非线性微分方程的若干解析解方法与可积系统
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摘要
基于计算机数学机械化思想和‘'AC=BD"统·理论模式，借助于现有的理论及相应的符号计算软件，本论文主要研究了孤子方程的AC=BD模式与卦理论，非线性微分方程的(Binary)Darboux与Backlund变换、微分变换及Hamiltonian可积簇，非线性微分方程的非局部分析，非线性微分方程、超对称和超离散方程的有限高亏格解与可积系统问题等.
第一章介绍计算机数学及计算机代数，孤子理论，非线性方程、超对称与超离散方程的机械化求解方法与可积系统问题等在国内外的历史发展概况，并介绍本论文的选题和主要工作.
第二章基于AC=BD模式及其C-D可积系统与C-D对,我们做了两方面的工作.在代数几何解中：推出了Dubrovin型方程,Its-Matveev公式及Super-Its-Matveev公式；在Sato理论中：分别给了Lax方程与Sato方程、Lax方程与反散射框架、Lax方程与Zarharov-Shabat方程、Sato方程与Hirota双线性方程等之间的关系,并揭示了这些方程解的统一模式可由Tau函数表示.另一方面,为了揭示可积系统的一般性结构,我们首次系统地提出了“卦理论”,包括“卦结构”和“卦恒等式”,并分别给出了一些内分解-和外分解-卦恒等式：Wronskian、Grammian、Pfaffian、Young图的Schur函数和特征多项式、Clifford-和Heisenberg-代数的Fock表示空间等,并首次阐明Clifford-(Heisenberg-)代数的Fock表示空间均为卦(同构卦)空间.最后给出了构造Tau函数与Theta函数之间关系的新方法,间接地建立了卦结构与代数几何解之间的联系.
第三章基于Lax谱理论、Painleve奇异流形理论,分别给出了一类微分方程的三类N-重Darboux变换,Auto-Backlund不(?)Binary Darboux变换,及其相应的周期波解和Grammian解.利用离散Lax谱问题,通过选取合适的谱Vn(m),给出了一类新的Hamiltonian Lattice簇的一些经典Lattice约化、multi-Hamiltonian(?)结构在对合意义下的可积性质、离散Darboux变换及其解析解.基于¨Sato理论框架并借助于限定的mKP方程,提出了一类自溶源rnKP (mKPESCSs)方程及其Lax谱问题；利用共轭Lax对,进而研究其向前、向后和N重Binary Darboux变换,其中Binary Darboux变换提供了两个不同次数的mKPESCSs之间的一个非自治Backlund变换；借助于这些变换可以得到mKPESCSs的一些新典型解如孤立子解、有理解、呼吸子解和指数解等.通过研究微分变换与Pade逼近理论,获得了著名的浅水波Camassa-Holm方程波峰连续与非连续解析近似解；与解析解比较,研究了其计算的有效性和高精度.
第四章借助于守恒律乘子，获得了一类微分方程的非局部分析其中包括非局部相关PDE系统、树形结构、非局部对称与守恒律等.借助于非局部对称，进而研究了原PDE系统的非局部线性化，并提出了广义不变解的一套新方法.对某一类PDE系统，给出了其非局部对称与Nonclassical方法在求解方而的关系
与George W. Bluman(?)教授等的合作中给出了著名的非线性Kompaneets(NLK)(?)方程的非局部分析；与近期(?)Ibragimov教授的工作相比，利用非局部分析中已得到的结论获得了NLK方程的更广义类型的解；这些新解不能由NLK方程局部对称的不变解所得到，并打破了NLK方程自1956年以来只有唯一一类局部平凡解析解的状况.特别地，得到了以前未知的五类涉及两参数的精确时间独立解析.有趣的是，这些解都可以用初等函数所表示，并且其中两类在有限时间内表现出爆破行为，另外三类则表现出静止行为.最后证明了所有的非平凡稳态解都具有不稳定性，并且他们相对于Dubinov教授所给出的隐式解是新的.
