基于符号计算的光纤通信等若干领域中变系数非线性模型的研究
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摘要
非线性模型在当前许多科学和工程领域的理论研究中具有非常重要的意义.它们可以用于描述光纤通信、流体力学、固体力学和等离子体物理等领域中的非线性现象.通过研究非线性模型的解析解及可积性质,我们可以更加深入地了解非线性模型所反映的相关动力机制的本质特征.近来,考虑传播介质的不均匀性与边界的不一致性等因素,变系数非线性模型被认为比常系数模型能够更实际地描述各种各样的非线性机制.计算机符号计算具有易于操作和实现的特点,能够精确地处理繁复冗长的表达式及微积分运算.符号计算软件强大的数学计算功能及绘图功能可以帮助我们处理变系数非线性模型解及相关性质的解析及可观测性研究,为变系数非线性模型的研究工作提供功能强大的辅助工具.
本文主要借助符号计算将某些适用于研究常系数非线性模型的解析方法进行推广并应用于光纤通信等领域中的若干变系数非线性模型.利用推广的Painlev(?)分析、双线性方法、AKNS方法、Wronskian技巧及Pfaffian方法解析地研究了广义变系数高阶nonlinearSchr(o｜¨)dinger(HNLS)方程、变系数(3+1)维Kadomstev-Petviashvili(KP)方程、变系数sine-Gordon(SG)方程、非均匀N耦合NLS方程及变系数KP方程的解及可积性质.这些变系数非线性模型在物理学和工程技术领域的不同分支中都有着广泛的应用,如光纤通信、等离子体、超导体、流体力学和非线性晶格等,特别是可用于描述带有非均匀边界条件或非均匀介质的物理背景中的各种非线性动力学机制.
本文的主要工作如下:
(Ⅰ)基于符号计算变系数非线性模型Painlev(?)性质的判定.Painlev(?)分析为判定非线性模型是否完全可积提供了必要条件.借助符号计算,本文对变系数(3+1)维KP方程及变系数SG方程进行了Painlev(?)分析.经检测发现变系数(3+1)维KP不具有Painlev(?)性质,而变系数SG方程是Painlev(?)可积的.
(Ⅱ)借助符号计算将双线性方法推广并应用于求解变系数非线性模型的各类精确解析解.本文利用截断的Painlev(?)展开式或在因变量变换中引入任意参数后修正的因变量变换,将变系数非线性模型双线性化并利用参数展开技巧得到变系数非线性模型的孤子型解.主要结果如下:(1)求得广义变系数HNLS方程的明多孤子型解,并分析参数对孤子性质的影响;(2)将广义变系数HNLS方程转化为相应的常系数HNLS方程,借助双线性方法求得广义变系数HNLS方程的暗孤子型解,然后利用符号计算及数值算法研究暗孤子型解的稳定性及传输特性;(3)求得非均匀N耦合NLS方程的显式孤子型解并研究孤子传播的性质;(4)求得变系数SG方程的多扭结孤子型解,并分析参数对解的性质的影响.
(Ⅲ)基于符号计算变系数非线性模型的可积性质如B(a｜¨)cklund变换及无穷多守恒律.本文利用两种不同的方法研究了变系数非线性模型的可积性质如B(a｜¨)cklund变换.利用变系数非线性模型的双线性形式,通过构造不同的形式B(a｜¨)cklund变换,本文得到广义变系数HNLS方程、变系数SG方程及非均匀Ⅳ耦合NLS方程的双线性B(a｜¨)cklund变换.借助变系数非线性模型对应的AKNS系统,求得了非均匀N耦合NLS方程及变系数SG方程Γ函数形式的B(a｜¨)cklund变换.此外,利用AKNS系统的F-Riccati形式求得了广义变系数HNLS方程、非均匀N耦合NLS方程及变系数SG方程的无穷多守恒律.
