应用符号计算研究光纤等领域中非线性方程的孤子解及畸形波解

发布时间：2018-07-02 15:14

本文选题：孤子 + 畸形波　；参考：《北京邮电大学》2017年博士论文

【摘要】：非线性发展方程可以用来描述光纤、Heisenberg铁磁体、流体以及等离子体等领域中的一些非线性现象。非线性发展方程中存在许多有理解,如孤子解、畸形波解等。孤子的产生源于非线性效应与色散效应的平衡。而作为一个在时间和空间局域化的有理解,Peregrine孤子可以作为畸形波的数学模型。本文基于一些非线性薛定谔类方程,并且利用符号计算,解析研究了光纤等领域中孤子和畸形波的性质。本文的研究内容主要有:(1)研究了一个2+1维的变系数耦合薛定谔方程。首先,通过寻找合适的有理变换将方程转化为双线性形式,进而得到了方程的明单孤子解和明双孤子解。根据所得到的孤子解,结合模拟的图像,分析了单孤子的传输和双孤子的碰撞等性质。(2)研究了一个四阶变系数薛定谔方程。在方程的可积条件下,利用Hirota双线性方法得到了该方程的暗单孤子解和暗双孤子解的表达式。基于得到的孤子解,模拟了单孤子传输和双孤子碰撞的图像,并分析了方程的一些物理参数对单孤子的传输和双孤子的碰撞的影响。(3)研究了一个变系数Kundu-Eckhaus方程。首先,利用该方程的Lax对,在其Darboux变换的基础上构造了广义的Darboux变换。然后,分别得到了该方程的一阶畸形波解和二阶畸形波解。最后,结合图像模拟,解析研究了方程中的非线性色散项对一阶畸形波和二阶畸形波的性质的影响。(4)研究了一个广义的非自治非线性方程。(a)在方程的可积条件下,利用合适的变换得到了该方程的双线性形式,进而得到了该方程的明单孤子解和明双孤子解。并且结合模拟的图像,研究了方程的系数对单孤子的传输和双孤子的碰撞产生的影响。另外,借助于分步Fourier方法,研究了孤子在有限初始扰动下的稳定性。(b)在方程的可积条件下,我们在Darboux变换的基础上构造了广义的Darboux变换,然后分别得到了该方程的一阶畸形波解和二阶畸形波解,并结合图像分析和研究了一阶畸形波和二阶畸形波的性质。(5)分别研究了一个常系数离散Ablowitz-Ladik方程和一个变系数离散Ablowitz-Ladik方程。首先,利用Hirota双线性方法分别得到了常系数离散Ablowitz-Ladik方程的暗孤子解的表达式和变系数离散Ablowitz-Ladik 方程的明孤子解的表达式。其次,对单孤子传输和双孤子碰撞进行了图像模拟,解析研究了单孤子传输的稳定性及双孤子碰撞的性质。(6)研究了一个耦合的三、五阶非线性薛定谔方程。首先,利用Hirota双线性方法,得到了该方程的明-明孤子解。然后结合模拟的图像,观察到了双孤子之间几种不同形式的碰撞:迎面碰撞、追赶碰撞以及束缚态等。(7)研究了一个变系数非线性系统。首先,在Darboux变换的基础上,构造了该系统的广义Darboux变换。然后,分别得到了系统的一阶畸形波解和二阶畸形波解。最后,利用图像模拟,分析了系统的参数对一阶畸形波和二阶畸形波的影响。
[Abstract]:The nonlinear evolution equation can be used to describe some nonlinear phenomena in the fields of optical fiber Heisenberg ferromagnet, fluid and plasma. There are many understanding in nonlinear evolution equation, such as soliton solution, malformed wave solution and so on. The soliton comes from the balance of nonlinear effect and dispersion effect. As an understanding Peregrine soliton localized in time and space, it can be used as a mathematical model of deformities. In this paper, based on some nonlinear Schrodinger class equations and symbolic computation, the properties of solitons and malformed waves in optical fiber and other fields are analytically studied. The main contents of this paper are as follows: (1) A 21 dimensional coupled Schrodinger equation with variable coefficients is studied. First of all, the equation is transformed into bilinear form by searching for proper rational transformation, and the open soliton solution and the open double soliton solution of the equation are obtained. Based on the obtained soliton solutions and the simulated images, the properties of the propagation of single soliton and the collision of double solitons are analyzed. (2) A fourth-order Schrodinger equation with variable coefficients is studied. Under the integrable condition of the equation, the expressions of the dark single soliton solution and the dark double soliton solution of the equation are obtained by using Hirota bilinear method. Based on the obtained soliton solution, the images of single soliton propagation and double soliton collision are simulated, and the effects of some physical parameters of the equation on the single soliton propagation and double soliton collision are analyzed. (3) A variable coefficient Kundu-Eckhaus equation is studied. Firstly, the generalized Darboux transformation is constructed on the basis of its Darboux transformation by using the lax pair of the equation. Then, the first and second order wave solutions of the equation are obtained, respectively. Finally, the influence of the nonlinear dispersion term in the equation on the properties of the first and second order deformities is analytically studied with image simulation. (4) A generalized nonautonomous nonlinear equation. (a) is studied under the integrable condition of the equation. The bilinear form of the equation is obtained by proper transformation, and the open soliton solution and the open double soliton solution of the equation are obtained. The effects of the coefficients of the equation on the propagation of single soliton and the collision of two solitons are studied. In addition, with the help of the step Fourier method, we study the stability of solitons under finite initial perturbations under the integrable condition of the equation. We construct the generalized Darboux transformation on the basis of the Darboux transformation. Then, the first and second order wave solutions of the equation are obtained, respectively. The properties of first-order and second-order deformities are analyzed and studied. (5) A constant coefficient discrete Ablowitz-Ladik equation and a variable coefficient discrete Ablowitz-Ladik equation are studied respectively. Firstly, by using Hirota bilinear method, the expressions of dark soliton solutions for discrete Ablowitz-Ladik equations with constant coefficients and open solitons solutions for discrete Ablowitz-Ladik equations with variable coefficients are obtained, respectively. Secondly, the image simulation of single soliton propagation and double soliton collision is carried out, and the stability of single soliton propagation and the properties of double soliton collision are analytically studied. (6) A coupled third and fifth order nonlinear Schrodinger equation is studied. Firstly, by using Hirota bilinear method, the open-open soliton solution of the equation is obtained. Then several different types of collisions between two solitons are observed, such as head-on collisions, chase collisions and bound states. (7) A nonlinear system with variable coefficients is studied. Firstly, based on the Darboux transformation, the generalized Darboux transformation of the system is constructed. Then, the first and second order wave solutions of the system are obtained, respectively. Finally, the effects of system parameters on the first and second order deformities are analyzed by image simulation.
【学位授予单位】：北京邮电大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175.29

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