第五章基于超空间,利用Hirota双线性和Riemann theta(?)函数的性质，分别研究了一类非线性微分方程和超对称方程的有限高亏格(?)的Riemann theta函数周期波解及其极限特性分析，并将其分别应用到了CDGSK方程、(2+1)-维DBS方程和超对称KdV-Burgers方程等.借助于theta函数的有理恒等式提出了求解一类离散方程的N-theta周期波解的方法：并将这类结论推广到了离散方程和(?)heta(?)函数的超离散化空间上，进而分别获得了相同亏格(?)的Ud-Riemann theta(?)函数周期波解.做为这种方法的应用，分别研究了离散修改的Korteweg-de Vires(mKdV)方程和广义的Toda lattice方程等
第六章借助于多维的Bell与super Bell多项式,分别研究了一类非线性微分方程和超对称方程的可积性分析，同时给出了可积判定条件，使此类方程(组)成为一类可积系统，并将其分别应用到了一类广义变系数的KP方程、5-阶KdV方程(?)IsKdV-Burgers方程等，获得了一些新的可积性结论.借助于超离散的‘'max-plus"代数理论及其Lax(?)(?)性系统的相容条件，提出了一般超离散方程的Lax可积定理和可解性定理：通过研究有限高亏格(?)的Riemann theta (?)函数的超离散化，进而获得了一类超离散化方程相同亏格(?)的Ud-Riemann theta函数解.最后将这一般的超离散化及其可积性理论分别应用到了离散的Lattice Krichever-Novikov方程、离散的mKdV方程和离散的Painleve方程等.
Based on the idea of computer mathematical mechanization and the unified theoretical of the model AC=BD, and by virtue of existing theories and some symbolic computation soft-wares, in this dissertation, we mainly focus on some topics from the view points of algebra and geometry, including the model AC=BD and "Trigram" structures of the soliton theory, the (Binary) Darboux and Backlund transformations、differential transform and nonlocal analysis of nonlinear differential equations and Hamiltonian integrable hierarchy, the finite genus so-lutions and integrable systems of nonlinear differential equations、supersymmetric equations and ultra-discrete equations, etc.
In Chapter1, we introduce the history and development of computer mathematics and computer algebra, soliton theory, the mechanical algorithm and integrable system of nonlin-ear differential equation、supersymmetric equation and ultra-discrete equation in summary at home and abroad. At last, we present an outline of this dissertation.
In Chapter2, based on the model AC=BD, the integrable system and the pair of C-D, we do some work in algebraic-geometry solution and Sato theory. Tn algebraic-geometry so-lution:we systematically derive the Its-Matveev formula, super-Its-Matveev formula. In Sato theory:we present the relationship between Lax equation and Sato equation, Lax equation and Zakharov-Shabat equation, Lax equation and inverse scattering scheme, Sato equation and Hirota's bilinear equation, respectively, which can be used to construct a unified model of solv-ing soliton equation by using Tau function. To construct a unified and fundamental structure of soliton equations, it is the first time to introduce our "Trigram theory" including "Trigram structures" and "Trigram identities", which present some exterior-and interior-decomposition Trigram identities, respectively, to reveal some integrable systems generated by Wronskian, Grammian, Pfaffian, Schur functions and characteristic polynomials of Young diagram, Fock representation of Clifford-and Heisenberg-algebra, etc, from which the Fock spaces of Clifford (Heisenberg) algebra is a (isomorphism) Trigram space. Finally, we present a new approach to construct the relationship between Tau function and Theta function, which indirectly present the relationship between Trigram structure and algebraic-geometry solution.
