耦合非线性薛定谔方程多孤子解及其传输特性研究论文设计.doc

中 文 摘 要由于群速度色散和自相位调制之间的相互平衡，光孤子可以在光纤中长距离传输且形状不发生改变，因为这一特性，孤子可以在光纤通信系统中实现远距离和大容量传输，并可以应用在很多领域当中，成为了很多学者研究的内容非线性薛定谔方程是描述光孤子传输的理想模型，是一类非常重要的非线性演化方程随着研究的进行，非线性薛定谔方程被推广到了变系数、复系数、多维、高阶、非局域和分数阶等包含各类物理效应的方程，通过对各种方程解的研究可以更好地理解各种非线性现象因此，基于非线性薛定谔方程研究孤子的传输特性以及潜在的一些应用是至关重要的，对孤子理论的发展和不同应用领域的发展具有一定的理论指导意义本文主要介绍了非线性薛定谔方程的研究背景和进展，孤子和呼吸子的由来和研究进展，在此基础上，采用Hirota双线性方法研究了孤子间的相互作用，具体的研究内容分为以下三个部分：（1） 基于自聚焦广义耦合非线性薛定谔方程，其中包含自相位调制、交叉相位调制和四波混频效应，采用Hirota双线性方法得到了该方程的4-亮-亮孤子解，并对孤子的碰撞动力学进行了详细地研究研究结果表明：特征值的虚部影响孤子的速度和脉冲宽度，特征值的实部对孤子的振幅有影响。

2） 基于包含四波混频效应的自散焦广义耦合非线性薛定谔方程，采用Hirota双线性方法得到了该方程的4-暗-暗孤子解，分析不同参数的取值范围，数值研究了其传输特性研究结果表明：通过调控参数，可以分别获得4-暗孤子、3-暗孤子及暗孤子-反暗孤子组合，但孤子间的相互作用依然是弹性碰撞3） 基于变系数耦合非线性薛定谔方程，利用Hirota双线性方法得到了该方程的2-孤子解，该解包括三个亮孤子分量和一个暗孤子分量孤子特征值取复数时，通过计算给出满足弹性碰撞的两种条件，一种为常规的弹性碰撞，另一种为一个亮孤子消失的弹性碰撞当弹性碰撞条件不满足时，三个亮孤子分量中的两个孤子相互作用为非弹性碰撞，参数取值不同，孤子间的能量交换不同，而暗孤子分量中的两个孤子依然保持弹性碰撞孤子特征值取常数时，出现束缚态孤子结果表明：合理选择参数，可以获得二孤子解的弹性碰撞、非弹性碰撞和束缚态传输等情况关键词： 耦合非线性薛定谔方程； 孤子解； Hirota双线性方法；ABSTRACTDue to the balance between group velocity dispersion and self-frequency shift modulation, optical solitons can travel long distances in the fiber without changing the shape. Because of this characteristic, solitons can be transmitted over large distances and large capacity in optical fiber communication systems. It can be applied in many fields and has become the content of many scholars' research. The nonlinear Schrödinger equation is an ideal model for describing the soliton transmission. It is a very important nonlinear evolution equation. As the research progresses, the nonlinear Schrödinger equation is extended to variable coefficients, complex coefficients, high-dimensional, high-order, non-local and fractional equations include various types of physical effects. By studying various equations, we can better understand many nonlinear phenomena. Therefore, it is very important to study the transmission characteristics of soliton and some potential applications through the nonlinear Schrödinger equation. It has certain theoretical guiding significance for the development of soliton theory and the development of different application fields.This article mainly introduces the research background and progress of nonlinear Schrödinger equation, the origin and research progress of solitons and breathers. Based on these, the interactions between solitons are studied by using the Hirota bilinear method. The specific research contents are as follows:(1) Based on the self-focusing generalized coupled nonlinear Schrödinger equation, which includes self-phase modulation, cross-phase modulation, and four-wave mixing effects, the 4-bright-bright soliton solution of the equation is obtained by using the Hirota bilinear method. The soliton collision dynamics have been studied in detail. The results show that the imaginary part of the eigenvalue affects the speed and pulse width of the soliton, and the real part of the eigenvalue change the amplitude of the soliton.(2) Based on the self-defocusing generalized coupled nonlinear Schrödinger equation including the four-wave mixing effect, the 4-dark-dark soliton solution of the equation is obtained bt using the Hirota bilinear method. The value range of different parameters are analyzed, and numerical research is performed. The results show that 4-dark solitons, 3-dark solitons and dark-soliton-anti-dark soliton combinations can be obtained by adjusting of parameters, but the interaction between solitons is still an elastic collision.(3)Based on the variable coefficient coupled nonlinear Schrödinger equation, the 2-soliton solution is obtained by using Hirota bilinear method. The solution includes three bright soliton components and one dark soliton component. When the eigenvalue of the soliton is complex, two elastic collision conditions are obtained that one is the conventional elastic collision and the other is the elastic collision with a bright soliton disappearing. When the conditions of elastic collision are not satisfied, the two solitons’ interactions in the three bright soliton components are inelastic with different parameter values and different energy exchanges between the solitons, while the two solitons in the dark soliton component still maintain the elastic collision. . When soliton eigenvalues are constant, bound state solitons can be obtained. The results show that reasonable selection of parameters can obtain the two-soliton solutions for elastic collisions, inelastic collisions, and bound state transmission.Key words: coupled nonlinear Schrödinger equation; Soliton solution; Hirota bilinear method;第一章 绪论1.1. 引言在物理学中，存在着很多非线性物理系统，可以用非线性模型来描述非线性物理现象，所以非线性模型起着十分重要的作用。

孤子由于在光纤通信系统[1]、应用物理[2]、光电子器件[3]等多个领域中的应用，在光孤子领域有一类重要的非线性偏微分方程，即非线性薛定谔方程[4,5]该方程具有非常重要的研究价值，吸引了国内外各个领域中大量的学者去研究它在非线性光学中，非线性薛定谔方程可以用来描述孤子的传输特性，孤子间的相互作用等随着技术的不断提高，更加复杂的物理现象可以用非线性薛定谔方程来解释其现象的本质，因此，研究非线性薛定谔方程对理论和实践都具有非常重要的意义迄今为止，关于非线性薛定谔方程已经做了很多研究，比如变系数非线性薛定谔方程[6,7]、高阶非线性薛定谔方程[8,9]、分数阶非线性薛定谔方程[10-12]等，针对这些方程主要研究了。