符号计算在非线性数学模型中的应用研究
|
摘要
本论文的研究工作主要是基于计算机符号计算技术,并结合微分方程、代数及算子等相关数学理论,跨学科地研究了现代科技中一些重要的非线性数学模型。这些模型的应用涉及到光孤子通信、生物体中能量传递、空间等离子体、玻色爱因斯坦凝聚态物质等诸多领域。
非线性科学是当前前沿科学中最热门的研究领域之一,它的研究方向包括孤子,混沌和分形三个分支。本论文主要是以高阶、耦合及其变系数等较复杂的、与孤子相关的非线性数学模型为主体研究对象,通过强大的符号计算技术,结合双线性方法、B(?)cklund变换、Wronskian技术和无穷守恒律等理论对这些模型进行解析研究。
本论文的主要工作如下:
(一)基于双线性方法的基本思想,通过符号计算技术,推导了通用的由原非线性微分方程转化为相应双线性方程的变换公式。利用所得的变换公式,分别把非等谱、高阶和高阶耦合三个不同类型的非线性Schr(o|¨)dinger方程从原方程转化为了双线性形式的方程,为进一步通过小参数展开法构造多孤子解和研究双线性B(?)cklund变换提供了基础。
(二)根据B(?)cklund变换的基本思想,通过符号计算技术,推导了一系列B(?)cklund变换所需要的双线性交换公式(交换公式的推导是B(?)cklund变换的难点之一)。通过选择合适的双线性交换公式,成功地推导了非等谱、高阶和高阶耦合三个不同类型非线性Schr(o|¨)dinger方程的B(?)cklund变换。然后通过符号计算和B(?)cklund变换,利用平凡解获得了在物理上具有重要意义的解析孤子解,并以非等谱的非线性Schr(o|¨)dinger方程为例详细探讨了所获得孤子解的性质和意义。在另一方面,作者利用B(?)cklund变换还推导出了高阶非线性Schr(o|¨)dinger方程的一个反散射变换方案。
(三)基于Wronskian技术的定义和性质,利用符号计算,提出了非等谱、高阶和高阶耦合三个不同类型非线性Schr(o|¨)dinger类型方程的Wronskian行列式解,并把这些解直接代入了原双线性方程进行验证。在另一方面,作者还证明了B(?)cklund变换,在参数满足一定约束的条件下,正好是Wronskian形式的N-1孤子解到N孤子解的一个变换。在对孤子问题的研究中,多孤子碰撞是研究的一个重要课题。利用Wronskian形式的N孤子解可以直接给出解析的多孤子解的显示形式,并用于研究多孤子碰撞问题,以非等谱非线性发展方程为例,详细分析和研究了双孤子解的一些碰撞性质。
(四)通过符号计算,作者提出了一个广义的变系数Miura变换,它关联着两个非线性发展方程的解,即原推广的变系数Korteweg-deVries方程和修正的变系数Korteweg-de Vries方程。然后,通过这个广义的Miura变换和Galilean不变变换证明了推广的具有外力项和扰动/色散项的变系数Korteweg-de Vries模型在广义的Painlev(?)可积条件下存在无穷守恒律。
本论文研究的非线性数学模型在现代科技领域都有很重要的意义和广泛的应用,作者希望本文基于计算机符号计算技术,结合相应微分方程、代数及算子等相关数学理论所获得的结果对于非线性领域中孤子的研究有所贡献,为现实世界中光孤子通信,生物体中能量传递,空间等离子体,玻色爱因斯坦凝聚态物质等诸多领域提供理论指导,为微分方程与符号计算技术的发展有所帮助!
Based on symbolic computation and the knowledge of differential equations, algebra and operators, this dissertation presents an interdisciplinary study of some important nonlinear mathematical models, arising from such applications as optical soliton communications, transfers of energy in biophysics, space plasmas and Bose-Einstein condensates.
The nonlinear science is one of the most popular fields of current scientific researches, and its branches include soliton, chaos and fractal. The main objective of this dissertation focuses on higher-order, couple and variable coefficient nonlinear mathematical models which are applicable to the description of various soliton phenomena. By virtue of the powerful symbolic computation technology, an analytic investigation is performed on these models from the aspects of bilinear representation, Backlund transformation, Wronskian solution and infinite conservation laws.
The results obtained in this dissertation are as follows:
( I ) Based on the basic principles of the bilinear method, via symbolic computation, we deduce some formulas which can conveniently transform original nonlinear differential equations into the corresponding bilinear ones. Then using these transformation formulas, we respectively transform three different types of nonlinear Schrodinger equations, i.e., nonisospectral, higher-order and couple higher-order ones, into the corresponding bilinear forms. From these bilinear equations, we can get multi-soliton solutions by employing the expansion method of small parameter. Furthermore, these bilinear equations also provide a basis for deriving bilinear Backlund transformations.
(II) Based on the basic idea of Backlund transformation, via symbolic computation, we derive a series of the bilinear exchange formulas which are used to deduce Backlund transformation. It is mentioned that the derivation of the exchange formulas is necessary but very difficult for constructing the bilinear Backlund transformation. Using a suitable exchange formula, we successfully derive Backlund transformations for three different types of nonlinear Schrodinger equations, i.e., nonisospectral, higher-order and couple higher-order ones. Via symbolic computation and starting from a trivial solution, the Backlund transformation yields the analytical soliton solution of interest in physics. As an example, we detailedly illustrate and discuss the soliton features of a nonisospectral nonlinear Schrodinger equation. On the other hand, based on the Backlund transformation, the author also derives the inverse scattering transform scheme for a higher-order nonlinear Schrodinger equation.
(III) Based on the definition and theory of Wronskian determinant, via symbolic computation, we present the Wronskian solutions of three different types of nonlinear Schrodinger equations, i.e., nonisospectral, higher-order and couple higher-order ones, and verify them by direct substitution into the bilinear equations. On the other hand, the authors also verify that the (N-1)- and N-soliton solutions satisfy the Backlund transformation with sets of parametric conditions. In the study of solitons, the multi-soliton collision is an important issue. The soliton in Wronskian form can directly give explicit multi-soliton solutions, so it becomes easier to study the dynamics of the multi-soliton collision. As an example, we give a detailed analysis on characteristics of the two-soliton solution of a nonisospectral nonlinear Schrodinger equation.
(IV) With the aid of symbolic computation, the author proposes a generalized Miura transformation which relates the solutions of the variable-coefficient Korteweg-de Vries equation to those of a variable-coefficient modified Korteweg-de Vries equation. Then by using such a Miura transformation and the Galilean invariant transformation, the author proves the existence of infinite conservation laws under the Painleve integrable condition.
The nonlinear mathematical models investigated in this dissertation have a wide range of applications in modern science and technology. The author hopes that the results, obtained by symbolic computation and the theory of differential equations, algebra, operators and other relevant mathematical theory, have some contribution to the study of nonlinear science. It is expected that our work can provide the theoretical guidance in the real world for optical soliton communications, transfers of energy in biophysics, space plasmas, Bose-Einstein condensates, etc., and well be of value in the development of differential equations and symbolic computation technique.
