【摘要】：孤立子理論在非線性科學(xué)研究領(lǐng)域里占有很重要的地位,在研究它的過程中發(fā)現(xiàn)了一大批的非線性發(fā)展方程,為了能更深入的了解這些非線性發(fā)展方程的實(shí)際意義,最為重要的一步就是獲得大量的新解。由于非線性發(fā)展方程的復(fù)雜性質(zhì),目前為止大量的非線性發(fā)展方程還沒有一個(gè)統(tǒng)一的求解方法。在所有非線性發(fā)展方程的求解方法中,輔助方程法是一種比較直接有效的方法。本文主要是給出函數(shù)變換與輔助方程相結(jié)合的方法,利用符號計(jì)算系統(tǒng)Mathematica,構(gòu)造了幾種變系數(shù)(常系數(shù))非線性發(fā)展方程(組)的復(fù)合型新解。這些解包括了 Airy函數(shù)、Jacobi橢圓函數(shù)、雙曲函數(shù)、三角函數(shù)和有理函數(shù)組合的復(fù)合型新解。第一章中簡述孤立子理論產(chǎn)生的歷史背景,并介紹了非線性發(fā)展方程的幾種求解方法以及本文的主要工作內(nèi)容。第二章中通過函數(shù)變換,將變系數(shù)sine-Gordon方程的求解問題轉(zhuǎn)化為二維線性波動(dòng)方程的求解問題。然后,利用波動(dòng)方程的解,構(gòu)造了變系數(shù)sine-Gordon方程的新解,并通過解的圖像研究了解的一些性質(zhì)。第三章中通過函數(shù)變換,將mKdV方程、Sharma-Tasso-Olver(STO)方程和mZK方程的求解問題化為Airy方程的求解問題。在此基礎(chǔ)上,利用Airy方程的解,得到了 mKdV方程等非線性發(fā)展方程的Airy函數(shù)解,并通過解的圖像研究了解的一些性質(zhì)。第四章中做了三項(xiàng)工作。1.利用第二種橢圓方程的已知解與解的非線性疊加公式,構(gòu)造了耦合KdV方程組的由Jacobi橢圓函數(shù)解、雙曲函數(shù)和三角函數(shù)兩兩組合的無窮序列復(fù)合型新解,并通過解的圖像研究了解的一些性質(zhì)。2.利用函數(shù)變換與二階齊次線性常微分方程(或Riccati方程)相結(jié)合的方法,構(gòu)造了變系數(shù)(3+1)維破碎孤子方程的復(fù)合型新解,并通過解的圖像研究了解的一些性質(zhì)。3.通過函數(shù)變換,將變形Boussinesq方程組的求解問題化為一階齊次線性常微分方程和二階齊次線性常微分方程的求解問題。在此基礎(chǔ)上,構(gòu)造了變形Boussinesq方程組的無窮序列復(fù)合型新解,并分析了解的性質(zhì)。
[Abstract]:The soliton theory plays an important role in the field of nonlinear science. In order to understand the practical significance of these nonlinear evolution equations, a large number of nonlinear evolution equations have been found in the course of its research. The most important step is to get a lot of new solutions. Due to the complex properties of nonlinear evolution equations, a large number of nonlinear evolution equations have not been solved by a unified method. Among all the methods for solving nonlinear evolution equations, the auxiliary equation method is a more direct and effective method. In this paper, the method of combining the function transformation with the auxiliary equation is given. By using the symbolic computing system Mathematica, the complex new solutions of several nonlinear evolution equations with variable coefficients (constant coefficients) are constructed. These solutions include composite new solutions of Airy function, Jacobi elliptic function, hyperbolic function, trigonometric function and rational function. In the first chapter, the historical background of soliton theory is briefly introduced, and several methods of solving nonlinear evolution equations and the main work of this paper are introduced. In the second chapter, the problem of solving sine-Gordon equation with variable coefficients is transformed into the solution of two-dimensional linear wave equation by means of function transformation. Then, by using the solution of the wave equation, the new solution of the variable coefficient sine-Gordon equation is constructed, and some properties of the solution are studied by the image of the solution. In chapter 3, the problem of solving mKdV equation, Sharma-Tasso-Olver (STO) equation and mZK equation is transformed into Airy equation by function transformation. On this basis, the Airy function solutions of nonlinear evolution equations such as mKdV equation are obtained by using the solution of Airy equation, and some properties of the solution are studied by the image of the solution. Chapter four has done three tasks. 1. Using the nonlinear superposition formula of known solutions and solutions of the second kind of elliptic equations, a new composite solution of infinite sequence by Jacobi elliptic function, hyperbolic function and trigonometric function is constructed. And through the solution of the image study of some properties of the solution. 2. By combining the function transformation with the second order homogeneous linear ordinary differential equation (or Riccati equation), a new complex solution of the (31) dimensional broken soliton equation with variable coefficients is constructed, and some properties of the solution are studied by the image of the solution. The problem of solving deformed Boussinesq equations is transformed into the first order homogeneous ordinary differential equation and the second order homogeneous linear ordinary differential equation by function transformation. On this basis, the infinite sequence complex solution of deformed Boussinesq equations is constructed, and the properties of the solution are analyzed.
【學(xué)位授予單位】：內(nèi)蒙古師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175.29

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