發(fā)布時(shí)間：2019-02-24 10:14

【摘要】：隨著現(xiàn)代科學(xué)技術(shù)的蓬勃發(fā)展,非線性科學(xué)成為了近代科學(xué)技術(shù)發(fā)展的一個(gè)新標(biāo)志,非線性科學(xué)滲透到各個(gè)學(xué)科和領(lǐng)域,如地球物理、海洋和氣象預(yù)測(cè)、流體力學(xué)、非線性光學(xué)、等離子體等。孤子作為非線性科學(xué)中的一個(gè)重要分支,擁有著寬廣的應(yīng)用性,受到了廣大學(xué)者的重視。非線性科學(xué)研究過程中的一項(xiàng)非常重要的工作,就是是非線性問題的求解。非線性偏微分方程存在許多解。其中,大部分解都能夠較好的描述實(shí)際物理問題,而這類非線性方程大多都是變系數(shù)方程,想要得到非線性方程的精確解是十分困難的。目前還僅僅停留在求數(shù)值解的水平。變系數(shù)薛定諤方程及其變形形式被認(rèn)為是應(yīng)用性最廣泛的數(shù)學(xué)理論模型,它可以由麥克斯韋方程組出發(fā)導(dǎo)出,在眾多鄰域都有著極其重要的地位。至今為止,已經(jīng)有許多種求解常系數(shù)非線性薛定諤的方程的方法,而對(duì)于變系數(shù)非線性薛定諤方程的精確孤子解的求法,還有待進(jìn)一步的研究和討論。本文主要圍繞高階非自治薛定諤型方程的畸形波動(dòng)力學(xué)和基帶調(diào)制不穩(wěn)定性展開研究,內(nèi)容主要包括以下幾個(gè)方面:(1)介紹孤子的起源、發(fā)展及研究現(xiàn)狀,孤子的形成機(jī)制,以及若干種求解孤子的有效方法。(2)采用相似變換的方法,研究了高階變系數(shù)非線性薛定諤方程(vc-HNLS)在連續(xù)波背景下的非自治孤子的解的情況,以及三階色散系數(shù)對(duì)孤子的壓縮效應(yīng),其中包括常數(shù)、三角函數(shù)、指數(shù)函數(shù)、和一次函數(shù)對(duì)孤子的影響。(3)當(dāng)三階色散系數(shù)為特殊的周期結(jié)構(gòu),并且選取合適的調(diào)制振幅和調(diào)制頻率時(shí),Peregeine Combs結(jié)構(gòu)出現(xiàn)。我們還分析了Peregeine Combs結(jié)構(gòu)空間性質(zhì),研究中發(fā)現(xiàn)當(dāng)調(diào)制振幅時(shí),Peregeine Combs結(jié)構(gòu)轉(zhuǎn)化成Peregeine Walls。(4)光波脈沖寬度達(dá)到飛秒量級(jí)時(shí),需要考慮高階效應(yīng)(三階色散、非線性色散、自徒峭以及增益(損耗)等)。這時(shí),非線性薛定諤方程變成高階非線性Hirota方程。本文簡(jiǎn)單介紹了調(diào)制不穩(wěn)定性,并且利用線性穩(wěn)定性分析的方法,研究了高階變系數(shù)Hirota方程的調(diào)制不穩(wěn)定。(5)對(duì)本文的研究結(jié)果進(jìn)行了簡(jiǎn)要的概括總結(jié),并對(duì)研究工作做了一些展望。
[Abstract]:With the vigorous development of modern science and technology, nonlinear science has become a new symbol of the development of modern science and technology. Nonlinear science has penetrated into various disciplines and fields, such as geophysics, marine and meteorological prediction, hydrodynamics, Nonlinear optics, plasma, etc. As an important branch of nonlinear science, soliton has wide application and has been paid attention to by many scholars. A very important work in the process of nonlinear scientific research is the solving of nonlinear problems. There are many solutions to nonlinear partial differential equations. Among them, most of the solutions can describe the practical physical problems well, but most of these nonlinear equations are variable coefficient equations, so it is very difficult to obtain the exact solutions of nonlinear equations. At present, it is only at the level of numerical solution. The variable coefficient Schrodinger equation and its deformation form are considered to be the most widely applied mathematical theoretical models. They can be derived from Maxwell equations and play an extremely important role in many neighborhoods. Up to now, there are many methods to solve the nonlinear Schrodinger equations with constant coefficients, but the exact soliton solutions of the nonlinear Schrodinger equations with variable coefficients need to be further studied and discussed. In this paper, the deformational wave mechanics and baseband modulation instability of higher order nonautonomous Schrodinger equation are studied. The main contents are as follows: (1) the origin, development and research status of soliton are introduced. The formation mechanism of solitons and some effective methods for solving solitons. (2) the solution of the nonlinear Schrodinger equation (vc-HNLS) with high order variable coefficients under the continuous wave background is studied by using the method of similarity transformation. And the squeezing effect of third-order dispersion coefficient on soliton, including the influence of constant, trigonometric function, exponential function, and first-order function on soliton. (3) when the third-order dispersion coefficient is a special periodic structure, The, Peregeine Combs structure appears when the appropriate modulation amplitude and frequency are selected. We also analyze the spatial properties of Peregeine Combs structures. It is found that the higher-order effects (third-order dispersion and nonlinear dispersion) need to be considered when the amplitude modulation of, Peregeine Combs is converted to Peregeine Walls. (_ 4) when the pulse width reaches the femtosecond order. Self kurtosis and gain (loss). In this case, the nonlinear Schrodinger equation becomes the higher order nonlinear Hirota equation. In this paper, modulation instability is briefly introduced, and the modulation instability of high-order variable coefficient Hirota equation is studied by means of linear stability analysis. (5) the results of this paper are summarized briefly. Some prospects for the research work are also given.
【學(xué)位授予單位】：華北電力大學(xué)(北京)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

【參考文獻(xiàn)】

相關(guān)期刊論文 前8條

1 冀慎統(tǒng);顏培根;劉學(xué)深;;Soliton breathers in spin-1 Bose Einstein condensates[J];Chinese Physics B;2014年03期

2 陶勇勝;賀勁松;K. Porsezian;;Deformed soliton,breather,and rogue wave solutions of an inhomogeneous nonlinear Schrdinger equation[J];Chinese Physics B;2013年07期

3 張解放;胡文成;;Controlling the propagation of optical rogue waves in nonlinear graded-index waveguide amplifiers[J];Chinese Optics Letters;2013年03期

4 阮航宇;(2+1)維Sawada-Kotera方程中兩個(gè)Y周期孤子的相互作用[J];物理學(xué)報(bào);2004年06期

5 MA WENXIU;ADJOINT SYMMETRY CONSTRAINTS OF MULTICOMPONENT AKNS EQUATIONS[J];Chinese Annals of Mathematics;2002年03期

6 程藝,阿妹;積分型Darboux變換[J];數(shù)學(xué)年刊A輯(中文版);1999年06期

7 耿獻(xiàn)國;二維Sawada-Kotera方程的Darboux變換[J];高校應(yīng)用數(shù)學(xué)學(xué)報(bào)A輯(中文版);1989年04期

8 陳登遠(yuǎn),曾云波,李翊神;AKNS類發(fā)展方程的遞推算子自身之間的轉(zhuǎn)換算子[J];高校應(yīng)用數(shù)學(xué)學(xué)報(bào)A輯(中文版);1989年01期

相關(guān)博士學(xué)位論文 前1條

1 李敏;高階非線性效應(yīng)下的超短脈沖光孤子的解析研究[D];北京郵電大學(xué);2013年

，

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