玻色—爱因斯坦凝聚和磁性薄膜中孤子特性研究
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摘要
由于孤子具有独特的特性和应用潜力,孤子动力学的研究是当今非线性科学研究的前沿课题之一。本论文在总结和分析国内外研究现状及介绍理论研究方法的基础上,从理论上研究了波色-爱因斯坦凝聚(BEC)中的孤子和磁性薄膜中的孤子的特性,得到了若干创新性的结论。论文由三个部分组成。
论文的第一部分介绍了BEC中的孤子和磁性薄膜中的微波静磁包络(MME)孤子传输和演化所遵循的非线性薛定谔方程(NLSE);介绍了用于分析NLSE的孤子解的理论工具-变分法;讨论了NLSE的数值解法-分步傅立叶法。
无论是从理解孤子系统的特性还是从他们的实际应用的角度看,非线性系统中孤子的稳定性是关键的特性,因此,研究BEC孤子系统的稳定条件及控制因素是重要的。论文的第二部分运用变分法和分步傅立叶法研究了BEC孤子系统的物理行为。这些研究包括以下三个方面。
通过在常规势阱中引入局部畸变,系统研究了BEC孤子的演化。研究表明BEC孤子系统的行为极大地依赖于势阱的局部畸变;合适的畸变不仅能被用来维护孤子的稳定,而且可通过适当的方式被用来调控BEC孤子;畸变对BEC孤子演化的影响也与囚禁电势的形式、BEC孤子系统的参数以及初始条件有关。
借用等效势的概念研究了单个和耦合BEC孤子的演化特征。通过变分法得到了外部囚禁电势的等效势和耦合BEC孤子间相互作用的等效势的表达式;揭示了等效势对BEC孤子运动的影响和耦合BEC孤子间相互作用的特点,得到了耦合BEC孤子系统出现定态和发生自陷的临界条件和相关的判据;发现了耦合BEC孤子间原子迁移的特点和影响因素。
研究了原子间三体作用对BEC孤子演化的影响,研究表明较小的三体作用的效应会极大地影响BEC孤子系统的行为。通过在圆柱体对称的磁阱中的某一方向引入光格电势,研究了二维“饼”状BEC孤子的演化,发现光格不仅使BEC孤子在该方向上趋向稳定,而且通过相互耦合作用也能影响孤子在其它方向上的稳定性,从而使二维“饼”状BEC孤子整体的稳定性增强。
论文的第三部分通过引入高阶色散项,采用变分法,研究了窄MME孤子的传输特性.。理论上证实在合适的条件下,具有高阶色散的媒质中可以存在稳定的MME孤子,得到了高阶色散效应下MME亮孤子参数的演化和传输速度的解析表达式;发现了高阶色散效应可以改变MME孤子包络的形状,而MME孤子的演化与相位无关;发现了较大的三阶色散能导致MME孤子三峰分裂,提出了通过引入初始相位调制来抑制MME孤子裂变的方法。
Solitons show unique properties and high potential for applications and, hence, have been one of the most exciting topics in modern nonlinear science. This thesis is devoted to the theoretical investigation of soliton dynamics in Bose-Einstein Condensates (BECs) and magnetic thin films. The work is based on the nonlinear Schr(?)dinger equation (NLSE) and the use of the variational appraoach and the split-step Fourier method.
The thesis consists of three main parts. In the first part, the nonlinear Schr?dinger equation (NLSE) is briefly introduced. This equation can be used to describe the evolution dynamics of both the solitons in the BEC systems and the microwave magnetic envelope (MME) solitons in magnetic films. The variational approach and the Split-step Fourier method are also introduced. The first one is a typical theoretical tool for the analyses of the soliton solution of the NLSE under restricted conditions. The later one is often used for the numerical simulation of the NLSE.
From both the fundamental and practical points of view, the stability of solitons in a nonlinear system is critical. The second part of the thesis is devoted to the stability of the BEC solitons and its control methods as well as the physical behaviors of BEC solitons under a variety of different trapping potentials. Three main works are as follows.
One work is on the dynamics of one and several BEC solitons in a typical trap that is perturbed by a local impurity. It is found that the dynamics of the solitons in such a trap strongly depends on the properties of the trapping potential, the parameters of the BEC system, and the other initial conditions. It is also found that an appropriate potential perturbation can not only enhance the stability of the solitons, but also can be used to control the evolution of the solitons.
The second work is on the evolution of single and coupled BEC solitons. In this work, the concept of effective potential was introduced to the analyses of the BEC solitons. The effective potential for the external trapping potential and that from the interaction of coupled solitons were obtained. From these effective potentials, one can know the evolution dynamics of the BEC solitons and the interaction of the coupled solitons. One can also know the thresholds for the stationary and self-trapping states of the solitons. In addition, one can know the properties for the atom transfer between coupled solitons.
The effects of three-body interactions on BEC soliton dynamics are also studied. It is found that even a weak three-body interaction can lead to a significant chang in soliton dynamics. The dynamics of 2D cake-shaped solitons was also studied. This work was done for a cylindral potential modified by an optical lattice. It is found the introduction of an optical lattice can not only enhance the stability of the solitons in the direction of the opdtical lattice, but also improve the stability in other directions.
The third part focuses on the MME solitons in magnetic thin films. The propagation dynamics of narrow MME solitons was studied. This work was performed with the use of a NLSE that contained high-order dispersion terms. It is found that, under certain conditions, the nonlinear systems with high-order dispersion can support narrow MME solitons. It is also found that, with high order dispersion, the soliton dynamics is independent of the soliton initial phase. Strong third-order dispersion effect can also lead to the break-up of an MME soliton, but this break-up can be controlled through the introduction of an initial phase.
