若干非线性数学物理问题的解析和数值研究
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摘要
在线性理论日臻完善的今天，非线性数学物理问题愈发为科学家所关注。无论是基础科学还是工程技术领域，研究者面对的大多是复杂的非线性问题。如何处理这些问题是对数学家和物理学家的重大挑战。本论文的主要工作沿两条线索展开：一方面，充分利用一些现有的数学工具研究几个重要的非线性模型，得到一些新颖的结果；另一方面，根据对一些已有严格解方法和近似方法的总结和反思，提出一种研究非线性数学物理问题的新型近似方法
本文第一章作为绪论部分概括地介绍了非线性科学的内涵、意义和研究现状，重点介绍了本论文涉及的一些非线性问题和非线性方程的数学及物理背景，概述了研究非线性数学物理问题的主要数学工具及其发展历程，同时阐明了本论文的主要工作
第二章从流体力学的基本模型β面上的无辐散正压位涡方程出发，推导出变系数KdV方程及变系数mKdV方程；从另一个研究分层流体的基本方程二层流体模型出发，推导出耦合变系数mKdV方程，然后通过构造这些变系数方程的严格解充分阐释了大气系统中一种重要的非线性现象—大气阻塞产生、发展和衰减的动力学机理。在这一章里，基于多重尺度展开法，我们提出了一个新的推导变系数近似方程的系统方法。同时，为了严格求解推导出的变系数方程，我们发展了一种在变系数方程和相应的常系数方程之间建立Backlund变换的直接方法。该直接方法作为求解变系数方程的一个基本方法还将应用于本论文的其他章节。
第三章通过施加四个合理化条件(单涡旋的局域性、短程相互作用、两体相互作用以及多涡旋的完整性)，将β面上的无辐散正压位涡方程转化为一个多涡旋相互作用模型。我们充分讨论了该模型的经典李对称和守恒律，给出了模型的几种严格解，包括涡旋源解和Bessel涡旋解。利用该模型，结合孤立子理论以及数值模拟方法，我们解释了大气系统中热带气旋之间一些重要的相互作用特征，包括气旋间互旋、两个气旋在临界距离处的合并或分离、强气旋对弱气旋的吸收，以及台风对副热带高压的环绕运动。
第四章利用本论文第一章发展的构造变系数方程与相应常系数方程之间Backlund变换的直接方法，严格求解了一个变系数3+1维NLS方程和一个耦合变系数1+1维NLS方程，给出了前者的孤波解和后者的向量孤波解，同时给出了这些严格解存在时变系数之间所要满足的关系。这两类方程在玻色一爱因斯坦凝聚和非线性光学领域都有重要的理论意义。同时，在这一章里，我们还利用经典李群法研究了一种和NLS方程相关的新型非线性方程一共振DS方程的对称性，并且得到了该方程的三类严格解。
与第二、三、四章主要处理具体非线性问题不同，第五章则花一章的篇幅着力于方法的研究。通过对现有非线性方程研究中经常使用的一些严格解方法及近似方法进行总结和提炼，我们在同伦分析方法的基础上提出了一种新的近似方法一不敏感性同伦方法。该方法通过建立线性或非线性同伦，将难以直接求解的原始模型和一个有严格解的简化模型联系起来，并且引入控制收敛的辅助参数，从而不依赖小的微扰参量即可得到原始模型的高精度近似解。与同伦分析法相比，一方面，该方法突破了线性同伦的局限，展示出完全可以用非线性同伦连接原始模型和简化模型；另一方面，明确给出了一个指导确定合理辅助参数的原则——不敏感性原理，并且通过构造不敏感量，给出了确定辅助参数以提高近似解精度的具体步骤。该方法的有效性通过非线性微分方程的求解和非简谐振子能量本征值的计算得到了检验，它既可以作为近似求解非线性方程的工具，也可作为物理学中一种有效的非微扰方法来使用。
第六章对论文的主要内容进行了总结和讨论，并且对未来工作可能的新发展和新方向做了展望。
Nonlinear problems in mathematics and physics are arousing great interests from scientists as the theories on linear problems have been well developed. Most of the problems in both basic science and engineering-oriented fields are nonlin-ear, and how to handle these complicated problems is a significant challenge for mathematicians and physicists. In this dissertation, two topics are discussed. On one hand, we study a few important nonlinear problems and obtain some meaningful results by using some known mathematical tools. On the other hand, we also propose a new approximate approach for nonlinear problems by abstract-ing and summarizing the basic idea which underlies some well-known exact and approximate mathematical methods.