In Chapter3, based on the theories of Lax spectral problem and painleve singular-manifold method, we present three new kinds of N-fold Darboux transformations, auto-Backlund and bi-nary Darboux transformations, the corresponding periodic wave solutions and grammian solu-tions for a kind of differential equations. Using the discrete Lax spectral problem and choosing the appropriate spectral Vn(m), we present a new kind of Hamiltonian Lattice hierarchy, and fur-ther investigate some classical Lattice reductions, integrable in involutory Lax's sense of the multi-Hamiltonian structure, discrete Darboux transformation and its analytic solutions. Based on the framework of Sato's theory, a mKP equation with self-consistent sources (mKPFSC-Ss) and its Lax pairs are structured by virtue of the constrained mKP equation. Using the conjugate Lax pairs, we construct the forward, the backward and the N-fold binary Darboux transformation for the mKPESCSs which offers a non-auto-Backlund transformation between two mKPFSCSs with different degrees of sources. With the help of these transformations, some new classical solutions for the mKPFSCSs such as soliton solutions, rational solutions, breather type solutions and exponential solutions are found. Through research the theories of the dif-ferential transformation and the Pade approach technique method, we investigate the solutions with and without continuity at crest of Camassa-Holm equation. Compared to exact solutions, we also research the computational efficiency、high accuracy of the method.
In Chapter4, by virtue of conservation law multiplier, we present the nonlocal analysis for a kind of differential equations, which include nonlocally related PDF systems, tree structure, nonlocal symmetry and conservation law etc. By virtue of the nonlocal symmetry, we further investigate the nonlocal linearization and propose a new generalized algorithm of solving in-variant solution. For special kind of PDEs, the relationship between nonlocal symmetry and nonclassical method on solving solutions are presented.
We investigate the nonlocal analysis of famous nonlinear Koinpaneets equation(NLK) joint work with Professor George. W. Bluman etc. Using the result obtained in nonlocal analysis, we obtain a wider classes of previously unknown solutions of the NLK equation beyond those ob-tained by Professor Ibragimov solution. These new solutions do not arise as invariant solutions of the NLK equation with respect to its local symmetries, which are breaking the status of ex-isting only one class of trivial local solution for NLK equation since1956. In particular, for live classes of initial conditions, each involving two parameters, previously unknown explicit time-dependent solutions are obtained. Interestingly, each of these solutions is expressed in terms of elementary functions. The two classes exhibit blow up behavior in finite time, and the other three classes exhibit quiescent behavior. As a consequence, it is shown that the corresponding nonlrivial stationary solutions are unstable. The nontrivial stationary solutions are also beyond those obtained by Professor Dubinov.
In Chapter5, based on the theories of superspace, and by means of the properties be-tween Hirota bilinear and Riemann theta function, we present some Riemann theta function periodic wave solutions with the finite genus (?) and the analysis of limiting characteristics for a kind of nonlinear differential equations and supersymmetry equations, respectively. Us-ing these methods, we investigate Caudrey-Dodd-Gibbon-Sawada-Kotera(CDGSK) equation,(2+1)-dimensional breaking soliton(DBS) equation and supersymmetric Korteweg-de Vries-Burgers(sKdV-Burgers) equation, etc. By virtue of the rational identities of theta functions, we present some N-theta function periodic wave solutions of a kind of discrete soliton equations. This method can be extended to the ultra-discrete space of the discrete equations and the theta functions, based on which, we can further obtain Ud-Riemann theta function periodic wave solutions with the same genus (?) for the corresponding udtra-discrete equations. As its applica-tion, we investigate a discrete modified Korteweg-de Vires(mKdV) equation and a generalized Toda lattice equation, etc.
In Chapter6, by virtue of multi-dimensional Bell polynomials and super Bell polynomials, we present the integrability analysis of a kind of nonlinear differential equations and supersym-metry equations, respectively. While we also present the judgment conditions of integrability to these equation(s) to make be a kind of integrable system(s). Using these methods, we inves-tigate a kind of generalized variable-coefficient Kadomtsev-Petviashvili equation, fifth-order Korteweg-de Vries equation and sKdV-Burgers equation, etc, and further obtain some new re-sults of integrability. Moreover, using the theory of "max-plus" algebra and the compatibility condition of Lax pair system, we propose the ultra-discrete equation and its Lax integrability theorem and solvability theorem. Through researching the ultra-discrete process of Riemann theta function with finite genus (?) we further present the same genus of Ud-Riemann theta function solutions of a kind of ud-discrete equations. Finally, the generalized theories of Ultra-discretization and its integrability are applied to discrete Lattice Krichever-Novikov equation, discrete mKdV equation and discrete Painleve equation, etc.
