非线性演化方程精确解构造性理论与算法研究
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摘要
非线性动力学是非线性科学的一个重要分支,非线性动力学研究对象主要包括分叉、混沌、分形、孤立子等新的现象。非线性演化方程精确解的求解方法是孤立子理论的一个主要内容。随着计算机技术的发展,特别是计算机符号计算软件的出现,非线性演化方程精确解构造性理论及算法研究已成为非线性科学中的前沿研究课题和新的热点。虽然目前已经提出和发展了许多构造非线性演化方程精确解的理论和算法,但是由于构造非线性演化方程精确解没有也不可能有统一而普适的方法,因此继续寻找一些行之有效的求解方法依然是一项十分重要和极有价值的工作。本文在归纳和总结现有各种构造非线性演化方程精确解的主要方法的基础上,对非线性演化方程精确解的构造理论及算法进行了较为系统和深入的研究,提出和拓展了几种构造非线性演化方程精确解的新方法,并将这些方法应用到了许多力学和物理学中非常重要的非线性演化方程,结果不仅获得了这些方程已有的精确解,而且得到了许多新解。因此本文的工作丰富和发展了非线性演化方程精确解的构造理论及算法,具有较大的理论意义和应用价值。
全文共分七章。第一章为绪论,对非线性演化方程精确解的构造理论和算法进行综述,说明课题的来源,介绍本文的研究目的、主要内容和创新点。第二章用力学的方法简单地导出了几个重要的非线性演化方程,说明本文研究中所涉及到的非线性演化方程是有力学或物理学背景的。第三章利用多项式判别系统把范式代数方法扩展到其引入的常微分方程的未知函数的最高次幂大于4的情形,并用该方法构造了KdV方程和变形Boussinesq方程的一系列精确行波解,包括有理函数解,孤波解,三角函数周期解,Jacobi周期函数解和隐函数解。第四章引入了一个低阶辅助常微分方程,并获得该方程的一些新解,基于该辅助方程建立了一个辅助方程法的算法和直接求解的算法。利用辅助方程法构造了组合KdV-mKdV方程,(2+1)维Broer-Kaup-Kupershmidt方程,和两类变系数KdV方程的孤波解和三角函数解。使用直接解法构造了两个非线性演化方程:(1+1)维Klein-Gordon方程,(3+1)维Kadomtsev-Petviashvili方程,和两个非线性耦合演化方程组:(2+1)维耦合色散长波方程,耦合Klein-Gordon-Zakharov方程的孤波解和三角函数解。第五章获得了广义Lienard方程的一些新解,并利用它构造了一维广义Klein-Gordon方程,广义Ablowitz方程和广义Gerdjikov-Ivanov方程的精确解。然后引入一个带任意次幂非线性项的辅助常微分方程,利用任意次幂辅助方程法构造了广义Zakharov方程和广义Benjamin-Bona-Mahony方程的精确解。第六章通过一般化sinh-Gordon方程和构造所研究的方程新的试探解来扩展sinh-Gordon方程展开法。并用它构造了(2+1)维Konopelchenko–Dubrovsky方程,KdV-mKdV方程,双sine-Gordon方程和BBM方程的行波解,包括Jacobi椭圆函数双周期解,孤波解,三角函数解。第七章提出了利用耦合的Riccati方程组构造非线性微分-差分方程精确解的一种新的算法—离散耦合Riccati方程展开法,利用该算法获得了一般格子方程,Toda格子方程和(2+1)维Toda格子方程的扭结孤波解和复数解。最后对本文的工作进行了总结,并对今后的研究方向作了展望。
Nonlinear dynamics is one of the important branches in nonlinear science, and its main research objects are bifurcation, chaos, fractal and solitons, etc.whereas study on how to solve nonlinear evolution equations is one of the main contents in soliton theory. With the development of computer technology, especially the emergence of computer symbol computation softwares, study on the theories and algorithmes for constructing exact solutions of nonlinear evolution equations has become a leading subject and new interest in nonliner science. At present, although a number of algorithmes are proposed and developed to construct exact solutions of nonlinear evolution equations, unfortunately, not all these approaches are universally applicable for solving all kinds of nonlinear evolution equations directly. As a consequence, it is still a very significant task to go on searching for various powerful and efficient approaches to solve nonlinear evolution equations. In this thesis, based on a complete summary and examination of the main methods for constructing exact of nonlinear evolution equations, a systematic investigation into the fundamental theories and algorithmes of looking for exact solutions of nonlinear evolution equations has been studied, and a few new approaches are proposed and developed to seek exact solutions of nonlinear evolution equations. By making use of these approaches proposed, a variety of exact solutions to many physically significant nonlinear evolution equations are obtained. Among them, some are in agreement with those obtained in some literatures, others are new ones which can not be found in the existing literatures to the best of our knowledge. Our studies enrich and develop the theories and algorithmes of constructing exact solutions of nonlinear evolution equations, and have more profound theoretical significance and important application value.
