本文關(guān)鍵詞： 行波解 非線性波方程 動(dòng)力系統(tǒng) 分支理論 不變流形　出處：《浙江理工大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：行波解是一種廣泛存在于各類非線性方程中的一種相似解,其典型的特征是這類解在空間傳播中能夠保持平移不變。許多在實(shí)驗(yàn)下觀察到的物理、化學(xué)和生物現(xiàn)象都可以用方程的行波解來(lái)描述,而這些方程往往都是非線性方程,比如用來(lái)描述我們?cè)谌粘Ｉ钪薪?jīng)常會(huì)遇到的各種各樣的水波、聲波、電磁波等非線性波方程的行波解描述了這些波在媒介中的傳播過(guò)程。認(rèn)識(shí)和發(fā)現(xiàn)這些非線性方程所蘊(yùn)含的各種波的內(nèi)在機(jī)理成為當(dāng)今非線性波研究領(lǐng)域中理論研究與數(shù)值分析的重要課題。在非線性科學(xué)研究中,動(dòng)力系統(tǒng)理論和方法由于其理論的深刻性與應(yīng)用的廣泛性已成為相關(guān)非線性科學(xué)領(lǐng)域中非�；钴S的前沿方向之一,因此把動(dòng)力系統(tǒng)理論和方法應(yīng)用于非線性波方程行波解的研究具有廣闊前景。本文首先研究了一類Ito五階mKdV方程,其對(duì)應(yīng)的行波方程只能約化為含有參數(shù)的四階常微分方程,對(duì)于所對(duì)應(yīng)的四階行波系統(tǒng),借助計(jì)算機(jī)符號(hào)計(jì)算,在某些參數(shù)條件下我們得到了這個(gè)含參四維系統(tǒng)的由平面動(dòng)力系統(tǒng)確定的二維不變流形,通過(guò)利用平面動(dòng)力定性分析和分支理論研究確定了這個(gè)二維不變流形的二維系統(tǒng)在各類參數(shù)條件下的分支和精確解,從而得到了這類高階非線性波方程的各類光滑的有界行波解,其中包括孤波解、扭波解和不同振幅的周期波解。其次,研究了一類復(fù)mKdV方程,利用平移、旋轉(zhuǎn)和尺度對(duì)稱將方程轉(zhuǎn)化為在一定參數(shù)條件下實(shí)的參數(shù)方程,該方程是一個(gè)二階常微分方程,我們通過(guò)利用動(dòng)力系統(tǒng)定性和分支理論分析了該二階常微分方程所對(duì)應(yīng)平面系統(tǒng)的相圖和分支,得到了該方程在各種參數(shù)條件下的有界解,從而得到了該復(fù)mKdV方程的各類行波解,其中包含了振幅為周期函數(shù)的包絡(luò)解。
[Abstract]:Traveling wave solutions are similar solutions widely found in all kinds of nonlinear equations. The typical characteristic of traveling wave solutions is that they can keep their translation invariant in space propagation. Both chemical and biological phenomena can be described by traveling wave solutions to equations that are often nonlinear, such as the various water waves, sound waves that we often encounter in our daily lives. The traveling wave solutions of nonlinear wave equations such as electromagnetic waves describe the propagation process of these waves in media. Understanding and discovering the intrinsic mechanism of various waves contained in these nonlinear equations has become a theoretical study in the field of nonlinear wave research today. Research and numerical analysis. In nonlinear scientific research, The theory and method of dynamic systems have become one of the most active frontier directions in the field of nonlinear science because of its deep theory and extensive application. Therefore, the application of dynamic system theory and method to the study of traveling wave solutions of nonlinear wave equations has broad prospects. In this paper, we first study a class of Ito fifth order mKdV equations, the corresponding traveling wave equations can only be reduced to fourth order ordinary differential equations with parameters. For the corresponding fourth-order traveling wave system, with the aid of the computer symbolic calculation, we obtain the two-dimensional invariant manifold of the four-dimensional system with parameters determined by the plane dynamic system. By using the qualitative analysis of plane dynamics and bifurcation theory, the bifurcation and exact solutions of this two-dimensional invariant manifold system under various parameter conditions are determined. Thus, all kinds of smooth and bounded traveling wave solutions of this kind of high order nonlinear wave equations are obtained, including solitary wave solutions, torsional wave solutions and periodic wave solutions with different amplitudes. Secondly, we study a class of complex mKdV equations, using translation, Rotation and scale symmetry transform the equation into a real parametric equation under certain parameter conditions, which is a second-order ordinary differential equation. By using the qualitative and bifurcation theory of dynamical system, we analyze the phase diagram and bifurcation of the plane system corresponding to the second order ordinary differential equation, and obtain the bounded solution of the equation under various parameter conditions. The traveling wave solutions of the complex mKdV equation are obtained, including the envelope solution of the periodic function of the amplitude.
【學(xué)位授予單位】：浙江理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175.29

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本文編號(hào)：1546430