高阶非自治薛定谔型方程的畸形波动力学和基带调制不稳定性
发布时间：2019-02-24 10:14
【摘要】：随着现代科学技术的蓬勃发展,非线性科学成为了近代科学技术发展的一个新标志,非线性科学渗透到各个学科和领域,如地球物理、海洋和气象预测、流体力学、非线性光学、等离子体等。孤子作为非线性科学中的一个重要分支,拥有着宽广的应用性,受到了广大学者的重视。非线性科学研究过程中的一项非常重要的工作,就是是非线性问题的求解。非线性偏微分方程存在许多解。其中,大部分解都能够较好的描述实际物理问题,而这类非线性方程大多都是变系数方程,想要得到非线性方程的精确解是十分困难的。目前还仅仅停留在求数值解的水平。变系数薛定谔方程及其变形形式被认为是应用性最广泛的数学理论模型,它可以由麦克斯韦方程组出发导出,在众多邻域都有着极其重要的地位。至今为止,已经有许多种求解常系数非线性薛定谔的方程的方法,而对于变系数非线性薛定谔方程的精确孤子解的求法,还有待进一步的研究和讨论。本文主要围绕高阶非自治薛定谔型方程的畸形波动力学和基带调制不稳定性展开研究,内容主要包括以下几个方面:(1)介绍孤子的起源、发展及研究现状,孤子的形成机制,以及若干种求解孤子的有效方法。(2)采用相似变换的方法,研究了高阶变系数非线性薛定谔方程(vc-HNLS)在连续波背景下的非自治孤子的解的情况,以及三阶色散系数对孤子的压缩效应,其中包括常数、三角函数、指数函数、和一次函数对孤子的影响。(3)当三阶色散系数为特殊的周期结构,并且选取合适的调制振幅和调制频率时,Peregeine Combs结构出现。我们还分析了Peregeine Combs结构空间性质,研究中发现当调制振幅时,Peregeine Combs结构转化成Peregeine Walls。(4)光波脉冲宽度达到飞秒量级时,需要考虑高阶效应(三阶色散、非线性色散、自徒峭以及增益(损耗)等)。这时,非线性薛定谔方程变成高阶非线性Hirota方程。本文简单介绍了调制不稳定性,并且利用线性稳定性分析的方法,研究了高阶变系数Hirota方程的调制不稳定。(5)对本文的研究结果进行了简要的概括总结,并对研究工作做了一些展望。
[Abstract]:With the vigorous development of modern science and technology, nonlinear science has become a new symbol of the development of modern science and technology. Nonlinear science has penetrated into various disciplines and fields, such as geophysics, marine and meteorological prediction, hydrodynamics, Nonlinear optics, plasma, etc. As an important branch of nonlinear science, soliton has wide application and has been paid attention to by many scholars. A very important work in the process of nonlinear scientific research is the solving of nonlinear problems. There are many solutions to nonlinear partial differential equations. Among them, most of the solutions can describe the practical physical problems well, but most of these nonlinear equations are variable coefficient equations, so it is very difficult to obtain the exact solutions of nonlinear equations. At present, it is only at the level of numerical solution. The variable coefficient Schrodinger equation and its deformation form are considered to be the most widely applied mathematical theoretical models. They can be derived from Maxwell equations and play an extremely important role in many neighborhoods. Up to now, there are many methods to solve the nonlinear Schrodinger equations with constant coefficients, but the exact soliton solutions of the nonlinear Schrodinger equations with variable coefficients need to be further studied and discussed. In this paper, the deformational wave mechanics and baseband modulation instability of higher order nonautonomous Schrodinger equation are studied. The main contents are as follows: (1) the origin, development and research status of soliton are introduced. The formation mechanism of solitons and some effective methods for solving solitons. (2) the solution of the nonlinear Schrodinger equation (vc-HNLS) with high order variable coefficients under the continuous wave background is studied by using the method of similarity transformation. And the squeezing effect of third-order dispersion coefficient on soliton, including the influence of constant, trigonometric function, exponential function, and first-order function on soliton. (3) when the third-order dispersion coefficient is a special periodic structure, The, Peregeine Combs structure appears when the appropriate modulation amplitude and frequency are selected. We also analyze the spatial properties of Peregeine Combs structures. It is found that the higher-order effects (third-order dispersion and nonlinear dispersion) need to be considered when the amplitude modulation of, Peregeine Combs is converted to Peregeine Walls. (_ 4) when the pulse width reaches the femtosecond order. Self kurtosis and gain (loss). In this case, the nonlinear Schrodinger equation becomes the higher order nonlinear Hirota equation. In this paper, modulation instability is briefly introduced, and the modulation instability of high-order variable coefficient Hirota equation is studied by means of linear stability analysis. (5) the results of this paper are summarized briefly. Some prospects for the research work are also given.
【学位授予单位】：华北电力大学(北京)
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
本文编号：2429454
[Abstract]:With the vigorous development of modern science and technology, nonlinear science has become a new symbol of the development of modern science and technology. Nonlinear science has penetrated into various disciplines and fields, such as geophysics, marine and meteorological prediction, hydrodynamics, Nonlinear optics, plasma, etc. As an important branch of nonlinear science, soliton has wide application and has been paid attention to by many scholars. A very important work in the process of nonlinear scientific research is the solving of nonlinear problems. There are many solutions to nonlinear partial differential equations. Among them, most of the solutions can describe the practical physical problems well, but most of these nonlinear equations are variable coefficient equations, so it is very difficult to obtain the exact solutions of nonlinear equations. At present, it is only at the level of numerical solution. The variable coefficient Schrodinger equation and its deformation form are considered to be the most widely applied mathematical theoretical models. They can be derived from Maxwell equations and play an extremely important role in many neighborhoods. Up to now, there are many methods to solve the nonlinear Schrodinger equations with constant coefficients, but the exact soliton solutions of the nonlinear Schrodinger equations with variable coefficients need to be further studied and discussed. In this paper, the deformational wave mechanics and baseband modulation instability of higher order nonautonomous Schrodinger equation are studied. The main contents are as follows: (1) the origin, development and research status of soliton are introduced. The formation mechanism of solitons and some effective methods for solving solitons. (2) the solution of the nonlinear Schrodinger equation (vc-HNLS) with high order variable coefficients under the continuous wave background is studied by using the method of similarity transformation. And the squeezing effect of third-order dispersion coefficient on soliton, including the influence of constant, trigonometric function, exponential function, and first-order function on soliton. (3) when the third-order dispersion coefficient is a special periodic structure, The, Peregeine Combs structure appears when the appropriate modulation amplitude and frequency are selected. We also analyze the spatial properties of Peregeine Combs structures. It is found that the higher-order effects (third-order dispersion and nonlinear dispersion) need to be considered when the amplitude modulation of, Peregeine Combs is converted to Peregeine Walls. (_ 4) when the pulse width reaches the femtosecond order. Self kurtosis and gain (loss). In this case, the nonlinear Schrodinger equation becomes the higher order nonlinear Hirota equation. In this paper, modulation instability is briefly introduced, and the modulation instability of high-order variable coefficient Hirota equation is studied by means of linear stability analysis. (5) the results of this paper are summarized briefly. Some prospects for the research work are also given.
【学位授予单位】：华北电力大学(北京)
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
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