非线性科学是当前前沿科学中最热门的研究领域之一,它的研究方向包括孤子,混沌和分形三个分支。本论文主要是以高阶、耦合及其变系数等较复杂的、与孤子相关的非线性数学模型为主体研究对象,通过强大的符号计算技术,结合双线性方法、B(?)cklund变换、Wronskian技术和无穷守恒律等理论对这些模型进行解析研究。
本论文的主要工作如下:
(一)基于双线性方法的基本思想,通过符号计算技术,推导了通用的由原非线性微分方程转化为相应双线性方程的变换公式。利用所得的变换公式,分别把非等谱、高阶和高阶耦合三个不同类型的非线性Schr(o|¨)dinger方程从原方程转化为了双线性形式的方程,为进一步通过小参数展开法构造多孤子解和研究双线性B(?)cklund变换提供了基础。
(二)根据B(?)cklund变换的基本思想,通过符号计算技术,推导了一系列B(?)cklund变换所需要的双线性交换公式(交换公式的推导是B(?)cklund变换的难点之一)。通过选择合适的双线性交换公式,成功地推导了非等谱、高阶和高阶耦合三个不同类型非线性Schr(o|¨)dinger方程的B(?)cklund变换。然后通过符号计算和B(?)cklund变换,利用平凡解获得了在物理上具有重要意义的解析孤子解,并以非等谱的非线性Schr(o|¨)dinger方程为例详细探讨了所获得孤子解的性质和意义。在另一方面,作者利用B(?)cklund变换还推导出了高阶非线性Schr(o|¨)dinger方程的一个反散射变换方案。
(三)基于Wronskian技术的定义和性质,利用符号计算,提出了非等谱、高阶和高阶耦合三个不同类型非线性Schr(o|¨)dinger类型方程的Wronskian行列式解,并把这些解直接代入了原双线性方程进行验证。在另一方面,作者还证明了B(?)cklund变换,在参数满足一定约束的条件下,正好是Wronskian形式的N-1孤子解到N孤子解的一个变换。在对孤子问题的研究中,多孤子碰撞是研究的一个重要课题。利用Wronskian形式的N孤子解可以直接给出解析的多孤子解的显示形式,并用于研究多孤子碰撞问题,以非等谱非线性发展方程为例,详细分析和研究了双孤子解的一些碰撞性质。
(四)通过符号计算,作者提出了一个广义的变系数Miura变换,它关联着两个非线性发展方程的解,即原推广的变系数Korteweg-deVries方程和修正的变系数Korteweg-de Vries方程。然后,通过这个广义的Miura变换和Galilean不变变换证明了推广的具有外力项和扰动/色散项的变系数Korteweg-de Vries模型在广义的Painlev(?)可积条件下存在无穷守恒律。
本论文研究的非线性数学模型在现代科技领域都有很重要的意义和广泛的应用,作者希望本文基于计算机符号计算技术,结合相应微分方程、代数及算子等相关数学理论所获得的结果对于非线性领域中孤子的研究有所贡献,为现实世界中光孤子通信,生物体中能量传递,空间等离子体,玻色爱因斯坦凝聚态物质等诸多领域提供理论指导,为微分方程与符号计算技术的发展有所帮助!
Based on symbolic computation and the knowledge of differential equations, algebra and operators, this dissertation presents an interdisciplinary study of some important nonlinear mathematical models, arising from such applications as optical soliton communications, transfers of energy in biophysics, space plasmas and Bose-Einstein condensates.
The nonlinear science is one of the most popular fields of current scientific researches, and its branches include soliton, chaos and fractal. The main objective of this dissertation focuses on higher-order, couple and variable coefficient nonlinear mathematical models which are applicable to the description of various soliton phenomena. By virtue of the powerful symbolic computation technology, an analytic investigation is performed on these models from the aspects of bilinear representation, Backlund transformation, Wronskian solution and infinite conservation laws.
The results obtained in this dissertation are as follows:
( I ) Based on the basic principles of the bilinear method, via symbolic computation, we deduce some formulas which can conveniently transform original nonlinear differential equations into the corresponding bilinear ones. Then using these transformation formulas, we respectively transform three different types of nonlinear Schrodinger equations, i.e., nonisospectral, higher-order and couple higher-order ones, into the corresponding bilinear forms. From these bilinear equations, we can get multi-soliton solutions by employing the expansion method of small parameter. Furthermore, these bilinear equations also provide a basis for deriving bilinear Backlund transformations.
(II) Based on the basic idea of Backlund transformation, via symbolic computation, we derive a series of the bilinear exchange formulas which are used to deduce Backlund transformation. It is mentioned that the derivation of the exchange formulas is necessary but very difficult for constructing the bilinear Backlund transformation. Using a suitable exchange formula, we successfully derive Backlund transformations for three different types of nonlinear Schrodinger equations, i.e., nonisospectral, higher-order and couple higher-order ones. Via symbolic computation and starting from a trivial solution, the Backlund transformation yields the analytical soliton solution of interest in physics. As an example, we detailedly illustrate and discuss the soliton features of a nonisospectral nonlinear Schrodinger equation. On the other hand, based on the Backlund transformation, the author also derives the inverse scattering transform scheme for a higher-order nonlinear Schrodinger equation.
(III) Based on the definition and theory of Wronskian determinant, via symbolic computation, we present the Wronskian solutions of three different types of nonlinear Schrodinger equations, i.e., nonisospectral, higher-order and couple higher-order ones, and verify them by direct substitution into the bilinear equations. On the other hand, the authors also verify that the (N-1)- and N-soliton solutions satisfy the Backlund transformation with sets of parametric conditions. In the study of solitons, the multi-soliton collision is an important issue. The soliton in Wronskian form can directly give explicit multi-soliton solutions, so it becomes easier to study the dynamics of the multi-soliton collision. As an example, we give a detailed analysis on characteristics of the two-soliton solution of a nonisospectral nonlinear Schrodinger equation.
(IV) With the aid of symbolic computation, the author proposes a generalized Miura transformation which relates the solutions of the variable-coefficient Korteweg-de Vries equation to those of a variable-coefficient modified Korteweg-de Vries equation. Then by using such a Miura transformation and the Galilean invariant transformation, the author proves the existence of infinite conservation laws under the Painleve integrable condition.
The nonlinear mathematical models investigated in this dissertation have a wide range of applications in modern science and technology. The author hopes that the results, obtained by symbolic computation and the theory of differential equations, algebra, operators and other relevant mathematical theory, have some contribution to the study of nonlinear science. It is expected that our work can provide the theoretical guidance in the real world for optical soliton communications, transfers of energy in biophysics, space plasmas, Bose-Einstein condensates, etc., and well be of value in the development of differential equations and symbolic computation technique.
引文
1.Barnett,M.P.,Capitani,J.F.,von zur Gathen,J.,et al.,Symbolic Calculation in Chemistry:Selected Examples.International Journal of Quantum Chemistry.100(2),2004,80-104.
