可积系统的多孤立子解及其符号计算研究
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摘要
本文以符号计算为工具利用N重Darboux阵方法、可对角化的Darboux阵方法、Hirota直接方法和Wronskian行列式技巧研究了可积系统的多孤立子解以及解的性质.另外利用李代数的半直和思想和变分恒等式构造了耦合KdV方程族的可积耦合系统及双Hamilton结构.
第一章是与本文相关的研究背景,简要综述了孤立子与可积系统理论的发展进程.针对性地介绍了近年来国内外在孤立子与可积系统方面的研究成果和发展状况.
第二章中,利用N重Darboux阵方法,首次构造了一类等谱问题统一形式的Darboux变换,应用所得到的Darboux变换于联系广义Broer-Kaup-Kupershmidt与Boussinesq-Burgers(BKK-BB)谱问题的孤立子方程族中的不同方程,获得了它们的形式各异的新N-孤立子解,其中包括了一些多峰状的双向孤立子解.将N重Darboux阵方法与约化、分解技巧相结合,获得了一系列非线性演化方程的N孤立子解和N-complexiton解.我们利用AKNS谱问题Darboux变换的一种约化,求得复mKdV方程的多孤立子解;利用分解技巧,获得一个(3+1)维非线性演化方程的多种解和(2+1)维KP方程的新多孤立子解.
第三章扩展可对角化的Darboux阵方法并将它应用到一个新谱问题、Boiti-Tu谱问题和一种广义Kaup-Newell谱问题上,成功构造出这些谱问题的Darboux变换,获得了一个无色散可积耦合方程的N-孤立子解,一个广义耦合mKdV方程的一系列孤立波解和一个广义导数非线性Schr(o|¨)dinger方程的一系列周期波解.基于可对角化的Darboux阵方法,我们给出了构造Darboux变换的一种算法.并在计算机代数系统Maple12上实现了该算法.
第四章推广了Hirota直接方法,将Hirota直接方法求解过程中的实参数推广到共轭复数范围,给出了单、双complexiton解和N-complexiton解的一般表达式.通过对参数的适当选取,N-complexiton解可退化到标准Hirota直接方法的N-孤立子解.我们给出了一系列非线性演化方程的非奇异的新多complexiton解.
第五章介绍了求解非线性演化方程的Wronskian行列式技巧.在本章中,我们给出一个(3+1)维非线性演化方程的广义Wronskian解公式,其中包括了positon解、negaton解、soliton(孤立子)解、complexiton解以及相互作用解.我们还计算出该(3+1)维非线性演化方程的双Wronskian解公式,利用它给出了该方程的有理解.
第六章中,我们利用李代数的半直和思想构造了耦合KdV方程族的可积耦合系统,基于变分恒等式,进一步得到一个可积耦合系统的双Hamilton结构.另外,首次将N重Darboux阵方法成功应用于可积耦合系统中,构造出可积耦合系统的Darboux变换.
In this dissertation,with the help of symbolic computation,the multi-soliton solutions of integrable systems are obtained by the N fold Darboux matrix method,the diagonal Darboux matrix method(DDMM),the Hirota's direct method and the Wronskian technique.The properties of these solutions are also investigated.In addition,the integrable coupling system of the coupled KdV hierarchy and its bi-Hamiltonian structures are constructed by semi-direct sums of Lie algebra and the variational identity.
Chapter 1 is the research background related to the dissertation.We briefly outline the development of the theory of solitons and integrable systems.Subsequently,we summarize the recent development and achievement in the theory of solitons and integrable systems at home and abroad.
In chapter 2,we first provide a unified explicit form of N fold Darboux transformation for a class of isospectral problem via the N fold Darboux matrix method.Some new bidirectional multipeak N-soliton solutions of some soliton equations associated with the generalized Broer-Kaup-Kupershmidt and Boussinesq-Burgers(BKK-BB)spectral problem are presented by the Darboux transformation.In addition,with the help of the reduction technique and the decomposition technique, we can get a series of N-soliton and N-complexiton solutions of some nonlinear evolution equations via the N fold Darboux transformations.As an applications,some new multi-soliton and multi-complexiton solutions for the complex mKdV equation,a(3+1)-dimensional nonlinear evolution equation and the(2+1)-dimensional KP equation are explicitly given.
In chapter 3,the Darboux transformations for a new spectral problem,the Boiti-Tu spectral problem and a generalized Kaup-Newell spectral problem are constructed by the diagonal Darboux matrix method.As an applications,we present N-soliton solutions of a coupled integrable dispersionless equation,a series of solitary wave solutions of the generalized coupled mKdV equation and some new periodic solutions of the generalized derivative nonlinear Schrodinger equation. Based on the diagonal Darboux matrix method,a new efficient algorithm for constructing the Darboux transformation is presented.The algorithm has been implemented with Maple 12.
In chapter 4,extending the application of the Hirota's direct method in soliton equations, multi-complexiton solution formulae of bilinear soliton equations are derived by changing the real parameters into conjugated complex parameters in pairs.When the parameters are suitably chosen, we can obtain the N-soliton solutions from the N-complexiton solution formulae.Further,we derive a series of non-singular multi-complexiton solutions of some nonlinear evolution equations.
In chapter 5,we present a generalized Wronskian solution formula of a(3+l)-dimensional nonlinear evolution equation,in which positon solutions,negaton solutions,soliton solutions, complexiton solutions and interaction solutions are included.And we compute a double Wronskian solution formula of the(3+1)-dimensional nonlinear evolution equation,by which some rational solutions are obtained.
In chapter 6,we derive an integrable coupling system of the coupled KdV hierarchy by semidirect sums of Lie algebra.Based on the variational identity,we further construct bi-Hamiltonian structures of the integrable coupling system.The N fold Darboux matrix method is applied to the integrable coupling system associated with the coupled KdV hierarchy for the first time.Then N fold Darboux transformation of the integrable coupling system is constructed successfully.