第一章介绍计算机数学及计算机代数，孤子理论，非线性方程、超对称与超离散方程的机械化求解方法与可积系统问题等在国内外的历史发展概况，并介绍本论文的选题和主要工作.
第二章基于AC=BD模式及其C-D可积系统与C-D对,我们做了两方面的工作.在代数几何解中：推出了Dubrovin型方程,Its-Matveev公式及Super-Its-Matveev公式；在Sato理论中：分别给了Lax方程与Sato方程、Lax方程与反散射框架、Lax方程与Zarharov-Shabat方程、Sato方程与Hirota双线性方程等之间的关系,并揭示了这些方程解的统一模式可由Tau函数表示.另一方面,为了揭示可积系统的一般性结构,我们首次系统地提出了“卦理论”,包括“卦结构”和“卦恒等式”,并分别给出了一些内分解-和外分解-卦恒等式：Wronskian、Grammian、Pfaffian、Young图的Schur函数和特征多项式、Clifford-和Heisenberg-代数的Fock表示空间等,并首次阐明Clifford-(Heisenberg-)代数的Fock表示空间均为卦(同构卦)空间.最后给出了构造Tau函数与Theta函数之间关系的新方法,间接地建立了卦结构与代数几何解之间的联系.
第三章基于Lax谱理论、Painleve奇异流形理论,分别给出了一类微分方程的三类N-重Darboux变换,Auto-Backlund不(?)Binary Darboux变换,及其相应的周期波解和Grammian解.利用离散Lax谱问题,通过选取合适的谱Vn(m),给出了一类新的Hamiltonian Lattice簇的一些经典Lattice约化、multi-Hamiltonian(?)结构在对合意义下的可积性质、离散Darboux变换及其解析解.基于¨Sato理论框架并借助于限定的mKP方程,提出了一类自溶源rnKP (mKPESCSs)方程及其Lax谱问题；利用共轭Lax对,进而研究其向前、向后和N重Binary Darboux变换,其中Binary Darboux变换提供了两个不同次数的mKPESCSs之间的一个非自治Backlund变换；借助于这些变换可以得到mKPESCSs的一些新典型解如孤立子解、有理解、呼吸子解和指数解等.通过研究微分变换与Pade逼近理论,获得了著名的浅水波Camassa-Holm方程波峰连续与非连续解析近似解；与解析解比较,研究了其计算的有效性和高精度.
第四章借助于守恒律乘子，获得了一类微分方程的非局部分析其中包括非局部相关PDE系统、树形结构、非局部对称与守恒律等.借助于非局部对称，进而研究了原PDE系统的非局部线性化，并提出了广义不变解的一套新方法.对某一类PDE系统，给出了其非局部对称与Nonclassical方法在求解方而的关系
与George W. Bluman(?)教授等的合作中给出了著名的非线性Kompaneets(NLK)(?)方程的非局部分析；与近期(?)Ibragimov教授的工作相比，利用非局部分析中已得到的结论获得了NLK方程的更广义类型的解；这些新解不能由NLK方程局部对称的不变解所得到，并打破了NLK方程自1956年以来只有唯一一类局部平凡解析解的状况.特别地，得到了以前未知的五类涉及两参数的精确时间独立解析.有趣的是，这些解都可以用初等函数所表示，并且其中两类在有限时间内表现出爆破行为，另外三类则表现出静止行为.最后证明了所有的非平凡稳态解都具有不稳定性，并且他们相对于Dubinov教授所给出的隐式解是新的.
