非线性波与可积系统
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摘要
本文以构造性的变换及符号计算为工具，来研究非线性波和可积系统中的一些问题：精确解(如孤子解、周期解、有理解、dromion解及compacton解等)、Panileve可积性、Backlund变换、Darboux变换、对称(相似约化)、条件对称、Lax可积族、Liouville可积的N-Hamilton结构、约束流、对合系统、Lax表示、r-矩阵、变量分离及可积的耦合系统．
第二章和第三章考虑非线性偏微分方程的精确解的构造：首先给出了C-D对和C-D可积系统的基本理论，然后在第三章中具体研究了它们的应用：(1)基于两种Riccati方程，提出了两种新的求解非线性微分方程更多解的方法，利用其中的一种方法，得到了WBK方程的12组精确解；(2)对齐次子衡法进行改进，以致于获得了(2+1)-维Broer-Kaup方程的很多新解；(3)基于带有外力项的广义KdV方程的Riccati形式的非等谱Lax对，提出了该方程的一个新的Darboux变换，利用该变换，得到了新的类孤波解和有理解；(4)通过构造了带有外力项的变系数KdV方程的Darboux变换及叠加原理，获得(2+1)-维广义KP方程的新的类单孤波解、双类孤波解和有理解。
第四章讨论了非线性微分方程的Painleve可积性和Backlund变换。利用WTC方法证明了两类(2+1)-维广义Burgers方程是Painleve可积的，并经截断展开原理获得了它们的Backlund变换，其中Cole-Hopf变换是其特例。
第五章考虑了非线性偏微分方程的对称和精确解：(1)将C-K直接法推广到具有全色散项的新的Estevez-Mansfield-Clarkson(EMC)E(m,n)方程中，得到了五种新的对称，进而得到了E(1,n)方程的新的孤波解以及E(m,m-1)方程的新的类compacton解。特别地，获得了E(3,2)方程和E(2,1)方程的新的compacton解；(2)将C-K直接法推广高维情形-(2+1)-维广义KdV方程，获得了8种新的(1+1)-维型的对称。利用这些结果，进一步可知道该方程也可约化为P-Ⅰ型和P-Ⅱ型的方程。最后研究了该方程的cnoidal波和类dromion结构；(3)将C-K直接法推广到(2+1)-维广义Burgers方程中，获得了11种新的(1+1)-维型的对称和6种新的条件对称。
第六章研究了Lax可积的新的方程族和Liouville可积的N-Hamilton结构方面的问题：将屠格式推广到新的含有任意函数的广义Dirac族的谱问题、广义Kaup-Newell谱问题及含有五个位势函数的3×3谱问题，研究了它们的Lax可积的方程族和Liouville可积的Hamilton结构。
第七章讨论了高阶约束流、对合系统、r-矩阵和变量分离性：(1)给出了一个广义Dirac族的Bargman约束流的r-矩阵，一个新的对合系统和解的对合表示；(2)给出了与Guo族有关的高阶约束条件及其可积的约束流(Hamilton系统)，及其Lax表示和r-矩阵；(3)证明了Dirac族的第一约束流的可分离性，并且给出了它的分离方
大连理工大学博工学位论文
程．
第八章提出了一个隐式的IOOp代数及其一组基所满足的对易关系，基于其中的
新的等谱问题，获得了著名的TC族的新的含有任意函数的LaX可积的耦合系统．
In this dissertation, with the aid of many types of constructive transformations and symbolic computation, some topics in nonlinear waves and integrable system are studied, including exact solutions, Painleve integrability, Backlund transformation, Darboux transformation, symmetry (similarity reduction), conditional symmetry, Lax integrable hierarchy, Liouville integrable N-Hamilton structure, constraint flow, involutive system, Lax representation, r-matrix, separation of variables and integrable couplings.
Chapter 2 and 3 are devoted to investigating exact solutions of nonlinear wave equations: Firstly, the basic theories of C-D pair and C-D integrable system are presented. Secondly, we choose some examples to illustrate them in Chapter 3. (1) Based on two types of Riccati equations, two kinds of new methods are proposed to obtain solutions of nonlinear differential equations. Twelve families of exact solutions of WBK equation are found by using one of two methods; (2) The homogeneous balance method is improved cind investigated to (2+l)-dimensional Broer-Kaup equation such that many families of new solutions are derived. (4) Based on the isospectral Lax pair of Riccati form for generalized KdV equation with the force term, new Darboux transformation and solitary-like wave solutions and rational solutions are obtained; (4) By constructing Darboux transformation and the superposition formula of generalized variable coefficients KdV equation with the force term, new single solitary-like wave solutions, double solitary-like wave solutions and rational solutions are found for (2+l)-dimensional generalized KP equation.
