几种非线性发展方程的复合型解及其性质研究
发布时间：2018-11-15 06:27
【摘要】：孤立子理论在非线性科学研究领域里占有很重要的地位,在研究它的过程中发现了一大批的非线性发展方程,为了能更深入的了解这些非线性发展方程的实际意义,最为重要的一步就是获得大量的新解。由于非线性发展方程的复杂性质,目前为止大量的非线性发展方程还没有一个统一的求解方法。在所有非线性发展方程的求解方法中,辅助方程法是一种比较直接有效的方法。本文主要是给出函数变换与辅助方程相结合的方法,利用符号计算系统Mathematica,构造了几种变系数(常系数)非线性发展方程(组)的复合型新解。这些解包括了 Airy函数、Jacobi椭圆函数、双曲函数、三角函数和有理函数组合的复合型新解。第一章中简述孤立子理论产生的历史背景,并介绍了非线性发展方程的几种求解方法以及本文的主要工作内容。第二章中通过函数变换,将变系数sine-Gordon方程的求解问题转化为二维线性波动方程的求解问题。然后,利用波动方程的解,构造了变系数sine-Gordon方程的新解,并通过解的图像研究了解的一些性质。第三章中通过函数变换,将mKdV方程、Sharma-Tasso-Olver(STO)方程和mZK方程的求解问题化为Airy方程的求解问题。在此基础上,利用Airy方程的解,得到了 mKdV方程等非线性发展方程的Airy函数解,并通过解的图像研究了解的一些性质。第四章中做了三项工作。1.利用第二种椭圆方程的已知解与解的非线性叠加公式,构造了耦合KdV方程组的由Jacobi椭圆函数解、双曲函数和三角函数两两组合的无穷序列复合型新解,并通过解的图像研究了解的一些性质。2.利用函数变换与二阶齐次线性常微分方程(或Riccati方程)相结合的方法,构造了变系数(3+1)维破碎孤子方程的复合型新解,并通过解的图像研究了解的一些性质。3.通过函数变换,将变形Boussinesq方程组的求解问题化为一阶齐次线性常微分方程和二阶齐次线性常微分方程的求解问题。在此基础上,构造了变形Boussinesq方程组的无穷序列复合型新解,并分析了解的性质。
[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.
【学位授予单位】：内蒙古师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29
本文编号：2332376
[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.
【学位授予单位】：内蒙古师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29
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