本文關(guān)鍵詞：輔助方程法及一些非線性發(fā)展方程（組）的精確解　出處：《河南科技大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 輔助方程法 齊次平衡原則 非線性發(fā)展方程 精確解

【摘要】：隨著科技的不斷發(fā)展,在許多學(xué)科領(lǐng)域中存在著大量的非線性問(wèn)題,其中一部分非線性問(wèn)題是利用非線性微分方程來(lái)描述的。為了能深入地了解這些非線性微分方程的物理意義,獲得方程的精確解就成為最為重要的一步。到目前為止,由于非線性微分方程的本身復(fù)雜性,還沒(méi)有一個(gè)統(tǒng)一的方法來(lái)求得這些非線性微分方程的精確解。因此,非線性微分方程(組)的精確求解不論在理論上,還是在應(yīng)用領(lǐng)域里仍是一個(gè)非常有研究?jī)r(jià)值的課題。經(jīng)過(guò)數(shù)學(xué)家和物理學(xué)家不懈的努力,現(xiàn)已發(fā)展出一系列用于求精確解的方法,如反散射方法、Darboux變換方法、Backlund變換方法、雙線性方法、李群方法、齊次平衡法、Dressing方法、輔助方程法等等。在這些求解方法中,輔助方程法由于直接、簡(jiǎn)潔、有效,而廣受重視。本文主要借助于輔助方程法,對(duì)非線性發(fā)展方程求解問(wèn)題進(jìn)行了研究和探討,主要研究:(1)分別利用具單個(gè)高次項(xiàng)的輔助方程和具兩個(gè)高次項(xiàng)的輔助方程求解了gKdV-qRLW方程、gKawahara方程、廣義對(duì)稱正則長(zhǎng)波方程以及g Zakharov方程組和具任意次Klein-Gordon-Zakharov方程組。(2)將F/G-展開(kāi)法做了推廣,利用推廣的F/G-展開(kāi)法求解了變系數(shù)mKd V方程、變系數(shù)Kd V方程和(3+1)維三次-五次Gross-Pitaevskii方程,得到了方程的精確解。
[Abstract]:With the continuous development of science and technology in many fields there are many nonlinear problems, the nonlinear part of the problem is the use of nonlinear differential equations. In order to deeply understand the physical meaning of these nonlinear differential equations, exact solutions of equations obtained has become the most important step. So far, because the nonlinear differential equations of the complexity, there is not a unified method to obtain the exact solutions of these nonlinear differential equations. Therefore, nonlinear differential equation (Group) the exact solution either in theory or in the application area, is a very valuable research topic. Through the unremitting efforts of mathematicians and physicists, now the development of a series of methods for seeking exact solutions, such as inverse scattering method, Darboux transformation method, Backlund transformation method, bilinear method, Li Qun method, homogeneous Balance method, Dressing method and auxiliary equation method and so on. In this method, the auxiliary equation method due to the direct, concise, effective, and wide attention. This paper is mainly based on the auxiliary equation method, the problem of solving nonlinear evolution equations are studied and discussed, the main research: (1) respectively by using the auxiliary equation with a single high-order and two high-order auxiliary equation gKdV-qRLW equation, gKawahara equation, G equation and the generalized symmetric regularized long wave equations with arbitrary Zakharov and Klein-Gordon-Zakharov equations. (2) the F/G- expansion method was extended by using the extended F/G- expansion method to solve the mKd V equation with variable coefficients. The variable coefficient Kd equation and V (3+1) three - five dimensional Gross-Pitaevskii equation, exact solutions of these equations are obtained.

【學(xué)位授予單位】：河南科技大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O175.29

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