This dissertation consists of seven chapters. In chapter one, the main theories and algorithmes of constructing exact solutions of nonlinear evolution equations are summarized, and the source of the subject, the research purpose, the primary contents and innovations of this dissertation are reported as well. In chapter two, several important nonlinear evolution equations in mechanics are derived, and it indicate that the nonlinear evolution equations involved in our study have a mechanics or physics background. In chapter three, We extend Fan’s algebraic method to its auxiliary ordinary differential equation’s degree more than 4 by using a complete discrimination system for polynomials, and apply this method to construct a series of traveling wave solutions for KdV equation and the variant Boussinesq equations including rational solutions, solitary wave solutions, triangular periodic solutions, Jacobi periodic wave solutions and implicit function solutions. In chapter four, an auxiliary ordinary differential equation method and a direct method are presented by introducing an auxiliary ordinary differential equation whose some new exact solutions are obtained. By using the auxiliary equation method the Combined KdV-mKdV equation, (2+1)-dimensional Broer-Kaup-Kupershmidt system and two kinds of KdV equation with variable coefficients are investigated and abundant exact traveling wave solutions are obtained that include solitary wave solutions and triangular periodic wave solutions. By making use of the direct method, the same kinds of solutions of (1+1)-dimensional Klein-Gordon equation, (3+1)-dimensional Kadomtsev Petviashvili equation, the (2+1)-dimensional dispersive long wave equations and the Klein- Gordon-Zakharov equations are obtained. In chapter five, some new exact solutions of the generalized Lienard equation are obtained, and the solutions of the equation are applied to construct directly the exact solutions of the generalized one-dimensional Klein-Gordon equation, the generalized Ablowitz equation and the generalized Gerdjikov-Ivanov equation. Then based on an auxiliary ordinary differential equation with nonlinear terms of any order, a new approach called the auxiliary equation with nonlinear terms of any order method is proposed to construct the exact solutions to the generalized Zakharov equations and the generalized Benjamin-Bona-Mahony equation. In chapter six, the sinh-Gordon equation expansion method is further extended by generalizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the (2+1)-dimensional Konopelchenko–Dubrovsky equation, the KdV-mKdV equation, the double sine-Gordon equation and the BBM equation are investigated and abundant exact travelling wave solutions are explicitly obtained including Jacobi elliptic doubly periodic function solutions, solitary wave solutions and trigonometric function solutions. In chapter seven, a new discrete coupled Riccati equations expansion algorithm is presented to construct the exact solutions of nonlinear differential-difference equations by introducing the coupled Riccati equations. As an example, we apply this algorithm to the general lattice equation, the relativistic Toda lattice equations and the (2+1)dimensional Toda lattice equation. As a result, some kink solitary wave solutions and complexiton solutions of them are obtained with the help of symbolic system Mathematica. Finally, the summary of this dissertation and the prospect of study on constructing exact solutions to nonlinear evolution equations are given.
全文共分七章。第一章为绪论,对非线性演化方程精确解的构造理论和算法进行综述,说明课题的来源,介绍本文的研究目的、主要内容和创新点。第二章用力学的方法简单地导出了几个重要的非线性演化方程,说明本文研究中所涉及到的非线性演化方程是有力学或物理学背景的。第三章利用多项式判别系统把范式代数方法扩展到其引入的常微分方程的未知函数的最高次幂大于4的情形,并用该方法构造了KdV方程和变形Boussinesq方程的一系列精确行波解,包括有理函数解,孤波解,三角函数周期解,Jacobi周期函数解和隐函数解。第四章引入了一个低阶辅助常微分方程,并获得该方程的一些新解,基于该辅助方程建立了一个辅助方程法的算法和直接求解的算法。利用辅助方程法构造了组合KdV-mKdV方程,(2+1)维Broer-Kaup-Kupershmidt方程,和两类变系数KdV方程的孤波解和三角函数解。使用直接解法构造了两个非线性演化方程:(1+1)维Klein-Gordon方程,(3+1)维Kadomtsev-Petviashvili方程,和两个非线性耦合演化方程组:(2+1)维耦合色散长波方程,耦合Klein-Gordon-Zakharov方程的孤波解和三角函数解。第五章获得了广义Lienard方程的一些新解,并利用它构造了一维广义Klein-Gordon方程,广义Ablowitz方程和广义Gerdjikov-Ivanov方程的精确解。然后引入一个带任意次幂非线性项的辅助常微分方程,利用任意次幂辅助方程法构造了广义Zakharov方程和广义Benjamin-Bona-Mahony方程的精确解。第六章通过一般化sinh-Gordon方程和构造所研究的方程新的试探解来扩展sinh-Gordon方程展开法。并用它构造了(2+1)维Konopelchenko–Dubrovsky方程,KdV-mKdV方程,双sine-Gordon方程和BBM方程的行波解,包括Jacobi椭圆函数双周期解,孤波解,三角函数解。第七章提出了利用耦合的Riccati方程组构造非线性微分-差分方程精确解的一种新的算法—离散耦合Riccati方程展开法,利用该算法获得了一般格子方程,Toda格子方程和(2+1)维Toda格子方程的扭结孤波解和复数解。最后对本文的工作进行了总结,并对今后的研究方向作了展望。