(Ⅳ)将Wronskian技巧推广并应用于求解变系数非线性模型行列式形式的多孤子型解.构造非线性模型的Wronski行列式解的难点之一就是Wronski行列式元素所满足的性质.本文从三个不同的思路出发,利用Wronskian技巧研究了变系数非线性模型Wronski行列式形式的多孤子型解,并借助符号计算软件对解进行了分析:(1)利用平衡的思想构造了变系数(3+1)维KP方程的Wronski行列式解,并通过直接代入到双线性方程中对解进行验证;(2)利用B(a｜¨)cklund变换构造Wronski行列式元素所满足的性质,求得变系数SG方程的Wronski行列式解;(3)借助AKNS系统构造双Wronski行列式元素所满足的性质,求得广义变系数HNLS方程双Wronski行列式形式的多孤子型解并对解进行分析.
(Ⅴ)Pfaffian方法的变系数推广及应用.Pfaffian是行列式的进一步推广.借助Pfaffian形式,本文求得变系数(3+1)维KP方程的Gramm行列式解并进行验证.将构造新耦合孤子方程的Pfaffian程序推广并应用于变系数KP方程.本文借助Pfaffian程序的变系数推广构造出变系数KP方程的耦合形式,然后求得变系数耦合KP方程的Wronski和Gramm型Pfaffian解,并借助Pfaffian的性质将解直接代入双线性形式的耦合方程中进行了验证.
本文借助计算机符号计算将Painlev(?)分析、双线性方法、AKNS方法、Wronskian技巧及Pfaffian方法进行了变系数推广并应用于研究光纤通信、流体力学中的若干变系数非线性模型的孤子型解及可积性质.借助于符号计算软件的绘图功能,本文对所得变系数非线性模型孤子型解的性质进行了绘图及分析.本文所用的研究变系数非线性模型的方法有望用于研究其他领域中的各类变系数非线性模型,此外所得光纤通信等领域中各个模型的解析研究结果可望有助于进一步的理论及实验研究.
The nonlinear models play a significant role in many current theoretical studies in several science and engineering fields.They can be used to describe the nonlinear phenomena in the optical fiber、fluid dynamics、solid-state dynamics and plasma physics,etc.By investigating the analytic solutions and integrable properties of the nonlinear models, the essential characteristics of the dynamical mechanism which those nonlinear models describle can be understood much more thoroughly. Recently,considering the inhomogeneity of the media and the uniformity of the boundary,the variable-coefficient nonlinear models are regarded to describe various nonlinear mechanism realistically than the constant-coefficient ones.The computerized symbolic computation possesses the merit that it can operate and realize easily and handle the complex and lengthy expressions together with differential and integral opterations more accurately.The powerful mathematic calculational and graphical functions of symbolic computation software can help us to perform the analytical and observable study on the exact solutions and properties of the variable-coefficient nonlinear models,so that it provides an efficient assistant tool for investigating the variable-coefficient nonlinear models.
In this paper,employing symbolic computation,some analytical methods in the soliton theory which can be used to investigate the constant-coefficient nonlinear models are generalized,and then applied to study the variable-coefficient nonlinear models in several fields such as the optial fibers.Using the generalized Painlev(?) analysis,bilinear method, AKNS method,Wronskian technique and Pfaffian method,the exact solutions and integrable properties of the generalized variable-coefficient higher-order nonlinear Schr(o|¨)dinger(HNLS) equation,the variable-coefficient (3+1)-dimensional Kadomstev-Petviashvili(KP)equation,the variable-coeffcient sine-Gordon equation,the inhomogeneous N coupled NLS equation and the variable-coefficient KP equation are analytically investigated.These variable-coefficient nonlinear models can be widely applied to various branches in physics and engineering fields such as the optical fiber,plasma physics,superconduct,fluid dynamics and nonlinear lattice.Specially,they can describe various nonlinear dynamical mechanism in the background with inhomogeneous media and uniform boundary conditions.
The main contents of this paper are listed as follows:
(Ⅰ) Test the Painlev(?) property for the variable-coefficient nonlinear models based on symbolic computation.The Painlev(?) analysis provides a necessary condition on whether a nonlinear model is completely integrable or not.With the help of symbolic computation,the Painlev(?) analysis is performed on the variable-coefficient(3+1)-dimensional KP equation and the variable-coefficient SG equation.It is found that the variable-coefficient(3+1)-dimensional KP equation does not possess the Painlev(?) property,while the variable-coefficient SG equation is Painlev(?) integrable.
