离散非线性微分—差分晶格系统的孤立波和局域模分析
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摘要
非线性现象在自然界中既普遍又重要。非线性科学是研究非线性现象共性的一门学问,它的研究主体是孤立子、混沌和分形。许多非线性问题的研究最终可归结为非线性系统来描述。非线性微分-差分晶格系统的部分或全部空间变量是离散的,而通常时间变量是连续的,它是非线性系统中一类重要的系统。
非线性系统的精确解对研究相关的非线性问题非常重要。孤立子理论研究的一个主要内容,就是寻求非线性系统的解,特别是孤立波解(包括精确解和数值解)。在过去的大约50年中,非线性科学研究领域颇具特色的新成就之一就是创造了求非线性方程的解特别是孤波解的各种精巧方法。但相对于非线性偏微分方程,非线性微分-差分晶格孤子系统的求解起步较晚,成果也有局限。
本文结合孤立子理论的发展方向,围绕非线性微分-差分晶格孤子系统,以数学机械化思想为指导,以计算机代数系统软件和吴方法为工具,研究若干重要的离散非线性微分差分晶格系统的求解问题。提出和改进了一系列求解该类系统的方法,得到了一些具有重要意义的离散非线性微分差分晶格系统丰富的精确解。采用逆方法,建立长程关联离散晶格模型。推广连续Toda晶格模型,得到多种精确孤立波解。
首先介绍了孤立子理论的发展历程,微分-差分方程的研究状况。总结和分析了目前人们常用的构造精确解的方法。综述了离散可积Toda晶格和氢键晶格的非线性特性。对经典的Toda晶格孤立子作了详细介绍,分析了Toda双孤子的弹性碰撞。介绍了具次邻耦合的非线性晶格中的几种孤立子。
求离散非线性微分-差分晶格系统精确孤立波解的直接方法开始于tanh函数展开法,它假设晶格系统的孤立波解可以表示为tanh双曲正切函数的叠加和组合。本文推广了tanh函数展开法,取组合函数满足Riccati方程,且解的形式同时含有组合函数的正负幂次项,提出修正的F展开法和广义tanh-sech法;将椭圆函数展开法应用到求非线性微分-差分晶格系统孤立波求解中,提出扩展的Jacobi椭圆函数展开法;定义Fibonacci-sec函数,提出了Fibonacci tan-see展开法。应用这三种不同的方法以及扩展的Sine-Gordon展开法研究了各类Toda晶格、离散mKdv晶格,Hybrid晶格、Ablowitz-Ladik晶格和Volterra晶格等,得到了丰富的精确解。利用“辅助方程方法”,将双曲函数展开法推广应用变系数的微分-差分晶格系统,获得了丰富的精确解。
研究次邻耦合情形下的局域模是本文的另一方面。采用逆方法,建立具有次邻耦合的长程关联离散Klein-Cordon晶格模型,相对于仅有近邻耦合的离散晶格系统,次邻耦合晶格系统具有更多形式的N—格点compacton解,次邻耦合系数影响N—格点解的稳定性。离散呼吸解在耦合非线性离散Klein-Gordon晶格系统仍存在。通过数值模拟表明,次邻耦合系统中宽孤子稳定而窄孤立子不稳定。
在非线性发展方程研究方面,考虑不仅有横向、纵向的运动,并且有相互耦合作用的情形,推广了连续Toda晶格系统。假设横向与纵向运动处于同一量级,应用直接法得到连续Toda晶格系统的compacton,multiplecompacton,peakon等弱激发模式和拟紧孤子解。Compacton在空间有限区域外为零,peakon在其峰或谷处有尖点,而拟紧孤子解的宽度与振幅都取决于速度。在无横向与纵向线性关系假设下,应用扩展的sin-cos法,除一般的孤立波外还得到了特殊的行波解。利用数学软件给出了相应弱激发模式和孤立波的图形。
最后给出研究工作总结和后续研究展望。
Nonlinear Science, which has solition theory, fractal and chaos as its main parts, is the subject of studying the common futures of nonlinearity. Nonlinearity is universal and important. Most nonlinear problems can be described by nonlinear equations, which generally includes nonlinear ordinary differential equations, partial differential equations (PDE), difference equations, functional equations and differential-difference equations (lattice systems). Nonlinear differential-difference lattice system is featured by the discreteness of its part or all spatial variables and the continuity of its time variable.
How to obtain the exact solutions of nonlinear equations is of vital importance to the study of the corresponding problem. The key problem in soliton theory is to get solutions of the nonlinear evolution equations, including exact ones or numerical ones. During the past 50 years or so, the scientists have created various ingenious methods to construct exact solutions, especially soliton solutions of nonlinear equations. However, compared to nonlinear partial differential equations, the study of constructing solitary solutions of differential-difference lattice systems started latter and less progress has been made.
In this dissertation, along with the direction of the development of soliton theory, centered with discrete soliton lattice systems, under the guidance of mathematics mechanization and by means of computer algebraic system software and the Wu method, some problems of solving some important discrete nonlinear differential-difference lattice systems are discussed and some methods for constructing the exact explicit solutions of differential-difference lattice systems are presented and improved. Many explicit solutions for such lattice systems are obtained. By using the reverse method, we construct discrete models with long-range interactions. Several kind of exact solitary wave solutions are presented for the continuum Toda lattice equation.