论文的第一部分介绍了BEC中的孤子和磁性薄膜中的微波静磁包络(MME)孤子传输和演化所遵循的非线性薛定谔方程(NLSE);介绍了用于分析NLSE的孤子解的理论工具-变分法;讨论了NLSE的数值解法-分步傅立叶法。
无论是从理解孤子系统的特性还是从他们的实际应用的角度看,非线性系统中孤子的稳定性是关键的特性,因此,研究BEC孤子系统的稳定条件及控制因素是重要的。论文的第二部分运用变分法和分步傅立叶法研究了BEC孤子系统的物理行为。这些研究包括以下三个方面。
通过在常规势阱中引入局部畸变,系统研究了BEC孤子的演化。研究表明BEC孤子系统的行为极大地依赖于势阱的局部畸变;合适的畸变不仅能被用来维护孤子的稳定,而且可通过适当的方式被用来调控BEC孤子;畸变对BEC孤子演化的影响也与囚禁电势的形式、BEC孤子系统的参数以及初始条件有关。
借用等效势的概念研究了单个和耦合BEC孤子的演化特征。通过变分法得到了外部囚禁电势的等效势和耦合BEC孤子间相互作用的等效势的表达式;揭示了等效势对BEC孤子运动的影响和耦合BEC孤子间相互作用的特点,得到了耦合BEC孤子系统出现定态和发生自陷的临界条件和相关的判据;发现了耦合BEC孤子间原子迁移的特点和影响因素。
研究了原子间三体作用对BEC孤子演化的影响,研究表明较小的三体作用的效应会极大地影响BEC孤子系统的行为。通过在圆柱体对称的磁阱中的某一方向引入光格电势,研究了二维“饼”状BEC孤子的演化,发现光格不仅使BEC孤子在该方向上趋向稳定,而且通过相互耦合作用也能影响孤子在其它方向上的稳定性,从而使二维“饼”状BEC孤子整体的稳定性增强。
论文的第三部分通过引入高阶色散项,采用变分法,研究了窄MME孤子的传输特性.。理论上证实在合适的条件下,具有高阶色散的媒质中可以存在稳定的MME孤子,得到了高阶色散效应下MME亮孤子参数的演化和传输速度的解析表达式;发现了高阶色散效应可以改变MME孤子包络的形状,而MME孤子的演化与相位无关;发现了较大的三阶色散能导致MME孤子三峰分裂,提出了通过引入初始相位调制来抑制MME孤子裂变的方法。
Solitons show unique properties and high potential for applications and, hence, have been one of the most exciting topics in modern nonlinear science. This thesis is devoted to the theoretical investigation of soliton dynamics in Bose-Einstein Condensates (BECs) and magnetic thin films. The work is based on the nonlinear Schr(?)dinger equation (NLSE) and the use of the variational appraoach and the split-step Fourier method.
The thesis consists of three main parts. In the first part, the nonlinear Schr?dinger equation (NLSE) is briefly introduced. This equation can be used to describe the evolution dynamics of both the solitons in the BEC systems and the microwave magnetic envelope (MME) solitons in magnetic films. The variational approach and the Split-step Fourier method are also introduced. The first one is a typical theoretical tool for the analyses of the soliton solution of the NLSE under restricted conditions. The later one is often used for the numerical simulation of the NLSE.
From both the fundamental and practical points of view, the stability of solitons in a nonlinear system is critical. The second part of the thesis is devoted to the stability of the BEC solitons and its control methods as well as the physical behaviors of BEC solitons under a variety of different trapping potentials. Three main works are as follows.
One work is on the dynamics of one and several BEC solitons in a typical trap that is perturbed by a local impurity. It is found that the dynamics of the solitons in such a trap strongly depends on the properties of the trapping potential, the parameters of the BEC system, and the other initial conditions. It is also found that an appropriate potential perturbation can not only enhance the stability of the solitons, but also can be used to control the evolution of the solitons.
The second work is on the evolution of single and coupled BEC solitons. In this work, the concept of effective potential was introduced to the analyses of the BEC solitons. The effective potential for the external trapping potential and that from the interaction of coupled solitons were obtained. From these effective potentials, one can know the evolution dynamics of the BEC solitons and the interaction of the coupled solitons. One can also know the thresholds for the stationary and self-trapping states of the solitons. In addition, one can know the properties for the atom transfer between coupled solitons.
The effects of three-body interactions on BEC soliton dynamics are also studied. It is found that even a weak three-body interaction can lead to a significant chang in soliton dynamics. The dynamics of 2D cake-shaped solitons was also studied. This work was done for a cylindral potential modified by an optical lattice. It is found the introduction of an optical lattice can not only enhance the stability of the solitons in the direction of the opdtical lattice, but also improve the stability in other directions.
The third part focuses on the MME solitons in magnetic thin films. The propagation dynamics of narrow MME solitons was studied. This work was performed with the use of a NLSE that contained high-order dispersion terms. It is found that, under certain conditions, the nonlinear systems with high-order dispersion can support narrow MME solitons. It is also found that, with high order dispersion, the soliton dynamics is independent of the soliton initial phase. Strong third-order dispersion effect can also lead to the break-up of an MME soliton, but this break-up can be controlled through the introduction of an initial phase.
引文
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