(Ⅱ) Generalize the bilinear method and apply the generalization to solve various kinds of exact solutions for several variable-coefficient nonlinear models.In this paper,employing the truncated Painlev(?) expansion or introducing arbitrary parameters into the dependent variable transformation to get the modified one,several variable-coefficient nonlinear models are bilinearized and the solitonic solutions for those models are derived using the formal expansion technique.The main results are as follows:(1) obtain the bright multi-solitonic solutions of the generalized variable-coefficient HNLS equation,and then analyze the influence of the variable coefficients on the soliton properties;(2) transform the generalized variable-coefficient HNLS equation into the constant-coefficient one,obtain the dark solitonic solutions of the generalized variable-coefficient HNLS equation by virtue of the bilinear method and the transformation,and study the stability and propagation proporties of the dark solitons with the help of symbolic and numerical computation;(3) derive the explicit one- and two-solitonic solutions of the inhomogeneous N coupled NLS equation and investigate the properties of soliton propagation;(4) derive the multi-kink solitonic solutions of the variable-coefficient SG equation and analyze the influence of the parameters.
(Ⅲ) Investigate the int(?)grable properties such as the B(a|¨)cklund transformation and an infinite number of conservation laws for the variable-coefficient nonlinear models using symbolic computation.In this paper,two different methods are employed to derive the B(a|¨)cklund transformations of the variable-coefficient nonlinear models.Using their bilinearized forms to construct different formal B(a|¨)cklund transformations, the bilinear B(a|¨)cklund transformations of the generalized variable-coefficient HNLS equation,the variable-coefficient SG equation and the inhomogeneous N coupled NLS equation are obtained.Moreover,by the F-Riccati form of the AKNS system for the variable-coefficient nonlinear models,the B(a|¨)cklund transformations inΓfunction form for the variable-coefficient N coupled NLS equation and the variable-coefficient SG equation are derived.Furthermore,an infinite number of conservation laws for the generalized variable-coefficient HNLS equation,the variable -coefficient N coupled NLS equation and the variable-coefficient SG equation are gained using the correspondingΓ-Riccati equations.
(Ⅳ) Generalize the Wronskian technique and apply the generalization to obtain the multi-solitonic solutions in Wronski determinant form for variable-coefficient nonlinear models.One of the difficulties to construct the Wronski determinant solution of nonlinear models is finding the properties which the Wronski determinant elements satisfy.Starting from three different viewpoints,the Wronski determinant form multi-solitonic solutions for the variable-coefficient nonlinear models are constructed, and those solutions are analyzed with the help of symbolic computation: (1) construct and verify the Wronski determinant solution for the variable -coefficint(3+1)-dimensional KP equation based on the balancing idea; (2) construct the Wronski determinant solution for the variable-coefficient SG equation using the B(a|¨)cklund transformation for the properties which the determinant elements satisfy;(3) using the AKNS system,construct multi-solitonic solutions in double Wronski determinant form for the generalized variable-coefficient HNLS equation and analyze the properties of solutions.
(Ⅴ) Generalize the Pfaffian method and apply to the variable-coefficient nonlinear models.The Pfaffian can be regarded as the generalization of determinant.Using the properties of Pfaffian,the Gramm determinant solution for the variable-coefficient(3+1)-dimensional KP equation are constructed and verified.The Pfaffian procedure which can be used to construct new coupled soliton equations is generalized and applied to the variable-coefficient KP equation.Using the generalization of the Pfaffian procedure,the variable-coefficient coupled KP equations are generated, and then the Wronski- and Gramm-Pfaffian solutions for the coupled variable-coefficient KP equations are derived and verified by the Pfaffian properties.
