两类数学物理方程的整体解
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摘要
本文主要研究两类数学物理方程整体解的性态,即Minkowski时空中相对论膜的非线性动力学和水波理论中Nwogu型Boussinesq方程模型的孤立波。主要内容由以下几章组成。
第一章对所考虑的两类数学物理方程的研究现状做一个简单介绍,并阐述本文要研究的问题,叙述我们得到的主要结果。
第二章研究Minkowski空间中相对论膜的非线性动力学。通过变分法和几何方法,推导出Minkowski空间R1+n(n≥3)中相对论膜的运动方程。它是(1+2)维拟线性双曲型方程组,具有很多重要的性质,如非严格双曲性、常重特征、线性退化性和强零条件等；研究还发现,方程的平面波解都是类光极值子流形；反之,除了一类特殊的类光极值子流形外,其余所有的类光极值子流形都是方程的平面波解。
第三章进一步研究相对论膜的非线性动力学。主要研究Minkowski空间R1+n(n≥3)中,我们所推导的相对论膜的运动方程,与以往所给出的经典方程之间的区别和联系。我们证明它们是等价的,并且从Noether定理角度重新认识此方程。同时,对于时空中相对论弦的情形给出类似相应的讨论。
第四章研究水波理论中的Nwogu型Boussinesq方程模型的孤立波和周期波。此模型包含一个独立参数,这个参数与在不同水深时相应的水平速度有关。本章利用平面动力系统的分支方法,定性地研究此方程模型孤立波和周期波的存在条件。我们发现,在这个模型中,会出现一类新的尖峰波解—尖峰型周期波。
第五章进一步研究上述Nwogu型Boussinesq方程模型,考察相向而行的孤立波的对撞问题。通过摄动方法,首先得到了方程的近似解。其次分析孤立波对撞的力学特征。由于模型包含独立参数,着重分析这个独立参数对于对撞的相移和最大波幅的影响；并将所得结果与经典可积的Boussinesq方程进行了比较。
The thesis concerns with the properties of global solutions for two kinds of mathemat-ical physical equations, precisely speaking, studies the nonlinear dynamics of relativistic membrane in the Minkowski space and solitary waves of the Nwogu's Boussinesq equation in water wave theory. It is organized as follows.
Chapter 1 briefly recalls the present situation of the study on the two kinds of mathe-matical physical equations. The central problems under consideration and the main results obtained are stated.
Chapter 2 concentrates on the nonlinear dynamics of relativistic membrane. By variational method and geometrical method, the system of equations for the relativistic membrane in the Minkowski space R1+n (n≥3) is derived. It can also be reduced to a (1+2)-dimensional quasilinear hyperbolic system and possesses many important properties such as non-strict hyperbolicity, constant multiplicity of eigenvalues, linear degeneracy of all characteristic fields, strong null condition, etc. An interesting phenomenon is found and proved, that is all plane wave solutions to this system are light-like extremal sub-manifolds and vice versa except for a type of special solution.
Chapter 3 furthermore studies the nonlinear dynamics of relativistic membrane and mainly focus on the relationships between the equations for the relativistic membrane moving in the Minkowski space R1+n (n≥3) derived by us and that in canonical form in literature. The equivalence between them is proved and another geometric explanation about it through Noether's second theorem is also given. Moreover, for the motion of relativistic string in the Minkowski space, the similar argument about the equivalence is also presented.
Chapter 4 investigates the solitary waves and periodic waves of Nwogu's Boussinesq equation in water wave theory. This model contains one independent parameter which is related to that the horizontal velocities at what level are chosen as the horizontal velocity variables. We employ the bifurcation method to qualitatively analyze the existence conditions of solitary waves and periodic waves for this model. Meanwhile, we find that the cusp periodic waves appear.