2.Wester,M.J.,Computer Algebra Systems:A Practical Guide,Wiley,1999.
3.Graham,R.L.,Knuth,D.E.and Patashnik,O.,Concrete Mathematics:A Foundation for Computer Science,Addison-Wesley Professional,2002.
4.Enns,R.H.and McGuire,G.C.,Nonlinear Physics with Maple for Scientists and Engineers,Birkh(a丨¨)user,2000.
5.Robeva,R.S.,Kirkwood,J.R.,Davies,R.L.,et al.,An Invitation to Biomathematics,Academic Press,2007.
6.Giordano,F.R.,Weir,M.D.and Fox,W.P.,A First Course in Mathematical Modeling,Brooks,2002.
7.Sedgewick,R.and Flajolet,P.,An Introduction to the Analysis of Algorithms Addison-Wesley Publishing Company,1996.
8.Donald,B.,Kapur,D.and Mundy,J.,Symbolic and Numerical Computation for Artificial Intelligence,Academic Press,1992.
9.《现代数学手册》编纂委员会,现代数学手册·计算机数学卷,华中科技大学出版社,2001.
10.吴文俊,数学机械化,科学出版社,2003.
11.Veltman,M.,Schoonship,CERN preprint,1967.
12.Hunt,B.R.,Lipsman,R.L.,Rosenberg,J.M.,et al.,A Guide to Matlab:For Beginners and Experienced Users,Cambridge University Press,2006.
13.Wolfram,S.,The Mathematica Book,Cambridge University Press,1999.
14.Cyganowski,S.,Kloeden,P.and Ombach,J.,From Elementary Probability to Stochastic Differential Equations with Maple Springer,2001.
15.Corless,R.M.,Essential Maple:An Introduction for Scientific Programmers,Springer,1995.
16.Infeld,E.and Rowlands,G.,Nonlinear Waves,Solitons and Chaos,Cambridge University Press,2000.
17.杨伯君和赵玉芳,高等数学物理方法,北京邮电大学出版社,2003.
18.Scott,A.,Encyclopedia of Nonlinear Science,Routledge,2005.
19.Diacu,F.and Holmes,P.,Celestial Encounter:The Origins of Chaos and Stability,Princeton University Press,1996.
20.Strogatz,S.H.,Nonlinear Dynamics and Chaos:With Applications to Physics, Biology,Chemistry and Engineering,Westview Press,2001.
21.Peitgen,H.O.,J(u丨¨)rgens,H.and Saupe,D.,Chaos and Fractals:New Frontiers of Science,Springer,2004.
22.Russell,J.S.Report on Waves.in Report of the 14th meeting of British association for the advancement of science.John Murray,London.1844,311-390.
23.Korteweg,D.J.and de Vries,G.,On the Change of Form of Long Waves Advancing in a Rectangular Canal and on a New Type of Long Stationary Wave.Philosophy Magazine.39,1895,422-443.
24.王明亮,非线性发展方程与孤立子,兰州大学出版社,1990.
25.黄景宁,徐济仲和熊吟涛,孤子:概念、原理和应用,高等教育出版社,2004.
26.Zabusky,N.J.and Kruskal,M.D.,Interaction Of "Solitons" In a Collisionless Plasma and the Recurrence of Initial States.Physical Review Letters.15(6),1965,240.
27.Perring,J.K.and Skyrme,T.H.,A Model Unified Field Equation.Nuclear Physics.31,1962,550-555.
28.Agrawal,G.,Applications of Nonlinear Fiber Optics,Academic Press,2001.
29.Hirota,R.,Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons.Physical Review Letters.27(18),1971,1192.
30.Hirota,R.,Exact Solution of the Modified Korteweg-de Vries Equation for Multiple Collisions of Solitons.Journal of the Physical Society of Japan.33(5),1972,1456-1458.
31.Hirota,R.,Exact Solution of the Sine-Gordon Equation for Multiple Collisions of Solitons.Journal of the Physical Society of Japan.33(5),1972,1459-1463.
32.Hirota,R.and Satsuma,J.,N-Soliton Solution of the K-dV Equation with Loss and Nonuniformity Terms.Journal of the Physical Society of Japan.41,1976,2141-2142.
33.Hirota,R.Direct Method of Finding Exact Solutions of Nonlinear Evolution Equations.in B(a丨¨)cklund Transformations,the Inverse Scattering Method,Solitons and Their Applications.Berlin.1976,40-68.
34.Hirota,R.,A New Form of B(a丨¨)cklund Transformations and Its Relation to the Inverse Scattering Problem Progress of Theoretical Physics.52,1974,1498-1512.
35.Hirota,R.and Satsuma,J.,N-Soliton Solution of the K-dV Equation with Loss and Nonuniformity Terms.Journal of the Physical Society of Japan.41,1976,2141-2142.
36.Hirota,R.and Satsuma,J.,A Variety of Nonlinear Network Equations Generated from the B(a丨¨)cklund Transformation for the Toda Lattice.Progress of Theoretical Physics Supplement.59,1976,64-100.
37.Hirota,R.and Satsuma,J.,Nonlinear Evolution Equations Generated from the B(a丨¨)cklund Transformation for the Boussinesq Equation Progress of Theoretical Physics.57(3),1977,797-807.
38.Hirota,R.,The Direct Method in Soliton Theory,Cambridge University Press,2004.
39.Weiss,J.,Tabor,M.and Carnevale,G.,The Painlev(?) Property for Partial Differential Equations.Journal of Mathematical Physics.24(3),1983,522-526.
40.Weiss,J.,The Painlev(?) Property for Partial Differential Equations.Ⅱ:B(a丨¨)cklund Transformation,Lax Pairs,and the Schwarzian Derivative.Journal of Mathematical Physics.24(6),1983,1405-1413.
41.Steeb,W.H.and Euler,N.,Nonlinear Evolution Equations and Painlev(?) Test,Word Scientific,1988.
42.刘式适和刘式达,非线性大气动力学,国防工业出版社,1996.
43.Weiss,J.,On Classes of lntegrable Systems and the Painlev(?) Property.Journal of Mathematical Physics.25(1),1984,13-24.
44.Newell,A.C.,Tabor,M.and Zeng,Y.B.,A Unified Approach to Painlev(?)Expansions.Physica D.29(1-2),1987,1-68.
45.潘祖梁,非线性问题的数学方法及其应用,浙江大学出版社,2004.
46.Clarkson,P.A.and Kruskal,M.D.,New Similarity Reductions of the Boussinesq Equation.Journal of Mathematical Physics.30(10),1989,2201-2213.
47.谷超豪,胡和生和周子翔,孤立子理论中的达布变换及其几何应用,上海科学技术出版公司,2005.
48.田播,[学位论文]计算机符号计算在非线性研究中的若干应用,北京航空航天大学,2003.
49.范恩贵,可积系统与计算机代数,科学出版社,2004.
50.Zhu,H.-W.and Tian,B.,B(a丨¨)cklund Transformation in Bilinear Form for a Higher-Order Nonlinear Schr(o丨¨)dinger Equation.Nonlinear Analysis:Theory,Methods & Applications.In Press,Corrected Proof.