The history and development of the soliton theory as well as the re- search status of mathematics mechanization are reviewed at the beginning of this dissertation. Then the theory of constructing exact solutions of partial differential equations (PDEs) under the AC = BD theory is introduced. The methods known up to today for constructing exact solutions of nonlinear differential-difference lattice systems are summarized and analyzed. The nonlinear characteristics of discrete integrable Toda lattice are presented. The solitary wave solutions of classical Toda Lattice is introduced. The elastic collision between the double-soliton of Toda lattice is analyzed. Several kinds of solitons in nonlinear lattices with next-neighbor interactions are presented.
In the subject of discrete nonlinear differential-difference lattice systems, by introducing negative power terms, and enlarging the scope of the combination function to both hyperbolic functions and triangular functions, we give the modified F—expand, method and the extended tanh-sech method. A general Jacobi ellipse expansion method and the Fibonacci tan-sec-expand method are also presented in this dissertation. By using the above mentioned methods as well as the extended Sine-Gordon method, a lot of discrete equations, such as Toda lattice hierarchy, discrete mKdv lattice systems, Hybrid lattice , Ablowitz-Ladik lattice and Volterra lattice are studied and abundant explicit traveling wave solutions are obtained. By using auxiliary equations, the tanh-expand method is also extend to differential-difference lattice systems with variable coefficients and several exact solutions are given in this case.
The study of discrete compacton solutions is the second part of this dissertation. Two discrete Klein-Gordon lattice models with long-range interactions are established by reverse method. The localized modes with first-and-second-neighbor interactions are studied. Discrete breathers as well as more kinds of discrete N-site compactons under the condition of next neighbor interactions are presented. The next-neighbor coefficient has an influence on the stability of N-site compactons. By numerical simulation it is shown that the broad breathers are stable while the narrow ones are not.
In the case of nonlinear evolution equations, the continuums Toda lattice model is extended to include coupling of longitudinal and transversal interactions. Supposing that the transversal and the longitudinal strains be of the same order of magnitude , by using direct algebraic method, we obtain compactons, multiple compactons ,peakons and compacton-like solutions of the continuum Toda lattice system . Compactons are zero outside of a finite spatial region, while peakons have cusp at their peak of valley, and the width and the amplitude of compacton-like solutions are determined by the velocity. For the special case of the continuum Toda lattice system, by using the extended sin-cos expand method, more exact traveling wave solutions are given. Profiles of most of those weak excitations and solitary wave solutions are presented for a better understanding.
A brief summary is given in the last part of the dissertation, while a possible academic prospect is also opened.
非线性系统的精确解对研究相关的非线性问题非常重要。孤立子理论研究的一个主要内容,就是寻求非线性系统的解,特别是孤立波解(包括精确解和数值解)。在过去的大约50年中,非线性科学研究领域颇具特色的新成就之一就是创造了求非线性方程的解特别是孤波解的各种精巧方法。但相对于非线性偏微分方程,非线性微分-差分晶格孤子系统的求解起步较晚,成果也有局限。
本文结合孤立子理论的发展方向,围绕非线性微分-差分晶格孤子系统,以数学机械化思想为指导,以计算机代数系统软件和吴方法为工具,研究若干重要的离散非线性微分差分晶格系统的求解问题。提出和改进了一系列求解该类系统的方法,得到了一些具有重要意义的离散非线性微分差分晶格系统丰富的精确解。采用逆方法,建立长程关联离散晶格模型。推广连续Toda晶格模型,得到多种精确孤立波解。
首先介绍了孤立子理论的发展历程,微分-差分方程的研究状况。总结和分析了目前人们常用的构造精确解的方法。综述了离散可积Toda晶格和氢键晶格的非线性特性。对经典的Toda晶格孤立子作了详细介绍,分析了Toda双孤子的弹性碰撞。介绍了具次邻耦合的非线性晶格中的几种孤立子。
求离散非线性微分-差分晶格系统精确孤立波解的直接方法开始于tanh函数展开法,它假设晶格系统的孤立波解可以表示为tanh双曲正切函数的叠加和组合。本文推广了tanh函数展开法,取组合函数满足Riccati方程,且解的形式同时含有组合函数的正负幂次项,提出修正的F展开法和广义tanh-sech法;将椭圆函数展开法应用到求非线性微分-差分晶格系统孤立波求解中,提出扩展的Jacobi椭圆函数展开法;定义Fibonacci-sec函数,提出了Fibonacci tan-see展开法。应用这三种不同的方法以及扩展的Sine-Gordon展开法研究了各类Toda晶格、离散mKdv晶格,Hybrid晶格、Ablowitz-Ladik晶格和Volterra晶格等,得到了丰富的精确解。利用“辅助方程方法”,将双曲函数展开法推广应用变系数的微分-差分晶格系统,获得了丰富的精确解。
研究次邻耦合情形下的局域模是本文的另一方面。采用逆方法,建立具有次邻耦合的长程关联离散Klein-Cordon晶格模型,相对于仅有近邻耦合的离散晶格系统,次邻耦合晶格系统具有更多形式的N—格点compacton解,次邻耦合系数影响N—格点解的稳定性。离散呼吸解在耦合非线性离散Klein-Gordon晶格系统仍存在。通过数值模拟表明,次邻耦合系统中宽孤子稳定而窄孤立子不稳定。
在非线性发展方程研究方面,考虑不仅有横向、纵向的运动,并且有相互耦合作用的情形,推广了连续Toda晶格系统。假设横向与纵向运动处于同一量级,应用直接法得到连续Toda晶格系统的compacton,multiplecompacton,peakon等弱激发模式和拟紧孤子解。Compacton在空间有限区域外为零,peakon在其峰或谷处有尖点,而拟紧孤子解的宽度与振幅都取决于速度。在无横向与纵向线性关系假设下,应用扩展的sin-cos法,除一般的孤立波外还得到了特殊的行波解。利用数学软件给出了相应弱激发模式和孤立波的图形。
最后给出研究工作总结和后续研究展望。
Nonlinear Science, which has solition theory, fractal and chaos as its main parts, is the subject of studying the common futures of nonlinearity. Nonlinearity is universal and important. Most nonlinear problems can be described by nonlinear equations, which generally includes nonlinear ordinary differential equations, partial differential equations (PDE), difference equations, functional equations and differential-difference equations (lattice systems). Nonlinear differential-difference lattice system is featured by the discreteness of its part or all spatial variables and the continuity of its time variable.
