对双耦合非线性薛定谔方程畸形波解的分析和研究

发布时间：2018-04-27 00:06

本文选题：广义达布变换 + 高阶畸形波解　；参考：《北京邮电大学》2017年硕士论文

【摘要】：畸形波是一类危害水平高的波。它发生的机制以及发生概率还不明确,普遍以为调制不稳定性可能会产生畸形波,但并不是所有的调制不稳定性都能产生畸形波。薛定谔方程的准确度只可依赖试验来进行检测,为量子学中的假设,并且联合了物质波的观念和波动方程的观念。可以用它来对微观粒子的运动进行描述。不仅在流体力学和等离子物理等范畴中,薛定谔方程具有广泛的应用,非线性薛定谔方程还可以用来描述光学和海森堡铁磁自旋链问题中的非线性动力特征,这也是本文研究的重点。在本文中主要利用广义达布变换来求解两类非线性薛定谔方程中的高阶畸形波解,达布变换是一种可由一个种子解不断得到新解的手段,与规范变换密不可分,它是一种特殊的规范变换,而规范变换可概括为一个方程在一组变换中可保持方程的形式不变形,该变换则为规范变换。其不能直接用来求得高阶畸形波解,因此想得到高阶的畸形波解,需通过极限知识将原有的达布变换进行改进推广。在利用广义达布变换时,需要用到Lax对的解,然而Lax对是非线性偏微分方程组,对其的求解需借助数学软件或类似待定系数求解法来完成,在本文中首先通过一矩阵变换将原有的Lax对转变成线性偏微分方程组,继而通过简单的特征值和特征向量的代数知识可求得Lax更多形式的解。通过矩阵变换将Lax对转变成线性偏微分方程组的方法不仅可以运用在(1+1)维非线性薛定谔模型中,还可运用在(2+1)维非线性薛定谔模型中和变系数的mKdV-NLS方程中。在一些物理情况下,在多个领域中,具有不同频率或偏振的光的传播模型可由耦合方程来描述。在光纤中,其中一个例子是一个具有自相位调制,交叉相位调制和四波混频的耦合的非线性薛定谔系统,也就是本文第三章要讨论的一个多模光纤中的(1+1)维耦合性薛定谔方程。在本文第五章中,讨论了另外两类非线性(1+1)维耦合薛定谔方程的畸形波解,在第六章中,探究了(1+1)维耦合变系数的mKdV-NLS方程解的性质。非线性薛定谔方程不仅能用来描述流体、玻色-爱因斯坦和光纤维中的非线性动力特征,也可以用来描述海森堡-铁磁自旋链中的非线性动力特征。本文通过对一个多模光纤中的(1+1)维耦合性薛定谔方程和海森堡自旋链中的(2+1)维非线性薛定谔方程进行具体的分析来探究畸形波解的性质以及通过调制不稳定性分析来探究畸形波产生的原因和机制。
[Abstract]:The abnormal wave is a kind of wave with high hazard level. The mechanism and probability of its occurrence are not clear. It is generally believed that modulation instability may produce abnormal waves, but not all modulation instability can produce abnormal waves. The accuracy of Schrodinger equation can only be detected by experiments. It is a hypothesis in quantum science and combines the idea of matter wave with that of wave equation. It can be used to describe the motion of microscopic particles. The Schrodinger equation is widely used not only in hydrodynamics and plasma physics, but also in the nonlinear Schrodinger equation, which can be used to describe the nonlinear dynamic characteristics of the optical and Heisenberg ferromagnetic spin chain problems. This is also the focus of this paper. In this paper, we mainly use the generalized Daber transform to solve the higher-order wave solutions of two kinds of nonlinear Schrodinger equations. The Daber transform is a means to obtain new solutions from a seed solution, which is closely related to the gauge transformation. It is a special gauge transformation, which can be summarized as a set of transformations in which an equation can be preserved in the form of a set of equations, and the transformation is a gauge transformation. It can not be directly used to obtain the higher-order deformable wave solutions. Therefore, if we want to obtain the higher-order deformational wave solutions, we need to improve and generalize the original Darboux transform through the limit knowledge. The solution of the Lax pair is needed when using the generalized Darboux transform. However, the Lax pair is a system of nonlinear partial differential equations. The solution of the Lax pair needs to be completed by means of mathematical software or similar undetermined coefficient solution method. In this paper, the original Lax pairs are first transformed into linear partial differential equations by a matrix transformation, and then more forms of Lax solutions can be obtained by simple algebraic knowledge of eigenvalues and Eigenvectors. The method of transforming Lax pairs into linear partial differential equations by matrix transformation can be used not only in the nonlinear Schrodinger model of 1D, but also in the nonlinear Schrodinger model of 21D and the mKdV-NLS equation with variable coefficients. In some physical cases, in many fields, the propagation model of light with different frequencies or polarization can be described by coupling equations. One example of a fiber is a nonlinear Schrodinger system with self-phase modulation, cross-phase modulation and four-wave mixing coupling. In the third chapter, we discuss the Schrodinger equation in multimode fiber. In chapter 5, we discuss the deformable wave solutions of the other two kinds of nonlinear coupled Schrodinger equations. In chapter 6, we study the properties of the solutions of the mKdV-NLS equations with the coupling coefficients of 1 1). The nonlinear Schrodinger equation can be used not only to describe the nonlinear dynamic characteristics of fluid, Bose-Einstein and optical fibers, but also to describe the nonlinear dynamic characteristics of Heisenberg ferromagnetic spin chain. In this paper, we analyze the Schrodinger equation in a multimode fiber and the nonlinear Schrodinger equation in Heisenberg spin chain in order to explore the properties of the deformable wave solutions and the modulation instability. Sex analysis to explore the causes and mechanisms of abnormal waves.
【学位授予单位】：北京邮电大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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