Study On Solving Methods And Properties Of Solutions For Several Kinds Of Nonlinear PDEs

In this dissertation, based on the theory of differential equations and with the aid of computer symbolic computing system Maple and Mathematica, we have studied the Painleve property for the nonlinear vibrating string equation and a kind of variable coefficient Boussinesq equations, and disscuss similarity reductions for the vibrating string equation and the variable coefficient Boussinesq equations, as well as the exact solutions for a kind of reaction diffusion equations by the first integral method, some new and significative results are obtained.Along with the rapid development of production and practice, science and technology, the nonlinear science is widespread applied in various fields, and a series of remarkable achievements is obtained in recent years. Because of the nonlinear problems are often described with nonlinear 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 the research about 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 IST method often have remarkable properties, such as the Painleve property, Backlund and Darboux transformations, a Lax pair, and so on. But there is no systematic way to determine whether a PDE is solvable using the IST method. The WTC algorithm which was proposed by Weiss, Tabor and Camevale can examine a PDE (group) whether has Painleve property. If a PDE (group) pass Painleve test, it will satisfy the essential condition of completely integrable; otherwise, the PDE (group) is not completely integrable.One important method to obtain the explicit solutions of nonlinear PDEs is the classical Lie group method of infinitesimal transformation which leads the order of a PDE decrease one time. The CK direct method which was developed by Clarkson and Kruskal in studying similarity reductions of Boussinesq equation involves no Lie group theoretic techniques, and seeks the solutions as the form u(x, t)=U(x, t, w(z(x, t))) of a equation. Base on Division Theorem and Hilbert-Nullstellensatz Theorem, Z. S. Feng proposed a new approach which is currently called the first integral method to the compound Burgers-KdV equation in 2002.Following the above theory and methods, we have completed three aspects work in this paper. Firstly, we propose Painleve test for the nonlinear vibrating string equation and a kind of variable coefficient Boussinesq equations, draw the conclusion that the vibrating string equation owns Painleve property and the variable coefficient Boussinesq equations own Painleve property when f(x), g(x) satisfy a certain constraint condition.s, and obtain Backlund transformations for the two equations. Secondly, using the classical Lie group method, the vibrating string equation can be reduced to the third and the fourth type of elliptic equations; by means of the CK direct method, we obtain similarity reductions and similarity solutions for the vibrating string equation and the variable coefficient Boussinesq equations; and some explicit solutions also are presented for the two equations. Thirdly, applying the first integral method, we obtain exact solutions for a kind of reaction diffusion equations. Our achievements about this paper includes: the paper "Similarity reductions for the nonlinear vibrating string equation" which based on the second part is accepted by "Journal of Engineering Mathematics". The paper "Exact solutions for a kind of reaction diffusion equations" which based on the third part is accepted by "Journal of Mathematical Research and Exposition".