51.Zhu,H.-W.,Tian,B.,Meng,X.-H.,et al.,A Bilinear B(a丨¨)cklund Transformation and N Soliton Like Solution of the Three Coupled Higher-Order Nonlinear Schr(o丨¨)dinger Equations with Symbolic Computation.Communications in Theoretical Physics.In Press,Accepted Manuscript.
52.Tian,B.and Gao,Y.-T.,Spherical Kadomtsev-Petviashvili Equation and Nebulons for Dust Ion-Acoustic Waves with Symbolic Computation.Physics Letters A.340(1-4),2005,243-250.
53.Tian,B.and Gao,Y.-T.,Symbolic Computation Study of the Perturbed Nonlinear Schr(o丨¨)dinger Model in Inhomogeneous Optical Fibers.Physics Letters A.342(3), 2005,228-236.
54.Tian,B.and Gao,Y.-T.,Variable-Coefficient Higher-Order Nonlinear Schr(o|¨)dinger Model in Optical Fibers:New Transformation with Burstons,Brightons and Symbolic Computation.Physics Letters A.359(3),2006,241-248.
55.Tian,B.and Gao,Y.-T.,Symbolic Computation on Cylindrical-Modified Dust-Ion-Acoustic Nebulons in Dusty Plasmas.Physics Letters A.362(4),2007,283-288.
56.Chen,H.-H.and Liu,C.-S.,Solitons in Nonuniform Media.Physical Review Letters.37(11),1976,693.
57.Zhang,D.-J.and Chen,D.-Y.,Negatons,Positons,Rational-Like Solutions and Conservation Laws of the Korteweg-de Vries Equation with Loss and Non-Uniformity Terms.Journal of Physics A.37(3),2004,851-865.
58.Liu,C.S.,Advances in Plasma Physics,Wiley,1976.
59.Rosenbluth,M.N.,Parametric Instabilities in Inhomogeneous Media.Physical Review Letters.29(9),1972,565.
60.Ginzburg,V.,The Propagation of Electromagnetic Waves in Plasma,Pergamon,1964.
61.Liu,C.S.,Rosenbluth,M.N.and White,R.B.,Raman and Brillouin Scattering of Electromagnetic Waves in Inhomogeneous Plasmas.Physics of Fluids.17(6),1974,1211-1219.
62.Perkins,F.W.and Flick,J.,Parametric Instabilities in Inhomogeneous Plasmas.Physics of Fluids.14(9),1971,2012-2018.
63.Estabrook,K.G.,Valeo,E.J.and Kruer,W.L.,Two-Dimensional Relativistic Simulations of Resonance Absorption.Physics of Fluids.18(9),1975,1151-1159.
64.Lee,Y.C.and Kaw,P.K.,Temporal Electrostatic Instabilities in Inhomogeneous Plasmas.Physical Review Letters.32(4),1974,135.
65.Forslund,D.W.,Kindel,J.M.,Lee,K.,et al.,Absorption of Laser Light on Self-Consistent Plasma Density Profiles.Physical Review Letters.36(1),1976,35.
66.龚倩,徐荣,叶小华等,高速超长距离光传输技术,人民邮电出版社,2005.
67.Hasegawa,A.and Tappert,F.,Transmission of Stationary Nonlinear Optical Pulses in Dispersive Dielectric Fibers.I.Anomalous Dispersion.Applied Physics Letters.23(3),1973,142-144.
68.Mollenauer,L.F.and Smith,K.,Demonstration of Soliton Transmission over More Than 4000 Km in Fiber with Loss Periodically Compensated by Raman Gain.Optics Letters.13(8),1988,675.
69.Hasegawa,A.and Tappert,F.,Transmission of Stationary Nonlinear Optical Pulses in Dispersive Dielectric Fibers.Ⅱ.Normal Dispersion.Applied Physics Letters.23(4),1973,171-172.
70.Kodama,Y.and Hasegawa,A.,Nonlinear Pulse Propagation in a Monomode Dielectric Guide.Journal of Quantum Electronics.23(5),1987,510-524.
71.Gilson,C.,Hietarinta,J.,Nimmo,J.,et al.,Sasa-Satsuma Higher-Order Nonlinear Schr(o|¨)dinger Equation and Its Bilinearization and Multisoliton Solutions.Physical Review E.68(1),2003,016614.
72.Mahalingam,A.and Porsezian,K.,Propagation of Dark Solitons with Higher-Order Effects in Optical Fibers.Physical Review E.64(4),2001,046608.
73.Gedalin,M.,Scott,T.C.and Band,Y.B.,Optical Solitary Waves in the Higher Order Nonlinear Schr(o|¨)dinger Equation.Physical Review Letters.78(3),1997,448.
74.Brugarino,T.and Sciacca,M.,Singularity Analysis and Integrability for a Hnls Equation Governing Pulse Propagation in a Generic Fiber Optics.Optics Communications.262(2),2006,250-256.
75.Xu,Z.,Li,L.,Li,Z.,et al.,Modulation Instability and Solitons on a Cw Background in an Optical Fiber with Higher-Order Effects.Physical Review E.67(2),2003,026603.
76.LI,B.,Exact Soliton Solutions for the Higher-Order Nonlinear Schr(o|¨)dinger Equation.International Journal of Modem Physics C.16,2005,1225-1237.
77.Chow,K.W.,A Class of Doubly Periodic Waves for Nonlinear Evolution Equations.Wave Motion.35(1),2002,71-90.
78.Davydov,A.S.,Solitons in Molecular Systems,Reidel,1991.
79.Brizhik,L.,Eremko,A.,Piette,B.,et al.,Solitons in Alpha -Helical Proteins.Physical Review E.70(3),2004,031914.
80.Brizhik,L.,Eremko,A.,Piette,B.,et al.,Charge and Energy Transfer by Solitons in Low-Dimensional Nanosystems with Helical Structure.Chemical Physics.324(1),2006,259-266.
81.Daniel,M.and Deepamala,K.,Davydov Soliton in Alpha Helical Proteins:Higher Order and Discreteness Effects.Physica A.221(1-3),1995,241-255.
82.庞小峰,非线性量子力学理论,重庆出版社,1994。
83.Daniel,M.and Latha,M.M.,Soliton in Alpha Helical Proteins with Interspine Coupling at Higher Order.Physics Letters A.302(2-3),2002,94-104.
84.Nakkeeran,K.,Porsezian,K.,Sundaram,P.S.,et al.,Optical Solitons in N-Coupled Higher Order Nonlinear Schr(o|¨)dinger Equations.Physical Review Letters.80(7),1998,1425.
85.Mahalingam,A.and Porsezian,K.,Propagation of Dark Solitons in a System of Coupled Higher-Order Nonlinear Schr(o|¨)dinger Equations.Journal of Physics A.35(13),2002,3099-3109.