In this paper,the Painlev(?) analysis,bilinear method,AKNS method, Wronskian technique and Pfaffian method are generalized and applied to investigate the solitonic solutions and integrable properties for the variable-coefficient nonlinear models in several fields such as the optical fiber and fluid dynamics using symbolic computation.By virtue of the graphical function of the symbolic computation software,the solitonic solutions of those variable-coefficient nonlinear models are illustrated and their propagation properties are analyzed.It can be expected that these generalized methods in this paper can be used to study a wide class of variable-coefficient nonlinear models in other fields and the analytical research results in this paper for the variable-coefficient nonlinear models in several fields such as the optical fiber can be helpful for further theoretical and experimental researches.
本文主要借助符号计算将某些适用于研究常系数非线性模型的解析方法进行推广并应用于光纤通信等领域中的若干变系数非线性模型.利用推广的Painlev(?)分析、双线性方法、AKNS方法、Wronskian技巧及Pfaffian方法解析地研究了广义变系数高阶nonlinearSchr(o｜¨)dinger(HNLS)方程、变系数(3+1)维Kadomstev-Petviashvili(KP)方程、变系数sine-Gordon(SG)方程、非均匀N耦合NLS方程及变系数KP方程的解及可积性质.这些变系数非线性模型在物理学和工程技术领域的不同分支中都有着广泛的应用,如光纤通信、等离子体、超导体、流体力学和非线性晶格等,特别是可用于描述带有非均匀边界条件或非均匀介质的物理背景中的各种非线性动力学机制.
本文的主要工作如下:
(Ⅰ)基于符号计算变系数非线性模型Painlev(?)性质的判定.Painlev(?)分析为判定非线性模型是否完全可积提供了必要条件.借助符号计算,本文对变系数(3+1)维KP方程及变系数SG方程进行了Painlev(?)分析.经检测发现变系数(3+1)维KP不具有Painlev(?)性质,而变系数SG方程是Painlev(?)可积的.
(Ⅱ)借助符号计算将双线性方法推广并应用于求解变系数非线性模型的各类精确解析解.本文利用截断的Painlev(?)展开式或在因变量变换中引入任意参数后修正的因变量变换,将变系数非线性模型双线性化并利用参数展开技巧得到变系数非线性模型的孤子型解.主要结果如下:(1)求得广义变系数HNLS方程的明多孤子型解,并分析参数对孤子性质的影响;(2)将广义变系数HNLS方程转化为相应的常系数HNLS方程,借助双线性方法求得广义变系数HNLS方程的暗孤子型解,然后利用符号计算及数值算法研究暗孤子型解的稳定性及传输特性;(3)求得非均匀N耦合NLS方程的显式孤子型解并研究孤子传播的性质;(4)求得变系数SG方程的多扭结孤子型解,并分析参数对解的性质的影响.
(Ⅲ)基于符号计算变系数非线性模型的可积性质如B(a｜¨)cklund变换及无穷多守恒律.本文利用两种不同的方法研究了变系数非线性模型的可积性质如B(a｜¨)cklund变换.利用变系数非线性模型的双线性形式,通过构造不同的形式B(a｜¨)cklund变换,本文得到广义变系数HNLS方程、变系数SG方程及非均匀Ⅳ耦合NLS方程的双线性B(a｜¨)cklund变换.借助变系数非线性模型对应的AKNS系统,求得了非均匀N耦合NLS方程及变系数SG方程Γ函数形式的B(a｜¨)cklund变换.此外,利用AKNS系统的F-Riccati形式求得了广义变系数HNLS方程、非均匀N耦合NLS方程及变系数SG方程的无穷多守恒律.
(Ⅳ)将Wronskian技巧推广并应用于求解变系数非线性模型行列式形式的多孤子型解.构造非线性模型的Wronski行列式解的难点之一就是Wronski行列式元素所满足的性质.本文从三个不同的思路出发,利用Wronskian技巧研究了变系数非线性模型Wronski行列式形式的多孤子型解,并借助符号计算软件对解进行了分析:(1)利用平衡的思想构造了变系数(3+1)维KP方程的Wronski行列式解,并通过直接代入到双线性方程中对解进行验证;(2)利用B(a｜¨)cklund变换构造Wronski行列式元素所满足的性质,求得变系数SG方程的Wronski行列式解;(3)借助AKNS系统构造双Wronski行列式元素所满足的性质,求得广义变系数HNLS方程双Wronski行列式形式的多孤子型解并对解进行分析.