第一章是与本文相关的研究背景,简要综述了孤立子与可积系统理论的发展进程.针对性地介绍了近年来国内外在孤立子与可积系统方面的研究成果和发展状况.
第二章中,利用N重Darboux阵方法,首次构造了一类等谱问题统一形式的Darboux变换,应用所得到的Darboux变换于联系广义Broer-Kaup-Kupershmidt与Boussinesq-Burgers(BKK-BB)谱问题的孤立子方程族中的不同方程,获得了它们的形式各异的新N-孤立子解,其中包括了一些多峰状的双向孤立子解.将N重Darboux阵方法与约化、分解技巧相结合,获得了一系列非线性演化方程的N孤立子解和N-complexiton解.我们利用AKNS谱问题Darboux变换的一种约化,求得复mKdV方程的多孤立子解;利用分解技巧,获得一个(3+1)维非线性演化方程的多种解和(2+1)维KP方程的新多孤立子解.
第三章扩展可对角化的Darboux阵方法并将它应用到一个新谱问题、Boiti-Tu谱问题和一种广义Kaup-Newell谱问题上,成功构造出这些谱问题的Darboux变换,获得了一个无色散可积耦合方程的N-孤立子解,一个广义耦合mKdV方程的一系列孤立波解和一个广义导数非线性Schr(o|¨)dinger方程的一系列周期波解.基于可对角化的Darboux阵方法,我们给出了构造Darboux变换的一种算法.并在计算机代数系统Maple12上实现了该算法.
第四章推广了Hirota直接方法,将Hirota直接方法求解过程中的实参数推广到共轭复数范围,给出了单、双complexiton解和N-complexiton解的一般表达式.通过对参数的适当选取,N-complexiton解可退化到标准Hirota直接方法的N-孤立子解.我们给出了一系列非线性演化方程的非奇异的新多complexiton解.
第五章介绍了求解非线性演化方程的Wronskian行列式技巧.在本章中,我们给出一个(3+1)维非线性演化方程的广义Wronskian解公式,其中包括了positon解、negaton解、soliton(孤立子)解、complexiton解以及相互作用解.我们还计算出该(3+1)维非线性演化方程的双Wronskian解公式,利用它给出了该方程的有理解.
第六章中,我们利用李代数的半直和思想构造了耦合KdV方程族的可积耦合系统,基于变分恒等式,进一步得到一个可积耦合系统的双Hamilton结构.另外,首次将N重Darboux阵方法成功应用于可积耦合系统中,构造出可积耦合系统的Darboux变换.
In this dissertation,with the help of symbolic computation,the multi-soliton solutions of integrable systems are obtained by the N fold Darboux matrix method,the diagonal Darboux matrix method(DDMM),the Hirota's direct method and the Wronskian technique.The properties of these solutions are also investigated.In addition,the integrable coupling system of the coupled KdV hierarchy and its bi-Hamiltonian structures are constructed by semi-direct sums of Lie algebra and the variational identity.
Chapter 1 is the research background related to the dissertation.We briefly outline the development of the theory of solitons and integrable systems.Subsequently,we summarize the recent development and achievement in the theory of solitons and integrable systems at home and abroad.
In chapter 2,we first provide a unified explicit form of N fold Darboux transformation for a class of isospectral problem via the N fold Darboux matrix method.Some new bidirectional multipeak N-soliton solutions of some soliton equations associated with the generalized Broer-Kaup-Kupershmidt and Boussinesq-Burgers(BKK-BB)spectral problem are presented by the Darboux transformation.In addition,with the help of the reduction technique and the decomposition technique, we can get a series of N-soliton and N-complexiton solutions of some nonlinear evolution equations via the N fold Darboux transformations.As an applications,some new multi-soliton and multi-complexiton solutions for the complex mKdV equation,a(3+1)-dimensional nonlinear evolution equation and the(2+1)-dimensional KP equation are explicitly given.
In chapter 3,the Darboux transformations for a new spectral problem,the Boiti-Tu spectral problem and a generalized Kaup-Newell spectral problem are constructed by the diagonal Darboux matrix method.As an applications,we present N-soliton solutions of a coupled integrable dispersionless equation,a series of solitary wave solutions of the generalized coupled mKdV equation and some new periodic solutions of the generalized derivative nonlinear Schrodinger equation. Based on the diagonal Darboux matrix method,a new efficient algorithm for constructing the Darboux transformation is presented.The algorithm has been implemented with Maple 12.
In chapter 4,extending the application of the Hirota's direct method in soliton equations, multi-complexiton solution formulae of bilinear soliton equations are derived by changing the real parameters into conjugated complex parameters in pairs.When the parameters are suitably chosen, we can obtain the N-soliton solutions from the N-complexiton solution formulae.Further,we derive a series of non-singular multi-complexiton solutions of some nonlinear evolution equations.
In chapter 5,we present a generalized Wronskian solution formula of a(3+l)-dimensional nonlinear evolution equation,in which positon solutions,negaton solutions,soliton solutions, complexiton solutions and interaction solutions are included.And we compute a double Wronskian solution formula of the(3+1)-dimensional nonlinear evolution equation,by which some rational solutions are obtained.
In chapter 6,we derive an integrable coupling system of the coupled KdV hierarchy by semidirect sums of Lie algebra.Based on the variational identity,we further construct bi-Hamiltonian structures of the integrable coupling system.The N fold Darboux matrix method is applied to the integrable coupling system associated with the coupled KdV hierarchy for the first time.Then N fold Darboux transformation of the integrable coupling system is constructed successfully.
引文
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