Chapter 1 is an introduction which is devoted to reviewing the mathematical and physical backgrounds of some important nonlinear equations discussed in this dissertation. The significance and development of the nonlinear science are reviewed, too. The mathematical tools involved are introduced in this chapter, and wre also briefly report the main wrorks of this dissertation.
In chapter 2, we derive a variable coefficient KdV equation and a variable coefficient modified KdV(mKdV) equation from a nonlinear inviscid nondissipa-tive and equivalent barotropic vorticity equation in a beta plane. We also derive a coupled variable coefficient mKdV system from a two-layer model of stratified fluid. By constructing exact solutions of these derived equations, we explain a crucial nonlinear phenomena in atmospheric system, that is. the evolution of atmospheric blocking life cycles thoroughly. Based on the multiple scale expan-sion method, we propose a systemic approach for deriving variable coefficient equations from the original models. We also develop a direct method to con-struct Backlund transformations which connect the derived variable coefficient equations with their corresponding constant coefficient ones. Then taking ad-vantages of the Backlund transformations and the known results of the constant coefficient nonlinear equations, the solutions of the variable coefficient nonlinear systems can be generated. This direct method is also used in other chapters in this dissertation as a fundamental way to solve variable coefficient nonlinear equations.
In chapter 3, we derive a multiple vortex interaction model from the (2+1)-dimensional nonlinear inviscid nondissipative and equivalent barotropic vorticity equation in a beta plane by supposing four reasonable constraints (vortex local-ity, short range interaction, one-one interaction and total system). The classical Lie symmetries and conservation laws of the model are discussed. Exact so-lutions such as the vortex sources and Bessel vortex solutions are given, too. Furthermore, we numerically simulate the vortex interactions in the atmospheric systems without rotation based on the vortex interaction model. It is found that this model can generate several interaction patterns such as merging, separation, mutual orbiting and absorption. The results of these numerical simulations are well consistent with some known experiments and meteorologic observations. The model is also used to explain the interaction between a typhoon and a subtropical high.
In chapter 4, the direct method developed in chapter 2 is used to solve a (3+1)-dimensional variable coefficient nonlinear Schrodinger(NLS) equation and a coupled (1+1)-dimensional variable coefficient NLS system via constructing the Backlund transformations between the variable coefficient equations and the constant variable ones. Solitary wave solution and vector solitary wave solution are obtained for the two variable coefficient equations respectively, and the rela-tions between those variable coefficients under which the exact solutions exist are also revealed. These two variable coefficient nonlinear equations are theoretically meaningful in Bose-Einstein condensation and nonlinear optics. We also study a novel nonlinear equation, the resonant Davey-Stewartson(DS) equation, which is related to the NLS equation, and three types of exact solutions of the resonant DS equation are constructed via the classical Lie group theory.
Different with chapter 2,3 and 4, chapter 5 is devoted to developing a novel approximate method, nonsensitive homotopy approach, which is based on the homotopy analysis method. The nonsensitive homotopy approach builds linear or nonlinear homotopy relations between the hard-to-solve original model and a simplified model which has exact solutions, while some auxiliary parameters are also introduced to generate highly accurate approximate solutions. This ap-proach does not rely on small perturbation parameters. Compared with the known liomotopy analysis method, the nonsensitive homotopy approach shows that it is possible to introduce nonlinear homotopy relations, and it gives a non-sensitive principle to chose the reasonable auxiliary parameters. Furthermore, by constructing the nonsensitive quantities, an explicit process for choosing aux-iliary parameters is given. The nonsensitive homotopy approach is used to solve two nonlinear differential equations and to calculate the energy levels of several quantum anharmonic oscillators. The results of the calculations and the error analysis show the validity of the nonsensitive homotopy approach. This approach can be used to solve nonlinear differential equations approximately, and can also be used as an effective nonperturbative technique in physics.