How to obtain the exact solutions of nonlinear equations is of vital importance to the study of the corresponding problem. The key problem in soliton theory is to get solutions of the nonlinear evolution equations, including exact ones or numerical ones. During the past 50 years or so, the scientists have created various ingenious methods to construct exact solutions, especially soliton solutions of nonlinear equations. However, compared to nonlinear partial differential equations, the study of constructing solitary solutions of differential-difference lattice systems started latter and less progress has been made.
In this dissertation, along with the direction of the development of soliton theory, centered with discrete soliton lattice systems, under the guidance of mathematics mechanization and by means of computer algebraic system software and the Wu method, some problems of solving some important discrete nonlinear differential-difference lattice systems are discussed and some methods for constructing the exact explicit solutions of differential-difference lattice systems are presented and improved. Many explicit solutions for such lattice systems are obtained. By using the reverse method, we construct discrete models with long-range interactions. Several kind of exact solitary wave solutions are presented for the continuum Toda lattice equation.
The history and development of the soliton theory as well as the re- search status of mathematics mechanization are reviewed at the beginning of this dissertation. Then the theory of constructing exact solutions of partial differential equations (PDEs) under the AC = BD theory is introduced. The methods known up to today for constructing exact solutions of nonlinear differential-difference lattice systems are summarized and analyzed. The nonlinear characteristics of discrete integrable Toda lattice are presented. The solitary wave solutions of classical Toda Lattice is introduced. The elastic collision between the double-soliton of Toda lattice is analyzed. Several kinds of solitons in nonlinear lattices with next-neighbor interactions are presented.
In the subject of discrete nonlinear differential-difference lattice systems, by introducing negative power terms, and enlarging the scope of the combination function to both hyperbolic functions and triangular functions, we give the modified F—expand, method and the extended tanh-sech method. A general Jacobi ellipse expansion method and the Fibonacci tan-sec-expand method are also presented in this dissertation. By using the above mentioned methods as well as the extended Sine-Gordon method, a lot of discrete equations, such as Toda lattice hierarchy, discrete mKdv lattice systems, Hybrid lattice , Ablowitz-Ladik lattice and Volterra lattice are studied and abundant explicit traveling wave solutions are obtained. By using auxiliary equations, the tanh-expand method is also extend to differential-difference lattice systems with variable coefficients and several exact solutions are given in this case.
The study of discrete compacton solutions is the second part of this dissertation. Two discrete Klein-Gordon lattice models with long-range interactions are established by reverse method. The localized modes with first-and-second-neighbor interactions are studied. Discrete breathers as well as more kinds of discrete N-site compactons under the condition of next neighbor interactions are presented. The next-neighbor coefficient has an influence on the stability of N-site compactons. By numerical simulation it is shown that the broad breathers are stable while the narrow ones are not.
In the case of nonlinear evolution equations, the continuums Toda lattice model is extended to include coupling of longitudinal and transversal interactions. Supposing that the transversal and the longitudinal strains be of the same order of magnitude , by using direct algebraic method, we obtain compactons, multiple compactons ,peakons and compacton-like solutions of the continuum Toda lattice system . Compactons are zero outside of a finite spatial region, while peakons have cusp at their peak of valley, and the width and the amplitude of compacton-like solutions are determined by the velocity. For the special case of the continuum Toda lattice system, by using the extended sin-cos expand method, more exact traveling wave solutions are given. Profiles of most of those weak excitations and solitary wave solutions are presented for a better understanding.
A brief summary is given in the last part of the dissertation, while a possible academic prospect is also opened.
引文
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