(Ⅴ)Pfaffian方法的变系数推广及应用.Pfaffian是行列式的进一步推广.借助Pfaffian形式,本文求得变系数(3+1)维KP方程的Gramm行列式解并进行验证.将构造新耦合孤子方程的Pfaffian程序推广并应用于变系数KP方程.本文借助Pfaffian程序的变系数推广构造出变系数KP方程的耦合形式,然后求得变系数耦合KP方程的Wronski和Gramm型Pfaffian解,并借助Pfaffian的性质将解直接代入双线性形式的耦合方程中进行了验证.
本文借助计算机符号计算将Painlev(?)分析、双线性方法、AKNS方法、Wronskian技巧及Pfaffian方法进行了变系数推广并应用于研究光纤通信、流体力学中的若干变系数非线性模型的孤子型解及可积性质.借助于符号计算软件的绘图功能,本文对所得变系数非线性模型孤子型解的性质进行了绘图及分析.本文所用的研究变系数非线性模型的方法有望用于研究其他领域中的各类变系数非线性模型,此外所得光纤通信等领域中各个模型的解析研究结果可望有助于进一步的理论及实验研究.
The nonlinear models play a significant role in many current theoretical studies in several science and engineering fields.They can be used to describe the nonlinear phenomena in the optical fiber、fluid dynamics、solid-state dynamics and plasma physics,etc.By investigating the analytic solutions and integrable properties of the nonlinear models, the essential characteristics of the dynamical mechanism which those nonlinear models describle can be understood much more thoroughly. Recently,considering the inhomogeneity of the media and the uniformity of the boundary,the variable-coefficient nonlinear models are regarded to describe various nonlinear mechanism realistically than the constant-coefficient ones.The computerized symbolic computation possesses the merit that it can operate and realize easily and handle the complex and lengthy expressions together with differential and integral opterations more accurately.The powerful mathematic calculational and graphical functions of symbolic computation software can help us to perform the analytical and observable study on the exact solutions and properties of the variable-coefficient nonlinear models,so that it provides an efficient assistant tool for investigating the variable-coefficient nonlinear models.
In this paper,employing symbolic computation,some analytical methods in the soliton theory which can be used to investigate the constant-coefficient nonlinear models are generalized,and then applied to study the variable-coefficient nonlinear models in several fields such as the optial fibers.Using the generalized Painlev(?) analysis,bilinear method, AKNS method,Wronskian technique and Pfaffian method,the exact solutions and integrable properties of the generalized variable-coefficient higher-order nonlinear Schr(o|¨)dinger(HNLS) equation,the variable-coefficient (3+1)-dimensional Kadomstev-Petviashvili(KP)equation,the variable-coeffcient sine-Gordon equation,the inhomogeneous N coupled NLS equation and the variable-coefficient KP equation are analytically investigated.These variable-coefficient nonlinear models can be widely applied to various branches in physics and engineering fields such as the optical fiber,plasma physics,superconduct,fluid dynamics and nonlinear lattice.Specially,they can describe various nonlinear dynamical mechanism in the background with inhomogeneous media and uniform boundary conditions.
The main contents of this paper are listed as follows:
(Ⅰ) Test the Painlev(?) property for the variable-coefficient nonlinear models based on symbolic computation.The Painlev(?) analysis provides a necessary condition on whether a nonlinear model is completely integrable or not.With the help of symbolic computation,the Painlev(?) analysis is performed on the variable-coefficient(3+1)-dimensional KP equation and the variable-coefficient SG equation.It is found that the variable-coefficient(3+1)-dimensional KP equation does not possess the Painlev(?) property,while the variable-coefficient SG equation is Painlev(?) integrable.