The last chapter concerns the summary and discussion for the whole disser-tation, and the prospect for the further work is also discussed in this chapter.
本文第一章作为绪论部分概括地介绍了非线性科学的内涵、意义和研究现状，重点介绍了本论文涉及的一些非线性问题和非线性方程的数学及物理背景，概述了研究非线性数学物理问题的主要数学工具及其发展历程，同时阐明了本论文的主要工作
第二章从流体力学的基本模型β面上的无辐散正压位涡方程出发，推导出变系数KdV方程及变系数mKdV方程；从另一个研究分层流体的基本方程二层流体模型出发，推导出耦合变系数mKdV方程，然后通过构造这些变系数方程的严格解充分阐释了大气系统中一种重要的非线性现象—大气阻塞产生、发展和衰减的动力学机理。在这一章里，基于多重尺度展开法，我们提出了一个新的推导变系数近似方程的系统方法。同时，为了严格求解推导出的变系数方程，我们发展了一种在变系数方程和相应的常系数方程之间建立Backlund变换的直接方法。该直接方法作为求解变系数方程的一个基本方法还将应用于本论文的其他章节。
第三章通过施加四个合理化条件(单涡旋的局域性、短程相互作用、两体相互作用以及多涡旋的完整性)，将β面上的无辐散正压位涡方程转化为一个多涡旋相互作用模型。我们充分讨论了该模型的经典李对称和守恒律，给出了模型的几种严格解，包括涡旋源解和Bessel涡旋解。利用该模型，结合孤立子理论以及数值模拟方法，我们解释了大气系统中热带气旋之间一些重要的相互作用特征，包括气旋间互旋、两个气旋在临界距离处的合并或分离、强气旋对弱气旋的吸收，以及台风对副热带高压的环绕运动。
第四章利用本论文第一章发展的构造变系数方程与相应常系数方程之间Backlund变换的直接方法，严格求解了一个变系数3+1维NLS方程和一个耦合变系数1+1维NLS方程，给出了前者的孤波解和后者的向量孤波解，同时给出了这些严格解存在时变系数之间所要满足的关系。这两类方程在玻色一爱因斯坦凝聚和非线性光学领域都有重要的理论意义。同时，在这一章里，我们还利用经典李群法研究了一种和NLS方程相关的新型非线性方程一共振DS方程的对称性，并且得到了该方程的三类严格解。
与第二、三、四章主要处理具体非线性问题不同，第五章则花一章的篇幅着力于方法的研究。通过对现有非线性方程研究中经常使用的一些严格解方法及近似方法进行总结和提炼，我们在同伦分析方法的基础上提出了一种新的近似方法一不敏感性同伦方法。该方法通过建立线性或非线性同伦，将难以直接求解的原始模型和一个有严格解的简化模型联系起来，并且引入控制收敛的辅助参数，从而不依赖小的微扰参量即可得到原始模型的高精度近似解。与同伦分析法相比，一方面，该方法突破了线性同伦的局限，展示出完全可以用非线性同伦连接原始模型和简化模型；另一方面，明确给出了一个指导确定合理辅助参数的原则——不敏感性原理，并且通过构造不敏感量，给出了确定辅助参数以提高近似解精度的具体步骤。该方法的有效性通过非线性微分方程的求解和非简谐振子能量本征值的计算得到了检验，它既可以作为近似求解非线性方程的工具，也可作为物理学中一种有效的非微扰方法来使用。
第六章对论文的主要内容进行了总结和讨论，并且对未来工作可能的新发展和新方向做了展望。
Nonlinear problems in mathematics and physics are arousing great interests from scientists as the theories on linear problems have been well developed. Most of the problems in both basic science and engineering-oriented fields are nonlin-ear, and how to handle these complicated problems is a significant challenge for mathematicians and physicists. In this dissertation, two topics are discussed. On one hand, we study a few important nonlinear problems and obtain some meaningful results by using some known mathematical tools. On the other hand, we also propose a new approximate approach for nonlinear problems by abstract-ing and summarizing the basic idea which underlies some well-known exact and approximate mathematical methods.
