasjev V V, Chu P L, Kivshar Y S. Breathing spatial solitons in non-Kerr media. Opt. Lett.,22.1997,1388-1390.
[13]Dattoli G, Orsitto F P, Torre A. Evidence for multistability of light solitons in SF6 absorption measurements. Opt. Lett.,14,1989,456-458.
[14]Gedalin M, Scott T C, Band Y B. Optical solitons in the higher order nonlinear Schrodinger equation. Phys. Rev. Lett.,78,1997,448-451.
[15]Mihalache D, Truta N, Crasovan L C. Painleve analysis and bright solitary waves of the higher-order nonlinear Schrodinger equation containing third-order dispersion and self-steepening term. Phys. Rev. E,56,1997,1064-1070.
[16]Yan Z Y. Optical solitary wave solutions to nonlinear schrodinger equation with cubic-quintic nonlinearity in non-Kerr media. J. Phys. Soc. Jpn.,73,2004,2397-2401.
[17]Zong F D, Dai C Q, Zhang J F. Optical Solitary waves in fourth-order dispersive nonlinear Schrodinger equation with self-steepening and self-frequency shift. Com-mun. Theor. Phys.,45,2006,721-726.
[18]Kundu A. Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrodinger type equation. J. Math. Phys.,25,1984,3433-3438.
[19]Calogero F, Eckhaus W. Nonlinear evolution equations, rescalings, model PDES and their integrability:Ⅰ, Inv. Prob.,3,1987,229-262.
[20]Levi D, Scimiterna C. The Kundu-Eckhaus equation and its discretizations. J. Phys. A,42,2009,465203-465210.
[21]Wang M L, Zhang J L, Li X Z. Solitary wave solutions of a generalized derivative nonlinear Schrodinger equation,Commun. Theor. Phys.,50,2008,39-42.
[22]Clarkson P A, Tuszynski J A. Exact solutions of the multidimensional derivative nonlinear Schrodinger equation for many-body systems of criticality. J. Phys. A,23, 1990,4269-4288.
[23]Kodama Y. Optical solitons in a monomode fiber. J. Stat. Phys.,39,1985,597-614.
[24]Johnson R S. On the modulation of water waves in the neighbourhood of kh≈ 1.363. Proc. Roy. Soc. London A,357,1977,131-141.
[25]Albuch L, Malomed B A. Transitions between symmetric and asymmetric solitons in dual-core systems with cubic-quintic nonlinearity. Math. Commun. Simul.,74, 2007,312-322.
[26]Hisakado M, Wadati M. J. Gauge transformations among generalised nonlinear Schrodinger equations. Phys. Soc. Jpn.,63,1994,3962-3966.
[27]Hisakado M, Wadati M J. Integrable multi-component hybrid nonlinear Schrodinger equations. Phys. Soc. Jpn.,64,1995,408-413.
[28]Tian B, Gao Y T, Zhu H W. Variable-coefficient higher-order nonlinear Schrodinger model in optical fibers:Variable-coefficient bilinear form, Backlund transformation, brightons and symbolic computation. Phys. Lett. A,366,2007,223-229.
[29]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.
[30]Tian B, Gao Y T. Variable-coefficient higher-order nonlinear Schrodinger model in optical fibers:New transformation with burstons, brightons and symbolic computa-tion. Phys. Lett. A,359,2006,241-248.
[31]Tian B, Wei G M, Zhang C Y, et al. Transformations for a generalized variable-coefficient Korteweg-de Vries model from blood vessels, Bose-Einstein condensates, rods and positons with symbolic computation. Phys. Lett. A,356,2006,8-16.
[32]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.
[33]Gao Y T, Tian B. Cylindrical Kadomtsev-Petviashvili model, nebulous and sym-bolic computation for cosmic dust ion-acoustic waves. Phys. Lett. A,349,2006, 314-319.
[34]许韬.符号计算与若干非线性发展方程的解析研究[博士学位论文].北京邮电大学,2011.
[35]Gao Y T, Tian B. Cosmic dust-ion-acoustic waves, spherical modified Kadomtsev-Petviashvili model, and symbolic computation. Phys. Plasmas,13,2006,112901: 1-6.
[36]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, 120703:1-4.
[37]李丽莉.基于计算机符号计算的非线性模型孤子解研究[博士学位论文].北京邮电大学,2009.