第五章基于超空间,利用Hirota双线性和Riemann theta(?)函数的性质，分别研究了一类非线性微分方程和超对称方程的有限高亏格(?)的Riemann theta函数周期波解及其极限特性分析，并将其分别应用到了CDGSK方程、(2+1)-维DBS方程和超对称KdV-Burgers方程等.借助于theta函数的有理恒等式提出了求解一类离散方程的N-theta周期波解的方法：并将这类结论推广到了离散方程和(?)heta(?)函数的超离散化空间上，进而分别获得了相同亏格(?)的Ud-Riemann theta(?)函数周期波解.做为这种方法的应用，分别研究了离散修改的Korteweg-de Vires(mKdV)方程和广义的Toda lattice方程等
第六章借助于多维的Bell与super Bell多项式,分别研究了一类非线性微分方程和超对称方程的可积性分析，同时给出了可积判定条件，使此类方程(组)成为一类可积系统，并将其分别应用到了一类广义变系数的KP方程、5-阶KdV方程(?)IsKdV-Burgers方程等，获得了一些新的可积性结论.借助于超离散的‘'max-plus"代数理论及其Lax(?)(?)性系统的相容条件，提出了一般超离散方程的Lax可积定理和可解性定理：通过研究有限高亏格(?)的Riemann theta (?)函数的超离散化，进而获得了一类超离散化方程相同亏格(?)的Ud-Riemann theta函数解.最后将这一般的超离散化及其可积性理论分别应用到了离散的Lattice Krichever-Novikov方程、离散的mKdV方程和离散的Painleve方程等.
Based on the idea of computer mathematical mechanization and the unified theoretical of the model AC=BD, and by virtue of existing theories and some symbolic computation soft-wares, in this dissertation, we mainly focus on some topics from the view points of algebra and geometry, including the model AC=BD and "Trigram" structures of the soliton theory, the (Binary) Darboux and Backlund transformations、differential transform and nonlocal analysis of nonlinear differential equations and Hamiltonian integrable hierarchy, the finite genus so-lutions and integrable systems of nonlinear differential equations、supersymmetric equations and ultra-discrete equations, etc.
In Chapter1, we introduce the history and development of computer mathematics and computer algebra, soliton theory, the mechanical algorithm and integrable system of nonlin-ear differential equation、supersymmetric equation and ultra-discrete equation in summary at home and abroad. At last, we present an outline of this dissertation.
In Chapter2, based on the model AC=BD, the integrable system and the pair of C-D, we do some work in algebraic-geometry solution and Sato theory. Tn algebraic-geometry so-lution:we systematically derive the Its-Matveev formula, super-Its-Matveev formula. In Sato theory:we present the relationship between Lax equation and Sato equation, Lax equation and Zakharov-Shabat equation, Lax equation and inverse scattering scheme, Sato equation and Hirota's bilinear equation, respectively, which can be used to construct a unified model of solv-ing soliton equation by using Tau function. To construct a unified and fundamental structure of soliton equations, it is the first time to introduce our "Trigram theory" including "Trigram structures" and "Trigram identities", which present some exterior-and interior-decomposition Trigram identities, respectively, to reveal some integrable systems generated by Wronskian, Grammian, Pfaffian, Schur functions and characteristic polynomials of Young diagram, Fock representation of Clifford-and Heisenberg-algebra, etc, from which the Fock spaces of Clifford (Heisenberg) algebra is a (isomorphism) Trigram space. Finally, we present a new approach to construct the relationship between Tau function and Theta function, which indirectly present the relationship between Trigram structure and algebraic-geometry solution.
In Chapter3, based on the theories of Lax spectral problem and painleve singular-manifold method, we present three new kinds of N-fold Darboux transformations, auto-Backlund and bi-nary Darboux transformations, the corresponding periodic wave solutions and grammian solu-tions for a kind of differential equations. Using the discrete Lax spectral problem and choosing the appropriate spectral Vn(m), we present a new kind of Hamiltonian Lattice hierarchy, and fur-ther investigate some classical Lattice reductions, integrable in involutory Lax's sense of the multi-Hamiltonian structure, discrete Darboux transformation and its analytic solutions. Based on the framework of Sato's theory, a mKP equation with self-consistent sources (mKPFSC-Ss) and its Lax pairs are structured by virtue of the constrained mKP equation. Using the conjugate Lax pairs, we construct the forward, the backward and the N-fold binary Darboux transformation for the mKPESCSs which offers a non-auto-Backlund transformation between two mKPFSCSs with different degrees of sources. With the help of these transformations, some new classical solutions for the mKPFSCSs such as soliton solutions, rational solutions, breather type solutions and exponential solutions are found. Through research the theories of the dif-ferential transformation and the Pade approach technique method, we investigate the solutions with and without continuity at crest of Camassa-Holm equation. Compared to exact solutions, we also research the computational efficiency、high accuracy of the method.
