The Study On The Properties And Solving Methods For Some Nonlinear Evolution Equations

In this dissertation, based on the theory of partial differential equations and with the aid of computer symbolic computing system-Maple, the Painlevéproperty for the variable coefficient KdV equation have been studied, and the exact solutions for a kind of reaction diffusion equations are obtained by the first integral method, some results are new and significative.Along with the rapid development of science and technology, the nonlinear science is widespread applied in various fields of natural sicnce, and a series of remarkable achievements is obtained in recent years. Because of the nonlinear problems are often described with nonlinear partial differential equations (PDEs), the nonlinear PDEs are more and more close to connected with other subjects, such as physics, chemistry, biology and engineering. Solving the nonlinear PDEs and researching the properties for their solutions becomes a very important research topic in theory and the practice.For the complexity of nonlinear PDEs, there still have many PDEs whose exact solutions are unable to be obtained. Although exact solutions for a lot of PDEs have been studied, the solving process has each skill respectively, there is still no a general solving method. Thus, sometimes we don't solve the equations, but research about properties for their solutions directly.Completely integrable nonlinear PDEs which can be solvable by the Inverse scattering method(IST) often have remarkable properties, such as the Painlevéproperty, Backlund and Darboux transformations, a Lax pair, and so on. But there is no systematic way to determine whether a PDE can be solved by the IST method. The WTC algorithm which was proposed by Weiss, Tabor and Carnevale can examine a PDE (group) whether has Painlevéproperty. If a PDE (group) pass Painlevétest, it will satisfy the essential condition of completely integrable; otherwise, the PDE (group) is not completely integrable.There are some methods for solving nonlinear PDES, shch as hom-egeneous balancing method,Tanh function method,Backlund transformation,Inverse scattering method,Darboux transformation,Hirota bilinear method,similarity reduction eta.Base on Division Theorem and Hilbert-Nullstellensatz Theorem, Z. S. Feng proposed a new approach for studing the compound Burgers-KdV equation in 2002, which is currently called the first integral method and becoming an effective method for solving some nonlinear PDES.According to the above theory and methods, two aspects work are completed in this paper. Firstly, the Painlevétest is proposed for the variable coefficient combined KdV equation with forcedterm, and the conclusion that the variable coefficient combined KdV equation with forcedterm own Painlevéproperty when satisfy a certain constraint conditions is drawed, as well as auto-Backlund transformations for the variable coefficient combined KdV equation with forcedterm are obtained. Secondly, applying the first integral method, exact solutions for a kind of reaction diffusion equations are gained. The achievement about this paper includes: the paper "Exact solutions for the Burgers-Huxley equation" which based on the second part is accepted by "Journal of Shandong University of Technology".