86.Radhakrishnan,R.,Lakshmanan,M.and Daniel,M.,Bright and Dark Optical Solitons in Coupled Higher-Order Nonlinear Schr(o|¨)dinger Equations through Singularity Structure Analysis.Journal of Physics A.28(24),1995,7299-7314.
87.Xu,T.,Zhang,C.-Y.,Li,J.,et al.,Symbolic Computation on Generalized Hopf-Cole Transformation for a Forced Burgers Model with Variable Coefficients from Fluid Dynamics.Wave Motion.44(4),2007,262-270.
88.Meng,X.-H.,Liu,W.-J.,Zhu,H.-W.,et al.,Multi-Soliton Solutions and a B(a|¨)cklund Transformation for a Generalized Variable-Coefficient Higher-Order Nonlinear Schr(o|¨)dinger Equation with Symbolic Computation.Physica A.357(1),2008,97-107.
89.Meng,X.-H.,Zhang,C.-Y.,Li,J.,et al.,Analytic Multi-Solitonic Solutions of Variable-Coefficient Higher-Order Nonlinear Schr(o|¨)dinger Models by Modified Bilinear Method with Symbolic Computation.Zeitschrift Fur Naturforschung Section a-a Journal of Physical Sciences.62(1-2),2007,13-20.
90.L(u|¨),X.,Zhu,H.-W.,Meng,X.-H.,et al.,Soliton Solutions and a B(a|¨)cklund Transformation for a Generalized Nonlinear Schr6dinger Equation with Variable Coefficients from Optical Fiber Communications.Journal of Mathematical Analysis and Applications.336(2),2007,1305-1315.
91.Matsuno,Y.,Bilinear Transformation Method,Academic Press,1984.
92.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.Journal of Physics A.39(46),2006,14353-14362.
93.Rogers,C.and Shadwick,W.F.,B(a|¨)cklund Transformations and Their Applications,Academic Press,1982.
94.Tian,B.,Gao,Y.-T.and Zhu,H.-W.,Variable-Coefficient Higher-Order Nonlinear Schr(o|¨)dinger Model in Optical Fibers:Variable-Coefficient Bilinear Form,B(a|¨)cklund Transformation,Brightons and Symbolic Computation.Physics Letters A.366(3),2007,223-229.
95.Nimmo,J.J.C.and Freeman,N.C.,The Use of B(a|¨)cklund Transformations in Obtaining N-Soliton Solutions in Wronskian Form.Journal of Physics A.17(7),1984,1415-1424.
96.Satsuma,J.,A Wronskian Representation of N-Soliton Solutions of Nonlinear Evolution Equations Journal of the Physical Society of Japan.46(1),1979,359-360.
97.Freeman,N.C.,Soliton Solutions of Nonlinear Evolution Equations.IMA Journal of Applied Mathematics.32,1984.
98.L(u|¨),X.,Zhu,H.-W.,Yao,Z.-Z.,et al.,Multisoliton Solutions in Terms of Double Wronskian Determinant for a Generalized Variable-Coefficient Nonlinear Schr(o|¨)dinger Equation from Plasma Physics,Arterial Mechanics,Fluid Dynamics and Optical Communications.Annals of Physics.In Press,Corrected Proof.
99.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.Chinese Physics Letters.24(5),2007,1173-1176.
100.Zhang,C.,Zhu,H.-W.,Zhang,C.-Y.,et al.,N-Solitonic Solution in Terms of Wronskian Determinant for a Perturbed Variable-Coefficient Korteweg-de vries Equation.International Journal of Theoretical Physics.In Press,Corrected Proof.
101.Yao,Z.-Z.,Zhang,C.-Y.,Zhu,H.-W.,et al.,Wronskian and Grammian Determinant Solutions for a Variable-Coefficient Kadomtsev-Petviashvili Equation.Communications in Theoretical Physics.In Press,Corrected Proof.
102.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.Physics Letters A.356(1),2006,8-16.
103.Frantzeskakis,D.J.,Proukakis,N.P.and Kevrekidis,P.G.,Dynamics of Shallow Dark Solitons in a Trapped Gas of Impenetrable Bosons.Physical Review A.70(1),2004,015601.
104.Capasso,F.,Sirtori,C.,Faist,J.,et al.,Observation of an Electronic Bound State above a Potential Well.Nature.358(6387),1992,565-567.
105.Jaworski,M.and Zagrodzinski,J.,Positon and Positon-Like Solutions of the Korteweg-de Vries and Sine-Gordon Equations.Chaos,Solitons & Fractals.5(12),1995,2229-2233.
106.Huang,G.,Szeflel,J.and Zhu,S.,Dynamics of Dark Solitons in Quasi-One-Dimensional Bose-Einstein Condensates.Physical Review A.65(5),2002,053605.
107.Quarteroni,A.,Tuveri,M.and Veneziani,A.,Computational Vascular Fluid Dynamics:Problems,Models and Methods.Computing and Visualization in Science.2(4),2000,163-197.
108.Davis,K.B.,Mewes,M.O.,Andrews,M.R.,et al.,Bose-Einstein Condensation in a Gas of Sodium Atoms.Physical Review Letters.75(22),1995,3969.
109.Anderson,M.H.,Ensher,J.R.,Matthews,M.R.,et al.,Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor.Science.269(5221 ),1995, 198-201.
110.Dalfovo,F.,Giorgini,S.,Pitaevskii,L.P.,et al.,Theory of Bose-Einstein Condensation in Trapped Gases.Reviews of Modern Physics.71(3),1999,463.
111.Gao,Y.-T.and Tian,B.,On the Non-Planar Dust-Ion-Acoustic Waves in Cosmic Dusty Plasmas with Transverse Perturbations.Europhysics Letters.77(1),2007,15001.
112.Gao,Y.-T.and Tian,B.,Cosmic Dust-Ion-Acoustic Waves,Spherical Modified Kadomtsev-Petviashvili Model,and Symbolic Computation.Physics of Plasmas.13(11),2006,112901-112906.
113.Miura,R.M.,Gardner,C.S.and Kruskal,M.D.,Korteweg-de Vries Equation and Generalizations.Ⅱ.Existence of Conservation Laws and Constants of Motion.Journal of Mathematical Physics.9(8),1968,1204-1209.
114.Ablowitz,M.J.and Segur,H.,Solitons and the Inverse Scattering Transform,SIAM,1981.
115.Dauxois,T.and Peyrard,M.,Physics of Solitons,Cambridge University Press,2006.
116.Joshi,N.,Painlev(?) Property of General Variable-Coefficient Versions of the Korteweg-de Vries and Non-Linear Schr(o|¨)dinger Equations.Physics Letters A.125(9),1987,456-460.
117.Hlavat,L.,Painlev(?) Analysis of Nonautonomous Evolution Equations.Physics Letters A.128(6-7),1988,335-338.
118.Li,J.,Xu,T.,Meng,X.-H.,et al.,Symbolic Computation on Integrable Properties of a Variable-Coefficient Korteweg-de Vries Equation from Arterial Mechanics and Bose-Einstein Condensates.Physica Scripta.75(3),2007,278-284.