In Chapter4, by virtue of conservation law multiplier, we present the nonlocal analysis for a kind of differential equations, which include nonlocally related PDF systems, tree structure, nonlocal symmetry and conservation law etc. By virtue of the nonlocal symmetry, we further investigate the nonlocal linearization and propose a new generalized algorithm of solving in-variant solution. For special kind of PDEs, the relationship between nonlocal symmetry and nonclassical method on solving solutions are presented.
We investigate the nonlocal analysis of famous nonlinear Koinpaneets equation(NLK) joint work with Professor George. W. Bluman etc. Using the result obtained in nonlocal analysis, we obtain a wider classes of previously unknown solutions of the NLK equation beyond those ob-tained by Professor Ibragimov solution. These new solutions do not arise as invariant solutions of the NLK equation with respect to its local symmetries, which are breaking the status of ex-isting only one class of trivial local solution for NLK equation since1956. In particular, for live classes of initial conditions, each involving two parameters, previously unknown explicit time-dependent solutions are obtained. Interestingly, each of these solutions is expressed in terms of elementary functions. The two classes exhibit blow up behavior in finite time, and the other three classes exhibit quiescent behavior. As a consequence, it is shown that the corresponding nonlrivial stationary solutions are unstable. The nontrivial stationary solutions are also beyond those obtained by Professor Dubinov.
In Chapter5, based on the theories of superspace, and by means of the properties be-tween Hirota bilinear and Riemann theta function, we present some Riemann theta function periodic wave solutions with the finite genus (?) and the analysis of limiting characteristics for a kind of nonlinear differential equations and supersymmetry equations, respectively. Us-ing these methods, we investigate Caudrey-Dodd-Gibbon-Sawada-Kotera(CDGSK) equation,(2+1)-dimensional breaking soliton(DBS) equation and supersymmetric Korteweg-de Vries-Burgers(sKdV-Burgers) equation, etc. By virtue of the rational identities of theta functions, we present some N-theta function periodic wave solutions of a kind of discrete soliton equations. This method can be extended to the ultra-discrete space of the discrete equations and the theta functions, based on which, we can further obtain Ud-Riemann theta function periodic wave solutions with the same genus (?) for the corresponding udtra-discrete equations. As its applica-tion, we investigate a discrete modified Korteweg-de Vires(mKdV) equation and a generalized Toda lattice equation, etc.
In Chapter6, by virtue of multi-dimensional Bell polynomials and super Bell polynomials, we present the integrability analysis of a kind of nonlinear differential equations and supersym-metry equations, respectively. While we also present the judgment conditions of integrability to these equation(s) to make be a kind of integrable system(s). Using these methods, we inves-tigate a kind of generalized variable-coefficient Kadomtsev-Petviashvili equation, fifth-order Korteweg-de Vries equation and sKdV-Burgers equation, etc, and further obtain some new re-sults of integrability. Moreover, using the theory of "max-plus" algebra and the compatibility condition of Lax pair system, we propose the ultra-discrete equation and its Lax integrability theorem and solvability theorem. Through researching the ultra-discrete process of Riemann theta function with finite genus (?) we further present the same genus of Ud-Riemann theta function solutions of a kind of ud-discrete equations. Finally, the generalized theories of Ultra-discretization and its integrability are applied to discrete Lattice Krichever-Novikov equation, discrete mKdV equation and discrete Painleve equation, etc.
引文
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