本文選題：孤子 + 畸形波��； 參考：《北京郵電大學》2017年博士論文

【摘要】：非線性發(fā)展方程可以用來描述光纖、Heisenberg鐵磁體、流體以及等離子體等領域中的一些非線性現象。非線性發(fā)展方程中存在許多有理解,如孤子解、畸形波解等。孤子的產生源于非線性效應與色散效應的平衡。而作為一個在時間和空間局域化的有理解,Peregrine孤子可以作為畸形波的數學模型。本文基于一些非線性薛定諤類方程,并且利用符號計算,解析研究了光纖等領域中孤子和畸形波的性質。本文的研究內容主要有:(1)研究了一個2+1維的變系數耦合薛定諤方程。首先,通過尋找合適的有理變換將方程轉化為雙線性形式,進而得到了方程的明單孤子解和明雙孤子解。根據所得到的孤子解,結合模擬的圖像,分析了單孤子的傳輸和雙孤子的碰撞等性質。(2)研究了一個四階變系數薛定諤方程。在方程的可積條件下,利用Hirota雙線性方法得到了該方程的暗單孤子解和暗雙孤子解的表達式�；诘玫降墓伦咏�,模擬了單孤子傳輸和雙孤子碰撞的圖像,并分析了方程的一些物理參數對單孤子的傳輸和雙孤子的碰撞的影響。(3)研究了一個變系數Kundu-Eckhaus方程。首先,利用該方程的Lax對,在其Darboux變換的基礎上構造了廣義的Darboux變換。然后,分別得到了該方程的一階畸形波解和二階畸形波解。最后,結合圖像模擬,解析研究了方程中的非線性色散項對一階畸形波和二階畸形波的性質的影響。(4)研究了一個廣義的非自治非線性方程。(a)在方程的可積條件下,利用合適的變換得到了該方程的雙線性形式,進而得到了該方程的明單孤子解和明雙孤子解。并且結合模擬的圖像,研究了方程的系數對單孤子的傳輸和雙孤子的碰撞產生的影響。另外,借助于分步Fourier方法,研究了孤子在有限初始擾動下的穩(wěn)定性。(b)在方程的可積條件下,我們在Darboux變換的基礎上構造了廣義的Darboux變換,然后分別得到了該方程的一階畸形波解和二階畸形波解,并結合圖像分析和研究了一階畸形波和二階畸形波的性質。(5)分別研究了一個常系數離散Ablowitz-Ladik方程和一個變系數離散Ablowitz-Ladik方程。首先,利用Hirota雙線性方法分別得到了常系數離散Ablowitz-Ladik方程的暗孤子解的表達式和變系數離散Ablowitz-Ladik 方程的明孤子解的表達式。其次,對單孤子傳輸和雙孤子碰撞進行了圖像模擬,解析研究了單孤子傳輸的穩(wěn)定性及雙孤子碰撞的性質。(6)研究了一個耦合的三、五階非線性薛定諤方程。首先,利用Hirota雙線性方法,得到了該方程的明-明孤子解。然后結合模擬的圖像,觀察到了雙孤子之間幾種不同形式的碰撞:迎面碰撞、追趕碰撞以及束縛態(tài)等。(7)研究了一個變系數非線性系統。首先,在Darboux變換的基礎上,構造了該系統的廣義Darboux變換。然后,分別得到了系統的一階畸形波解和二階畸形波解。最后,利用圖像模擬,分析了系統的參數對一階畸形波和二階畸形波的影響。
[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.
【學位授予單位】：北京郵電大學
【學位級別】：博士
【學位授予年份】：2017
【分類號】：O175.29

【參考文獻】

相關博士學位論文 前3條

1 劉榮香;光纖通信等領域中非線性Schr(?)dinger類方程的解析研究[D];北京郵電大學;2014年

2 崔成;畸形波生成、演化及內部結構研究[D];大連理工大學;2013年

3 郭睿;基于符號計算的若干非線性模型可積性質及孤子解的研究[D];北京郵電大學;2012年

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本文編號：2090534