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The Study of Nonlinear Ordinary Differential Equations and Solitons

Author: ZhangDongHua
Tutor: TianBo
School: Beijing University of Posts and Telecommunications
Course: Applied Mathematics
Keywords: Nonlinear evolution equations Solitary wave solutions Similarity Reductions Hirota method Painlevé test
CLC: O175.29
Type: Master's thesis
Year: 2007
Downloads: 214
Quote: 0
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Abstract


With the development of nonlinear science, there has been a large number of nonlinear evolution equations plays an important role in different physical context. In order to explore the value of these equations in the application, the exact solution for solving nonlinear evolution equations or equations is a very important issue in nonlinear science. Although this is a hot topic, scientists have done a lot of work, giving a lot of the exact solution of the equation, but also get some very effective solution, but an equation or a class of equations, there is no system , unified solution. Because of this, so there is still a great research value of the nonlinear evolution equations, many scientists have been active in this field. The solitary wave solutions of nonlinear evolution equations meaningful solution, this is because it has important applications in many disciplines. This article is based on the solitary wave theory of nonlinear differential equations, explore several important solving the exact solution methods, and find some new nonlinear evolution equations single solitary wave solutions and solitary wave solutions. This chapter is organized as follows: The first chapter introduces the development status of the theoretical background of nonlinear evolution equations, as well as the concept of soliton generation, soliton theory in advanced disciplines. On this basis, the solitary wave solutions of nonlinear evolution equations theoretical basis and soliton nature. Chapter II of the similarity transformation theoretical ideas for solving nonlinear evolution equations, similarity transformation allows high-level or high-dimensional nonlinear evolution equations reduced-order or dimensionality reduction, into a low-level or low-dimensional partial differential or normally differential equation, easy to solve. Classical Lie group method, non-classical Lie group method, and CK Law, belong similarity transformation, focusing on the CK method and compared these methods. Finally, CK law about the fifth-order dispersion equation for ordinary differential equations. The third chapter the variable coefficients balanced role method for solving nonlinear evolution equations theoretical ideas, as well as the important role of this method. Through a balanced role to get the exact solution of nonlinear evolution equations with physical background, you can also get the Backlund transformation, and you can use it to find the nonlinear evolution equations similarity reduction, as well as multi-soliton solutions. Finally, with variable coefficients balanced role to obtain new exact solutions Huxley equation with variable coefficients. The fourth chapter is the focus of this study. The bilinear transform the 1970s developed by Hirota and accurate solution method for solving nonlinear evolution equations to solve the equation by nonlinear equations into operator bilinear form of the sub-form, complex differential equation with a simple bilinear operator form soliton solutions of nonlinear evolution equations, can be obtained in this way can also be used to find the Backlund transformation, lax on. This chapter explores the theoretical background of the method of Hirota bilinear operator defined Hirota method obtained the single soliton equations Cadrey-Dldd-Gibbon Kaeada and multi-soliton solutions. Chapter nonlinear evolution equations integrable test methods. That Painleve assumed. The singularity the Painleve nature and the Painleve ODE and PDE detection. And Chapter II CK method fifth-order dispersion equation about the nature of ordinary differential equations were Painleve ODE detected.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Partial Differential Equations > Nonlinear partial differential equations
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