[38]Das G, Sarma J. Response to "Comment on'A new mathematical approach for finding the solitary waves in dusty plasma'" [Phys. Plasmas 6.4392 (1999)]. Phys. Plasmas,6,1999,4394:1-4.
[39]Hong W P. Comment on:"Spherical Kadomtsev-Petviashvili equation and neb-ulons for dust ion-acoustic waves with symbolic computation" [Phys. Lett. A 340 (2005) 243]. Phys. Lett. A,361,2007,520-522.
[40]Tian B, Gao Y T. On the solitonic structures of the cylindrical dust-acoustic dust-ion-acoustic waves with symbolic computation. Phys. Lett. A,340,2005,449-455.
[41]Tian B, Gao Y T. Spherical nebulons and Backlund transformation for a space or laboratory un-magnetized dusty plasma with symbolic computation. Eur. Phys. J. D,33,2005,59-65.
[42]Tian B, Gao Y T. Comment on "Exact solutions of cylindrical and spherical dust ion acoustic waves" [Phys. Plasmas 10,4162 (2003)]. Phys. Plasmas,12,2005, 054701:1-4.
[43]Tian B, Gao Y T. Cylindrical nebulous, symbolic computation and Backlund trans-formation for the cosmic dust acoustic waves. Phys. Plasmas (Lett.),12,2005, 070703:1-4.
[44]Tian B, Gao Y T. Symbolic computation on cylindrical-modified dust-ion-acoustic nebulons in dusty plasmas. Phys. Lett. A.362,2007,283-288.
[45]Ablowitz M J, Kaup D J, Newell A C, et al. Nonlinear-evolution equations of physical significance. Phys. Rev. Lett.,31,1973,125-127.
[46]Li J, Zhang H Q, Xu T, et al. Soliton-like solutions of a generalized variable-coefficient higher order nonlinear Schrodinger equation from inhomogeneous optical fibers with symbolic computation. J. Phys. A,40,2007,13299-13309.
[47]Porsezian K, Nakkeeran K. Optical Solitons in Presence of Kerr Dispersion and Self-Frequency Shift. Phys. Rev. Lett.,76,1996,3955-3958.
[48]Mahalingam A, Porsezian K. Propagation of dark solitons with higher-order effects in optical fibers. Phys. Rev. E,64,2001,046608-046616.
[49]Ablowitz M J, Kaup D J, Newell A C, et al. Method for solving the Sine-Gordon equation. Phys. Rev. Lett.,30,1973,1262-1264.
[50]Wright O C, Forest M G. On the Backlund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrodinger system. Phys. D,141,2000,104-116.
[51]Li J, Zhang H Q, Xu T, et al. Symbolic computation on the multi-soliton-like solutions of the cylindrical Kadomtsev-Petviashvili equation from dusty plasmas. J. Phys. A,40,2007,7643-7657.
[52]Li Y S. Soliton and Integrable System. Shanghai Scientific and Technological Edu-cation Publishing,1999.
[53]Seong N H, Kim D Y. Experimental observation of stable bound solitons in a figure-eight fiber laser. Opt, Lett.,27,2002,1321-1323.
[54]Kim W S, Moon H T. Soliton-kink interactions in a generalized nonlinear Schrodinger system. Phys. Lett. A,266,2000,364-369.
[55]Malomed B A. Bound solitons in coupled nonlinear Schrodinger equations. Phys. Rev. A,45,1991,8321-8323.
[56]Haelterman M, Shepppard A. Bifurcation phenomena and multiple soliton-bound states in isotropie Kerr media. Phys. Rev. E,49,1994,3376-3381.
[57]Sun Z Y, Gao Y T, Yu X, et al. Bound vector solitons and soliton complexes for the coupled nonlinear Schrodinger equations. Phys. Rev. E,80,2009,066608-066618.
[58]Shan W R, Qi F H, Guo R, et al. Conservation laws and solitons for the coupled cubic-quintic nonlinear Schrodinger equations in nonlinear optics. Phys. Scr.,85, 2012,015002:1-9.
[59]Wadati M, Sanuki H, Konno K. Relationships among inverse method, Backlund transformation and an infinite number of conservation Laws. Prog. Theor. Phys.,53, 1975,419-436.
[60]Zhang D J, Chen D Y. The conservation laws of some discrete soliton systems. Chaos Solitons Frac., 14,2002,573-579.