Chapter 5 pays attention to the head-on collsions between two solitary waves of the Nwogu's Boussinesq equation. We first apply the perturbation method to this model and derive an approximate solution. Then the mechanics of the head-on collision, especially the impacts of the independent parameter on the phase shifts and the maximum run-up amplitude of two colliding waves is investigated. Comparison between our results and that of the integrable classical Boussinesq equation is also given
第一章对所考虑的两类数学物理方程的研究现状做一个简单介绍,并阐述本文要研究的问题,叙述我们得到的主要结果。
第二章研究Minkowski空间中相对论膜的非线性动力学。通过变分法和几何方法,推导出Minkowski空间R1+n(n≥3)中相对论膜的运动方程。它是(1+2)维拟线性双曲型方程组,具有很多重要的性质,如非严格双曲性、常重特征、线性退化性和强零条件等；研究还发现,方程的平面波解都是类光极值子流形；反之,除了一类特殊的类光极值子流形外,其余所有的类光极值子流形都是方程的平面波解。
第三章进一步研究相对论膜的非线性动力学。主要研究Minkowski空间R1+n(n≥3)中,我们所推导的相对论膜的运动方程,与以往所给出的经典方程之间的区别和联系。我们证明它们是等价的,并且从Noether定理角度重新认识此方程。同时,对于时空中相对论弦的情形给出类似相应的讨论。
第四章研究水波理论中的Nwogu型Boussinesq方程模型的孤立波和周期波。此模型包含一个独立参数,这个参数与在不同水深时相应的水平速度有关。本章利用平面动力系统的分支方法,定性地研究此方程模型孤立波和周期波的存在条件。我们发现,在这个模型中,会出现一类新的尖峰波解—尖峰型周期波。
第五章进一步研究上述Nwogu型Boussinesq方程模型,考察相向而行的孤立波的对撞问题。通过摄动方法,首先得到了方程的近似解。其次分析孤立波对撞的力学特征。由于模型包含独立参数,着重分析这个独立参数对于对撞的相移和最大波幅的影响；并将所得结果与经典可积的Boussinesq方程进行了比较。
The thesis concerns with the properties of global solutions for two kinds of mathemat-ical physical equations, precisely speaking, studies the nonlinear dynamics of relativistic membrane in the Minkowski space and solitary waves of the Nwogu's Boussinesq equation in water wave theory. It is organized as follows.
Chapter 1 briefly recalls the present situation of the study on the two kinds of mathe-matical physical equations. The central problems under consideration and the main results obtained are stated.
Chapter 2 concentrates on the nonlinear dynamics of relativistic membrane. By variational method and geometrical method, the system of equations for the relativistic membrane in the Minkowski space R1+n (n≥3) is derived. It can also be reduced to a (1+2)-dimensional quasilinear hyperbolic system and possesses many important properties such as non-strict hyperbolicity, constant multiplicity of eigenvalues, linear degeneracy of all characteristic fields, strong null condition, etc. An interesting phenomenon is found and proved, that is all plane wave solutions to this system are light-like extremal sub-manifolds and vice versa except for a type of special solution.
Chapter 3 furthermore studies the nonlinear dynamics of relativistic membrane and mainly focus on the relationships between the equations for the relativistic membrane moving in the Minkowski space R1+n (n≥3) derived by us and that in canonical form in literature. The equivalence between them is proved and another geometric explanation about it through Noether's second theorem is also given. Moreover, for the motion of relativistic string in the Minkowski space, the similar argument about the equivalence is also presented.
Chapter 4 investigates the solitary waves and periodic waves of Nwogu's Boussinesq equation in water wave theory. This model contains one independent parameter which is related to that the horizontal velocities at what level are chosen as the horizontal velocity variables. We employ the bifurcation method to qualitatively analyze the existence conditions of solitary waves and periodic waves for this model. Meanwhile, we find that the cusp periodic waves appear.
Chapter 5 pays attention to the head-on collsions between two solitary waves of the Nwogu's Boussinesq equation. We first apply the perturbation method to this model and derive an approximate solution. Then the mechanics of the head-on collision, especially the impacts of the independent parameter on the phase shifts and the maximum run-up amplitude of two colliding waves is investigated. Comparison between our results and that of the integrable classical Boussinesq equation is also given
引文
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