2.Wester,M.J.,Computer Algebra Systems:A Practical Guide,Wiley,1999.
3.Graham,R.L.,Knuth,D.E.and Patashnik,O.,Concrete Mathematics:A Foundation for Computer Science,Addison-Wesley Professional,2002.
4.Enns,R.H.and McGuire,G.C.,Nonlinear Physics with Maple for Scientists and Engineers,Birkh(a丨¨)user,2000.
5.Robeva,R.S.,Kirkwood,J.R.,Davies,R.L.,et al.,An Invitation to Biomathematics,Academic Press,2007.
6.Giordano,F.R.,Weir,M.D.and Fox,W.P.,A First Course in Mathematical Modeling,Brooks,2002.
7.Sedgewick,R.and Flajolet,P.,An Introduction to the Analysis of Algorithms Addison-Wesley Publishing Company,1996.
8.Donald,B.,Kapur,D.and Mundy,J.,Symbolic and Numerical Computation for Artificial Intelligence,Academic Press,1992.
9.《现代数学手册》编纂委员会,现代数学手册·计算机数学卷,华中科技大学出版社,2001.
10.吴文俊,数学机械化,科学出版社,2003.
11.Veltman,M.,Schoonship,CERN preprint,1967.
12.Hunt,B.R.,Lipsman,R.L.,Rosenberg,J.M.,et al.,A Guide to Matlab:For Beginners and Experienced Users,Cambridge University Press,2006.
13.Wolfram,S.,The Mathematica Book,Cambridge University Press,1999.
14.Cyganowski,S.,Kloeden,P.and Ombach,J.,From Elementary Probability to Stochastic Differential Equations with Maple Springer,2001.
15.Corless,R.M.,Essential Maple:An Introduction for Scientific Programmers,Springer,1995.
16.Infeld,E.and Rowlands,G.,Nonlinear Waves,Solitons and Chaos,Cambridge University Press,2000.
17.杨伯君和赵玉芳,高等数学物理方法,北京邮电大学出版社,2003.
18.Scott,A.,Encyclopedia of Nonlinear Science,Routledge,2005.
19.Diacu,F.and Holmes,P.,Celestial Encounter:The Origins of Chaos and Stability,Princeton University Press,1996.
20.Strogatz,S.H.,Nonlinear Dynamics and Chaos:With Applications to Physics, Biology,Chemistry and Engineering,Westview Press,2001.
21.Peitgen,H.O.,J(u丨¨)rgens,H.and Saupe,D.,Chaos and Fractals:New Frontiers of Science,Springer,2004.
22.Russell,J.S.Report on Waves.in Report of the 14th meeting of British association for the advancement of science.John Murray,London.1844,311-390.
23.Korteweg,D.J.and de Vries,G.,On the Change of Form of Long Waves Advancing in a Rectangular Canal and on a New Type of Long Stationary Wave.Philosophy Magazine.39,1895,422-443.
24.王明亮,非线性发展方程与孤立子,兰州大学出版社,1990.
25.黄景宁,徐济仲和熊吟涛,孤子:概念、原理和应用,高等教育出版社,2004.
26.Zabusky,N.J.and Kruskal,M.D.,Interaction Of "Solitons" In a Collisionless Plasma and the Recurrence of Initial States.Physical Review Letters.15(6),1965,240.
27.Perring,J.K.and Skyrme,T.H.,A Model Unified Field Equation.Nuclear Physics.31,1962,550-555.
28.Agrawal,G.,Applications of Nonlinear Fiber Optics,Academic Press,2001.
29.Hirota,R.,Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons.Physical Review Letters.27(18),1971,1192.
30.Hirota,R.,Exact Solution of the Modified Korteweg-de Vries Equation for Multiple Collisions of Solitons.Journal of the Physical Society of Japan.33(5),1972,1456-1458.
31.Hirota,R.,Exact Solution of the Sine-Gordon Equation for Multiple Collisions of Solitons.Journal of the Physical Society of Japan.33(5),1972,1459-1463.
32.Hirota,R.and Satsuma,J.,N-Soliton Solution of the K-dV Equation with Loss and Nonuniformity Terms.Journal of the Physical Society of Japan.41,1976,2141-2142.
33.Hirota,R.Direct Method of Finding Exact Solutions of Nonlinear Evolution Equations.in B(a丨¨)cklund Transformations,the Inverse Scattering Method,Solitons and Their Applications.Berlin.1976,40-68.
34.Hirota,R.,A New Form of B(a丨¨)cklund Transformations and Its Relation to the Inverse Scattering Problem Progress of Theoretical Physics.52,1974,1498-1512.
35.Hirota,R.and Satsuma,J.,N-Soliton Solution of the K-dV Equation with Loss and Nonuniformity Terms.Journal of the Physical Society of Japan.41,1976,2141-2142.
36.Hirota,R.and Satsuma,J.,A Variety of Nonlinear Network Equations Generated from the B(a丨¨)cklund Transformation for the Toda Lattice.Progress of Theoretical Physics Supplement.59,1976,64-100.
37.Hirota,R.and Satsuma,J.,Nonlinear Evolution Equations Generated from the B(a丨¨)cklund Transformation for the Boussinesq Equation Progress of Theoretical Physics.57(3),1977,797-807.
38.Hirota,R.,The Direct Method in Soliton Theory,Cambridge University Press,2004.
39.Weiss,J.,Tabor,M.and Carnevale,G.,The Painlev(?) Property for Partial Differential Equations.Journal of Mathematical Physics.24(3),1983,522-526.
40.Weiss,J.,The Painlev(?) Property for Partial Differential Equations.Ⅱ:B(a丨¨)cklund Transformation,Lax Pairs,and the Schwarzian Derivative.Journal of Mathematical Physics.24(6),1983,1405-1413.
41.Steeb,W.H.and Euler,N.,Nonlinear Evolution Equations and Painlev(?) Test,Word Scientific,1988.
42.刘式适和刘式达,非线性大气动力学,国防工业出版社,1996.
43.Weiss,J.,On Classes of lntegrable Systems and the Painlev(?) Property.Journal of Mathematical Physics.25(1),1984,13-24.
44.Newell,A.C.,Tabor,M.and Zeng,Y.B.,A Unified Approach to Painlev(?)Expansions.Physica D.29(1-2),1987,1-68.
45.潘祖梁,非线性问题的数学方法及其应用,浙江大学出版社,2004.
46.Clarkson,P.A.and Kruskal,M.D.,New Similarity Reductions of the Boussinesq Equation.Journal of Mathematical Physics.30(10),1989,2201-2213.
47.谷超豪,胡和生和周子翔,孤立子理论中的达布变换及其几何应用,上海科学技术出版公司,2005.
48.田播,[学位论文]计算机符号计算在非线性研究中的若干应用,北京航空航天大学,2003.
49.范恩贵,可积系统与计算机代数,科学出版社,2004.
50.Zhu,H.-W.and Tian,B.,B(a丨¨)cklund Transformation in Bilinear Form for a Higher-Order Nonlinear Schr(o丨¨)dinger Equation.Nonlinear Analysis:Theory,Methods & Applications.In Press,Corrected Proof.