Chapter 4 deals with Painleve integrability and Backlund transformation. Two types of (2+l)-dimensional generalized Burgers equations are shown to pass the Painleve test by using WTC method, and their Backlund transformations are obtained through Painleve truncating expansion. Cole-Hopf transformation is their special case.
In Chapter 5, symmetries and exact solutions of nonlinear partial differential equations are studied. (1) The C-K direct method is extended to new Estevez-Mansfield-Clarkson (EMC) equation with fully dispersion term, E(m, n) equation, such that five symmetries are found. Moreover new solitary wave solutions of ?l,n) equation and compacton-like of E(m,m - 1) equation are obtained. In particular, new compacton solutions of E(3,2) equation and E(2,1) equation are derived; (2) The C-K direct method is extended to higher-dimensional case-(2+1)-dimensional generalized KdV equation so that eight families of new (1+1)-dimensionaJ symmetries are found which leads to P-I equation and P-II equation. Finally the cnoidal waves and dromion-like structures are
considered; (3) The C-K direct method is extended to (2+l)-dimensional generalized Burgers equation such that eleven types of new (l + l)-dimensional symmetries and six kinds of new conditional symmetries are obtained.
Chapter 6 concentrates on new Lax integrabie hierarchies of equations and Liou-ville integrabie N-Hamilton structures. The Tu's scheme is extended to the generalized Dirac spectral problem with an arbitrary function, the generalized Kaup-Newell spectral problem and the new 3x3 spectral problem with five potentials such that their Lax integrabie integrabie hierarchies of equations and Liouville integrabie Hamilton structures are obtained.
Chapter 7 is devote to higher-order constraint flow, involutive system, Lax representation, r-matrix and the separation of variables. (1) r-rnatrix, new involutive system and involutive solutions of Bargman constraint flow of generalized Dirac hierarchy are found; (2) higher-order constraint conditions and integrabie constraint flows of Guo's hierarchy and their Lax representations and r-matrix axe given; (3) It is shown that first constraint flow of Dirac hierarchy is separability and its separation equation is presented.
In chapter 8, we propose a new implicitly loop algebra. New Lax integrabie couplings of the famous TC hierarchy are obtained by using the new spectral pro
第二章和第三章考虑非线性偏微分方程的精确解的构造：首先给出了C-D对和C-D可积系统的基本理论，然后在第三章中具体研究了它们的应用：(1)基于两种Riccati方程，提出了两种新的求解非线性微分方程更多解的方法，利用其中的一种方法，得到了WBK方程的12组精确解；(2)对齐次子衡法进行改进，以致于获得了(2+1)-维Broer-Kaup方程的很多新解；(3)基于带有外力项的广义KdV方程的Riccati形式的非等谱Lax对，提出了该方程的一个新的Darboux变换，利用该变换，得到了新的类孤波解和有理解；(4)通过构造了带有外力项的变系数KdV方程的Darboux变换及叠加原理，获得(2+1)-维广义KP方程的新的类单孤波解、双类孤波解和有理解。
第四章讨论了非线性微分方程的Painleve可积性和Backlund变换。利用WTC方法证明了两类(2+1)-维广义Burgers方程是Painleve可积的，并经截断展开原理获得了它们的Backlund变换，其中Cole-Hopf变换是其特例。
第五章考虑了非线性偏微分方程的对称和精确解：(1)将C-K直接法推广到具有全色散项的新的Estevez-Mansfield-Clarkson(EMC)E(m,n)方程中，得到了五种新的对称，进而得到了E(1,n)方程的新的孤波解以及E(m,m-1)方程的新的类compacton解。特别地，获得了E(3,2)方程和E(2,1)方程的新的compacton解；(2)将C-K直接法推广高维情形-(2+1)-维广义KdV方程，获得了8种新的(1+1)-维型的对称。利用这些结果，进一步可知道该方程也可约化为P-Ⅰ型和P-Ⅱ型的方程。最后研究了该方程的cnoidal波和类dromion结构；(3)将C-K直接法推广到(2+1)-维广义Burgers方程中，获得了11种新的(1+1)-维型的对称和6种新的条件对称。
第六章研究了Lax可积的新的方程族和Liouville可积的N-Hamilton结构方面的问题：将屠格式推广到新的含有任意函数的广义Dirac族的谱问题、广义Kaup-Newell谱问题及含有五个位势函数的3×3谱问题，研究了它们的Lax可积的方程族和Liouville可积的Hamilton结构。
第七章讨论了高阶约束流、对合系统、r-矩阵和变量分离性：(1)给出了一个广义Dirac族的Bargman约束流的r-矩阵，一个新的对合系统和解的对合表示；(2)给出了与Guo族有关的高阶约束条件及其可积的约束流(Hamilton系统)，及其Lax表示和r-矩阵；(3)证明了Dirac族的第一约束流的可分离性，并且给出了它的分离方
大连理工大学博工学位论文
程．
第八章提出了一个隐式的IOOp代数及其一组基所满足的对易关系，基于其中的
新的等谱问题，获得了著名的TC族的新的含有任意函数的LaX可积的耦合系统．
In this dissertation, with the aid of many types of constructive transformations and symbolic computation, some topics in nonlinear waves and integrable system are studied, including exact solutions, Painleve integrability, Backlund transformation, Darboux transformation, symmetry (similarity reduction), conditional symmetry, Lax integrable hierarchy, Liouville integrable N-Hamilton structure, constraint flow, involutive system, Lax representation, r-matrix, separation of variables and integrable couplings.