Nonlinear dynamics is one of the important branches in nonlinear science, and its main research objects are bifurcation, chaos, fractal and solitons, etc.whereas study on how to solve nonlinear evolution equations is one of the main contents in soliton theory. With the development of computer technology, especially the emergence of computer symbol computation softwares, study on the theories and algorithmes for constructing exact solutions of nonlinear evolution equations has become a leading subject and new interest in nonliner science. At present, although a number of algorithmes are proposed and developed to construct exact solutions of nonlinear evolution equations, unfortunately, not all these approaches are universally applicable for solving all kinds of nonlinear evolution equations directly. As a consequence, it is still a very significant task to go on searching for various powerful and efficient approaches to solve nonlinear evolution equations. In this thesis, based on a complete summary and examination of the main methods for constructing exact of nonlinear evolution equations, a systematic investigation into the fundamental theories and algorithmes of looking for exact solutions of nonlinear evolution equations has been studied, and a few new approaches are proposed and developed to seek exact solutions of nonlinear evolution equations. By making use of these approaches proposed, a variety of exact solutions to many physically significant nonlinear evolution equations are obtained. Among them, some are in agreement with those obtained in some literatures, others are new ones which can not be found in the existing literatures to the best of our knowledge. Our studies enrich and develop the theories and algorithmes of constructing exact solutions of nonlinear evolution equations, and have more profound theoretical significance and important application value.
This dissertation consists of seven chapters. In chapter one, the main theories and algorithmes of constructing exact solutions of nonlinear evolution equations are summarized, and the source of the subject, the research purpose, the primary contents and innovations of this dissertation are reported as well. In chapter two, several important nonlinear evolution equations in mechanics are derived, and it indicate that the nonlinear evolution equations involved in our study have a mechanics or physics background. In chapter three, We extend Fan’s algebraic method to its auxiliary ordinary differential equation’s degree more than 4 by using a complete discrimination system for polynomials, and apply this method to construct a series of traveling wave solutions for KdV equation and the variant Boussinesq equations including rational solutions, solitary wave solutions, triangular periodic solutions, Jacobi periodic wave solutions and implicit function solutions. In chapter four, an auxiliary ordinary differential equation method and a direct method are presented by introducing an auxiliary ordinary differential equation whose some new exact solutions are obtained. By using the auxiliary equation method the Combined KdV-mKdV equation, (2+1)-dimensional Broer-Kaup-Kupershmidt system and two kinds of KdV equation with variable coefficients are investigated and abundant exact traveling wave solutions are obtained that include solitary wave solutions and triangular periodic wave solutions. By making use of the direct method, the same kinds of solutions of (1+1)-dimensional Klein-Gordon equation, (3+1)-dimensional Kadomtsev Petviashvili equation, the (2+1)-dimensional dispersive long wave equations and the Klein- Gordon-Zakharov equations are obtained. In chapter five, some new exact solutions of the generalized Lienard equation are obtained, and the solutions of the equation are applied to construct directly the exact solutions of the generalized one-dimensional Klein-Gordon equation, the generalized Ablowitz equation and the generalized Gerdjikov-Ivanov equation. Then based on an auxiliary ordinary differential equation with nonlinear terms of any order, a new approach called the auxiliary equation with nonlinear terms of any order method is proposed to construct the exact solutions to the generalized Zakharov equations and the generalized Benjamin-Bona-Mahony equation. In chapter six, the sinh-Gordon equation expansion method is further extended by generalizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the (2+1)-dimensional Konopelchenko–Dubrovsky equation, the KdV-mKdV equation, the double sine-Gordon equation and the BBM equation are investigated and abundant exact travelling wave solutions are explicitly obtained including Jacobi elliptic doubly periodic function solutions, solitary wave solutions and trigonometric function solutions. In chapter seven, a new discrete coupled Riccati equations expansion algorithm is presented to construct the exact solutions of nonlinear differential-difference equations by introducing the coupled Riccati equations. As an example, we apply this algorithm to the general lattice equation, the relativistic Toda lattice equations and the (2+1)dimensional Toda lattice equation. As a result, some kink solitary wave solutions and complexiton solutions of them are obtained with the help of symbolic system Mathematica. Finally, the summary of this dissertation and the prospect of study on constructing exact solutions to nonlinear evolution equations are given.
引文
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