Chapter 1 is an introduction which is devoted to reviewing the mathematical and physical backgrounds of some important nonlinear equations discussed in this dissertation. The significance and development of the nonlinear science are reviewed, too. The mathematical tools involved are introduced in this chapter, and wre also briefly report the main wrorks of this dissertation.
In chapter 2, we derive a variable coefficient KdV equation and a variable coefficient modified KdV(mKdV) equation from a nonlinear inviscid nondissipa-tive and equivalent barotropic vorticity equation in a beta plane. We also derive a coupled variable coefficient mKdV system from a two-layer model of stratified fluid. By constructing exact solutions of these derived equations, we explain a crucial nonlinear phenomena in atmospheric system, that is. the evolution of atmospheric blocking life cycles thoroughly. Based on the multiple scale expan-sion method, we propose a systemic approach for deriving variable coefficient equations from the original models. We also develop a direct method to con-struct Backlund transformations which connect the derived variable coefficient equations with their corresponding constant coefficient ones. Then taking ad-vantages of the Backlund transformations and the known results of the constant coefficient nonlinear equations, the solutions of the variable coefficient nonlinear systems can be generated. This direct method is also used in other chapters in this dissertation as a fundamental way to solve variable coefficient nonlinear equations.
In chapter 3, we derive a multiple vortex interaction model from the (2+1)-dimensional nonlinear inviscid nondissipative and equivalent barotropic vorticity equation in a beta plane by supposing four reasonable constraints (vortex local-ity, short range interaction, one-one interaction and total system). The classical Lie symmetries and conservation laws of the model are discussed. Exact so-lutions such as the vortex sources and Bessel vortex solutions are given, too. Furthermore, we numerically simulate the vortex interactions in the atmospheric systems without rotation based on the vortex interaction model. It is found that this model can generate several interaction patterns such as merging, separation, mutual orbiting and absorption. The results of these numerical simulations are well consistent with some known experiments and meteorologic observations. The model is also used to explain the interaction between a typhoon and a subtropical high.
In chapter 4, the direct method developed in chapter 2 is used to solve a (3+1)-dimensional variable coefficient nonlinear Schrodinger(NLS) equation and a coupled (1+1)-dimensional variable coefficient NLS system via constructing the Backlund transformations between the variable coefficient equations and the constant variable ones. Solitary wave solution and vector solitary wave solution are obtained for the two variable coefficient equations respectively, and the rela-tions between those variable coefficients under which the exact solutions exist are also revealed. These two variable coefficient nonlinear equations are theoretically meaningful in Bose-Einstein condensation and nonlinear optics. We also study a novel nonlinear equation, the resonant Davey-Stewartson(DS) equation, which is related to the NLS equation, and three types of exact solutions of the resonant DS equation are constructed via the classical Lie group theory.
Different with chapter 2,3 and 4, chapter 5 is devoted to developing a novel approximate method, nonsensitive homotopy approach, which is based on the homotopy analysis method. The nonsensitive homotopy approach builds linear or nonlinear homotopy relations between the hard-to-solve original model and a simplified model which has exact solutions, while some auxiliary parameters are also introduced to generate highly accurate approximate solutions. This ap-proach does not rely on small perturbation parameters. Compared with the known liomotopy analysis method, the nonsensitive homotopy approach shows that it is possible to introduce nonlinear homotopy relations, and it gives a non-sensitive principle to chose the reasonable auxiliary parameters. Furthermore, by constructing the nonsensitive quantities, an explicit process for choosing aux-iliary parameters is given. The nonsensitive homotopy approach is used to solve two nonlinear differential equations and to calculate the energy levels of several quantum anharmonic oscillators. The results of the calculations and the error analysis show the validity of the nonsensitive homotopy approach. This approach can be used to solve nonlinear differential equations approximately, and can also be used as an effective nonperturbative technique in physics.
The last chapter concerns the summary and discussion for the whole disser-tation, and the prospect for the further work is also discussed in this chapter.
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