Chapter 2 and 3 are devoted to investigating exact solutions of nonlinear wave equations: Firstly, the basic theories of C-D pair and C-D integrable system are presented. Secondly, we choose some examples to illustrate them in Chapter 3. (1) Based on two types of Riccati equations, two kinds of new methods are proposed to obtain solutions of nonlinear differential equations. Twelve families of exact solutions of WBK equation are found by using one of two methods; (2) The homogeneous balance method is improved cind investigated to (2+l)-dimensional Broer-Kaup equation such that many families of new solutions are derived. (4) Based on the isospectral Lax pair of Riccati form for generalized KdV equation with the force term, new Darboux transformation and solitary-like wave solutions and rational solutions are obtained; (4) By constructing Darboux transformation and the superposition formula of generalized variable coefficients KdV equation with the force term, new single solitary-like wave solutions, double solitary-like wave solutions and rational solutions are found for (2+l)-dimensional generalized KP equation.
Chapter 4 deals with Painleve integrability and Backlund transformation. Two types of (2+l)-dimensional generalized Burgers equations are shown to pass the Painleve test by using WTC method, and their Backlund transformations are obtained through Painleve truncating expansion. Cole-Hopf transformation is their special case.
In Chapter 5, symmetries and exact solutions of nonlinear partial differential equations are studied. (1) The C-K direct method is extended to new Estevez-Mansfield-Clarkson (EMC) equation with fully dispersion term, E(m, n) equation, such that five symmetries are found. Moreover new solitary wave solutions of ?l,n) equation and compacton-like of E(m,m - 1) equation are obtained. In particular, new compacton solutions of E(3,2) equation and E(2,1) equation are derived; (2) The C-K direct method is extended to higher-dimensional case-(2+1)-dimensional generalized KdV equation so that eight families of new (1+1)-dimensionaJ symmetries are found which leads to P-I equation and P-II equation. Finally the cnoidal waves and dromion-like structures are
considered; (3) The C-K direct method is extended to (2+l)-dimensional generalized Burgers equation such that eleven types of new (l + l)-dimensional symmetries and six kinds of new conditional symmetries are obtained.
Chapter 6 concentrates on new Lax integrabie hierarchies of equations and Liou-ville integrabie N-Hamilton structures. The Tu's scheme is extended to the generalized Dirac spectral problem with an arbitrary function, the generalized Kaup-Newell spectral problem and the new 3x3 spectral problem with five potentials such that their Lax integrabie integrabie hierarchies of equations and Liouville integrabie Hamilton structures are obtained.
Chapter 7 is devote to higher-order constraint flow, involutive system, Lax representation, r-matrix and the separation of variables. (1) r-rnatrix, new involutive system and involutive solutions of Bargman constraint flow of generalized Dirac hierarchy are found; (2) higher-order constraint conditions and integrabie constraint flows of Guo's hierarchy and their Lax representations and r-matrix axe given; (3) It is shown that first constraint flow of Dirac hierarchy is separability and its separation equation is presented.
In chapter 8, we propose a new implicitly loop algebra. New Lax integrabie couplings of the famous TC hierarchy are obtained by using the new spectral pro
引文
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