(Ⅱ) Generalize the bilinear method and apply the generalization to solve various kinds of exact solutions for several variable-coefficient nonlinear models.In this paper,employing the truncated Painlev(?) expansion or introducing arbitrary parameters into the dependent variable transformation to get the modified one,several variable-coefficient nonlinear models are bilinearized and the solitonic solutions for those models are derived using the formal expansion technique.The main results are as follows:(1) obtain the bright multi-solitonic solutions of the generalized variable-coefficient HNLS equation,and then analyze the influence of the variable coefficients on the soliton properties;(2) transform the generalized variable-coefficient HNLS equation into the constant-coefficient one,obtain the dark solitonic solutions of the generalized variable-coefficient HNLS equation by virtue of the bilinear method and the transformation,and study the stability and propagation proporties of the dark solitons with the help of symbolic and numerical computation;(3) derive the explicit one- and two-solitonic solutions of the inhomogeneous N coupled NLS equation and investigate the properties of soliton propagation;(4) derive the multi-kink solitonic solutions of the variable-coefficient SG equation and analyze the influence of the parameters.
(Ⅲ) Investigate the int(?)grable properties such as the B(a|¨)cklund transformation and an infinite number of conservation laws for the variable-coefficient nonlinear models using symbolic computation.In this paper,two different methods are employed to derive the B(a|¨)cklund transformations of the variable-coefficient nonlinear models.Using their bilinearized forms to construct different formal B(a|¨)cklund transformations, the bilinear B(a|¨)cklund transformations of the generalized variable-coefficient HNLS equation,the variable-coefficient SG equation and the inhomogeneous N coupled NLS equation are obtained.Moreover,by the F-Riccati form of the AKNS system for the variable-coefficient nonlinear models,the B(a|¨)cklund transformations inΓfunction form for the variable-coefficient N coupled NLS equation and the variable-coefficient SG equation are derived.Furthermore,an infinite number of conservation laws for the generalized variable-coefficient HNLS equation,the variable -coefficient N coupled NLS equation and the variable-coefficient SG equation are gained using the correspondingΓ-Riccati equations.
(Ⅳ) Generalize the Wronskian technique and apply the generalization to obtain the multi-solitonic solutions in Wronski determinant form for variable-coefficient nonlinear models.One of the difficulties to construct the Wronski determinant solution of nonlinear models is finding the properties which the Wronski determinant elements satisfy.Starting from three different viewpoints,the Wronski determinant form multi-solitonic solutions for the variable-coefficient nonlinear models are constructed, and those solutions are analyzed with the help of symbolic computation: (1) construct and verify the Wronski determinant solution for the variable -coefficint(3+1)-dimensional KP equation based on the balancing idea; (2) construct the Wronski determinant solution for the variable-coefficient SG equation using the B(a|¨)cklund transformation for the properties which the determinant elements satisfy;(3) using the AKNS system,construct multi-solitonic solutions in double Wronski determinant form for the generalized variable-coefficient HNLS equation and analyze the properties of solutions.
(Ⅴ) Generalize the Pfaffian method and apply to the variable-coefficient nonlinear models.The Pfaffian can be regarded as the generalization of determinant.Using the properties of Pfaffian,the Gramm determinant solution for the variable-coefficient(3+1)-dimensional KP equation are constructed and verified.The Pfaffian procedure which can be used to construct new coupled soliton equations is generalized and applied to the variable-coefficient KP equation.Using the generalization of the Pfaffian procedure,the variable-coefficient coupled KP equations are generated, and then the Wronski- and Gramm-Pfaffian solutions for the coupled variable-coefficient KP equations are derived and verified by the Pfaffian properties.
In this paper,the Painlev(?) analysis,bilinear method,AKNS method, Wronskian technique and Pfaffian method are generalized and applied to investigate the solitonic solutions and integrable properties for the variable-coefficient nonlinear models in several fields such as the optical fiber and fluid dynamics using symbolic computation.By virtue of the graphical function of the symbolic computation software,the solitonic solutions of those variable-coefficient nonlinear models are illustrated and their propagation properties are analyzed.It can be expected that these generalized methods in this paper can be used to study a wide class of variable-coefficient nonlinear models in other fields and the analytical research results in this paper for the variable-coefficient nonlinear models in several fields such as the optical fiber can be helpful for further theoretical and experimental researches.
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.
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