51.Zhu,H.-W.,Tian,B.,Meng,X.-H.,et al.,A Bilinear B(a丨¨)cklund Transformation and N Soliton Like Solution of the Three Coupled Higher-Order Nonlinear Schr(o丨¨)dinger Equations with Symbolic Computation.Communications in Theoretical Physics.In Press,Accepted Manuscript.
52.Tian,B.and Gao,Y.-T.,Spherical Kadomtsev-Petviashvili Equation and Nebulons for Dust Ion-Acoustic Waves with Symbolic Computation.Physics Letters A.340(1-4),2005,243-250.
53.Tian,B.and Gao,Y.-T.,Symbolic Computation Study of the Perturbed Nonlinear Schr(o丨¨)dinger Model in Inhomogeneous Optical Fibers.Physics Letters A.342(3), 2005,228-236.
54.Tian,B.and Gao,Y.-T.,Variable-Coefficient Higher-Order Nonlinear Schr(o|¨)dinger Model in Optical Fibers:New Transformation with Burstons,Brightons and Symbolic Computation.Physics Letters A.359(3),2006,241-248.
55.Tian,B.and Gao,Y.-T.,Symbolic Computation on Cylindrical-Modified Dust-Ion-Acoustic Nebulons in Dusty Plasmas.Physics Letters A.362(4),2007,283-288.
56.Chen,H.-H.and Liu,C.-S.,Solitons in Nonuniform Media.Physical Review Letters.37(11),1976,693.
57.Zhang,D.-J.and Chen,D.-Y.,Negatons,Positons,Rational-Like Solutions and Conservation Laws of the Korteweg-de Vries Equation with Loss and Non-Uniformity Terms.Journal of Physics A.37(3),2004,851-865.
58.Liu,C.S.,Advances in Plasma Physics,Wiley,1976.
59.Rosenbluth,M.N.,Parametric Instabilities in Inhomogeneous Media.Physical Review Letters.29(9),1972,565.
60.Ginzburg,V.,The Propagation of Electromagnetic Waves in Plasma,Pergamon,1964.
61.Liu,C.S.,Rosenbluth,M.N.and White,R.B.,Raman and Brillouin Scattering of Electromagnetic Waves in Inhomogeneous Plasmas.Physics of Fluids.17(6),1974,1211-1219.
62.Perkins,F.W.and Flick,J.,Parametric Instabilities in Inhomogeneous Plasmas.Physics of Fluids.14(9),1971,2012-2018.
63.Estabrook,K.G.,Valeo,E.J.and Kruer,W.L.,Two-Dimensional Relativistic Simulations of Resonance Absorption.Physics of Fluids.18(9),1975,1151-1159.
64.Lee,Y.C.and Kaw,P.K.,Temporal Electrostatic Instabilities in Inhomogeneous Plasmas.Physical Review Letters.32(4),1974,135.
65.Forslund,D.W.,Kindel,J.M.,Lee,K.,et al.,Absorption of Laser Light on Self-Consistent Plasma Density Profiles.Physical Review Letters.36(1),1976,35.
66.龚倩,徐荣,叶小华等,高速超长距离光传输技术,人民邮电出版社,2005.
67.Hasegawa,A.and Tappert,F.,Transmission of Stationary Nonlinear Optical Pulses in Dispersive Dielectric Fibers.I.Anomalous Dispersion.Applied Physics Letters.23(3),1973,142-144.
68.Mollenauer,L.F.and Smith,K.,Demonstration of Soliton Transmission over More Than 4000 Km in Fiber with Loss Periodically Compensated by Raman Gain.Optics Letters.13(8),1988,675.
69.Hasegawa,A.and Tappert,F.,Transmission of Stationary Nonlinear Optical Pulses in Dispersive Dielectric Fibers.Ⅱ.Normal Dispersion.Applied Physics Letters.23(4),1973,171-172.
70.Kodama,Y.and Hasegawa,A.,Nonlinear Pulse Propagation in a Monomode Dielectric Guide.Journal of Quantum Electronics.23(5),1987,510-524.
71.Gilson,C.,Hietarinta,J.,Nimmo,J.,et al.,Sasa-Satsuma Higher-Order Nonlinear Schr(o|¨)dinger Equation and Its Bilinearization and Multisoliton Solutions.Physical Review E.68(1),2003,016614.
72.Mahalingam,A.and Porsezian,K.,Propagation of Dark Solitons with Higher-Order Effects in Optical Fibers.Physical Review E.64(4),2001,046608.
73.Gedalin,M.,Scott,T.C.and Band,Y.B.,Optical Solitary Waves in the Higher Order Nonlinear Schr(o|¨)dinger Equation.Physical Review Letters.78(3),1997,448.
74.Brugarino,T.and Sciacca,M.,Singularity Analysis and Integrability for a Hnls Equation Governing Pulse Propagation in a Generic Fiber Optics.Optics Communications.262(2),2006,250-256.
75.Xu,Z.,Li,L.,Li,Z.,et al.,Modulation Instability and Solitons on a Cw Background in an Optical Fiber with Higher-Order Effects.Physical Review E.67(2),2003,026603.
76.LI,B.,Exact Soliton Solutions for the Higher-Order Nonlinear Schr(o|¨)dinger Equation.International Journal of Modem Physics C.16,2005,1225-1237.
77.Chow,K.W.,A Class of Doubly Periodic Waves for Nonlinear Evolution Equations.Wave Motion.35(1),2002,71-90.
78.Davydov,A.S.,Solitons in Molecular Systems,Reidel,1991.
79.Brizhik,L.,Eremko,A.,Piette,B.,et al.,Solitons in Alpha -Helical Proteins.Physical Review E.70(3),2004,031914.
80.Brizhik,L.,Eremko,A.,Piette,B.,et al.,Charge and Energy Transfer by Solitons in Low-Dimensional Nanosystems with Helical Structure.Chemical Physics.324(1),2006,259-266.
81.Daniel,M.and Deepamala,K.,Davydov Soliton in Alpha Helical Proteins:Higher Order and Discreteness Effects.Physica A.221(1-3),1995,241-255.
82.庞小峰,非线性量子力学理论,重庆出版社,1994。
83.Daniel,M.and Latha,M.M.,Soliton in Alpha Helical Proteins with Interspine Coupling at Higher Order.Physics Letters A.302(2-3),2002,94-104.
84.Nakkeeran,K.,Porsezian,K.,Sundaram,P.S.,et al.,Optical Solitons in N-Coupled Higher Order Nonlinear Schr(o|¨)dinger Equations.Physical Review Letters.80(7),1998,1425.
85.Mahalingam,A.and Porsezian,K.,Propagation of Dark Solitons in a System of Coupled Higher-Order Nonlinear Schr(o|¨)dinger Equations.Journal of Physics A.35(13),2002,3099-3109.
