發(fā)布時間：2018-04-27 00:06

  本文選題：廣義達布變換 + 高階畸形波解��； 參考：《北京郵電大學》2017年碩士論文

【摘要】：畸形波是一類危害水平高的波。它發(fā)生的機制以及發(fā)生概率還不明確,普遍以為調(diào)制不穩(wěn)定性可能會產(chǎn)生畸形波,但并不是所有的調(diào)制不穩(wěn)定性都能產(chǎn)生畸形波。薛定諤方程的準確度只可依賴試驗來進行檢測,為量子學中的假設(shè),并且聯(lián)合了物質(zhì)波的觀念和波動方程的觀念�？梢杂盟鼇韺ξ⒂^粒子的運動進行描述。不僅在流體力學和等離子物理等范疇中,薛定諤方程具有廣泛的應用,非線性薛定諤方程還可以用來描述光學和海森堡鐵磁自旋鏈問題中的非線性動力特征,這也是本文研究的重點。在本文中主要利用廣義達布變換來求解兩類非線性薛定諤方程中的高階畸形波解,達布變換是一種可由一個種子解不斷得到新解的手段,與規(guī)范變換密不可分,它是一種特殊的規(guī)范變換,而規(guī)范變換可概括為一個方程在一組變換中可保持方程的形式不變形,該變換則為規(guī)范變換。其不能直接用來求得高階畸形波解,因此想得到高階的畸形波解,需通過極限知識將原有的達布變換進行改進推廣。在利用廣義達布變換時,需要用到Lax對的解,然而Lax對是非線性偏微分方程組,對其的求解需借助數(shù)學軟件或類似待定系數(shù)求解法來完成,在本文中首先通過一矩陣變換將原有的Lax對轉(zhuǎn)變成線性偏微分方程組,繼而通過簡單的特征值和特征向量的代數(shù)知識可求得Lax更多形式的解。通過矩陣變換將Lax對轉(zhuǎn)變成線性偏微分方程組的方法不僅可以運用在(1+1)維非線性薛定諤模型中,還可運用在(2+1)維非線性薛定諤模型中和變系數(shù)的mKdV-NLS方程中。在一些物理情況下,在多個領(lǐng)域中,具有不同頻率或偏振的光的傳播模型可由耦合方程來描述。在光纖中,其中一個例子是一個具有自相位調(diào)制,交叉相位調(diào)制和四波混頻的耦合的非線性薛定諤系統(tǒng),也就是本文第三章要討論的一個多模光纖中的(1+1)維耦合性薛定諤方程。在本文第五章中,討論了另外兩類非線性(1+1)維耦合薛定諤方程的畸形波解,在第六章中,探究了(1+1)維耦合變系數(shù)的mKdV-NLS方程解的性質(zhì)。非線性薛定諤方程不僅能用來描述流體、玻色-愛因斯坦和光纖維中的非線性動力特征,也可以用來描述海森堡-鐵磁自旋鏈中的非線性動力特征。本文通過對一個多模光纖中的(1+1)維耦合性薛定諤方程和海森堡自旋鏈中的(2+1)維非線性薛定諤方程進行具體的分析來探究畸形波解的性質(zhì)以及通過調(diào)制不穩(wěn)定性分析來探究畸形波產(chǎn)生的原因和機制。
[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.
【學位授予單位】：北京郵電大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O175

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