86.Radhakrishnan,R.,Lakshmanan,M.and Daniel,M.,Bright and Dark Optical Solitons in Coupled Higher-Order Nonlinear Schr(o|¨)dinger Equations through Singularity Structure Analysis.Journal of Physics A.28(24),1995,7299-7314.
87.Xu,T.,Zhang,C.-Y.,Li,J.,et al.,Symbolic Computation on Generalized Hopf-Cole Transformation for a Forced Burgers Model with Variable Coefficients from Fluid Dynamics.Wave Motion.44(4),2007,262-270.
88.Meng,X.-H.,Liu,W.-J.,Zhu,H.-W.,et al.,Multi-Soliton Solutions and a B(a|¨)cklund Transformation for a Generalized Variable-Coefficient Higher-Order Nonlinear Schr(o|¨)dinger Equation with Symbolic Computation.Physica A.357(1),2008,97-107.
89.Meng,X.-H.,Zhang,C.-Y.,Li,J.,et al.,Analytic Multi-Solitonic Solutions of Variable-Coefficient Higher-Order Nonlinear Schr(o|¨)dinger Models by Modified Bilinear Method with Symbolic Computation.Zeitschrift Fur Naturforschung Section a-a Journal of Physical Sciences.62(1-2),2007,13-20.
90.L(u|¨),X.,Zhu,H.-W.,Meng,X.-H.,et al.,Soliton Solutions and a B(a|¨)cklund Transformation for a Generalized Nonlinear Schr6dinger Equation with Variable Coefficients from Optical Fiber Communications.Journal of Mathematical Analysis and Applications.336(2),2007,1305-1315.
91.Matsuno,Y.,Bilinear Transformation Method,Academic Press,1984.
92.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.Journal of Physics A.39(46),2006,14353-14362.
93.Rogers,C.and Shadwick,W.F.,B(a|¨)cklund Transformations and Their Applications,Academic Press,1982.
94.Tian,B.,Gao,Y.-T.and Zhu,H.-W.,Variable-Coefficient Higher-Order Nonlinear Schr(o|¨)dinger Model in Optical Fibers:Variable-Coefficient Bilinear Form,B(a|¨)cklund Transformation,Brightons and Symbolic Computation.Physics Letters A.366(3),2007,223-229.
95.Nimmo,J.J.C.and Freeman,N.C.,The Use of B(a|¨)cklund Transformations in Obtaining N-Soliton Solutions in Wronskian Form.Journal of Physics A.17(7),1984,1415-1424.
96.Satsuma,J.,A Wronskian Representation of N-Soliton Solutions of Nonlinear Evolution Equations Journal of the Physical Society of Japan.46(1),1979,359-360.
97.Freeman,N.C.,Soliton Solutions of Nonlinear Evolution Equations.IMA Journal of Applied Mathematics.32,1984.
98.L(u|¨),X.,Zhu,H.-W.,Yao,Z.-Z.,et al.,Multisoliton Solutions in Terms of Double Wronskian Determinant for a Generalized Variable-Coefficient Nonlinear Schr(o|¨)dinger Equation from Plasma Physics,Arterial Mechanics,Fluid Dynamics and Optical Communications.Annals of Physics.In Press,Corrected Proof.
99.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.Chinese Physics Letters.24(5),2007,1173-1176.
100.Zhang,C.,Zhu,H.-W.,Zhang,C.-Y.,et al.,N-Solitonic Solution in Terms of Wronskian Determinant for a Perturbed Variable-Coefficient Korteweg-de vries Equation.International Journal of Theoretical Physics.In Press,Corrected Proof.
101.Yao,Z.-Z.,Zhang,C.-Y.,Zhu,H.-W.,et al.,Wronskian and Grammian Determinant Solutions for a Variable-Coefficient Kadomtsev-Petviashvili Equation.Communications in Theoretical Physics.In Press,Corrected Proof.
102.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.Physics Letters A.356(1),2006,8-16.
103.Frantzeskakis,D.J.,Proukakis,N.P.and Kevrekidis,P.G.,Dynamics of Shallow Dark Solitons in a Trapped Gas of Impenetrable Bosons.Physical Review A.70(1),2004,015601.
104.Capasso,F.,Sirtori,C.,Faist,J.,et al.,Observation of an Electronic Bound State above a Potential Well.Nature.358(6387),1992,565-567.
105.Jaworski,M.and Zagrodzinski,J.,Positon and Positon-Like Solutions of the Korteweg-de Vries and Sine-Gordon Equations.Chaos,Solitons & Fractals.5(12),1995,2229-2233.
106.Huang,G.,Szeflel,J.and Zhu,S.,Dynamics of Dark Solitons in Quasi-One-Dimensional Bose-Einstein Condensates.Physical Review A.65(5),2002,053605.
107.Quarteroni,A.,Tuveri,M.and Veneziani,A.,Computational Vascular Fluid Dynamics:Problems,Models and Methods.Computing and Visualization in Science.2(4),2000,163-197.
108.Davis,K.B.,Mewes,M.O.,Andrews,M.R.,et al.,Bose-Einstein Condensation in a Gas of Sodium Atoms.Physical Review Letters.75(22),1995,3969.
109.Anderson,M.H.,Ensher,J.R.,Matthews,M.R.,et al.,Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor.Science.269(5221 ),1995, 198-201.
110.Dalfovo,F.,Giorgini,S.,Pitaevskii,L.P.,et al.,Theory of Bose-Einstein Condensation in Trapped Gases.Reviews of Modern Physics.71(3),1999,463.
111.Gao,Y.-T.and Tian,B.,On the Non-Planar Dust-Ion-Acoustic Waves in Cosmic Dusty Plasmas with Transverse Perturbations.Europhysics Letters.77(1),2007,15001.
112.Gao,Y.-T.and Tian,B.,Cosmic Dust-Ion-Acoustic Waves,Spherical Modified Kadomtsev-Petviashvili Model,and Symbolic Computation.Physics of Plasmas.13(11),2006,112901-112906.
113.Miura,R.M.,Gardner,C.S.and Kruskal,M.D.,Korteweg-de Vries Equation and Generalizations.Ⅱ.Existence of Conservation Laws and Constants of Motion.Journal of Mathematical Physics.9(8),1968,1204-1209.
114.Ablowitz,M.J.and Segur,H.,Solitons and the Inverse Scattering Transform,SIAM,1981.
115.Dauxois,T.and Peyrard,M.,Physics of Solitons,Cambridge University Press,2006.
116.Joshi,N.,Painlev(?) Property of General Variable-Coefficient Versions of the Korteweg-de Vries and Non-Linear Schr(o|¨)dinger Equations.Physics Letters A.125(9),1987,456-460.
117.Hlavat,L.,Painlev(?) Analysis of Nonautonomous Evolution Equations.Physics Letters A.128(6-7),1988,335-338.
118.Li,J.,Xu,T.,Meng,X.-H.,et al.,Symbolic Computation on Integrable Properties of a Variable-Coefficient Korteweg-de Vries Equation from Arterial Mechanics and Bose-Einstein Condensates.Physica Scripta.75(3),2007,278-284.