The Research Of Exact Solutions For Some Nonlinear Evolution Equations

The nonlinear evolution equation is very important in many areas, solving these equations is always a popular research subject. Nowadays, many scholars are interested in this problem. Lots of efficient methods have been proposed to solve the linear partial equations, such as wave equations, heat-conduct equations, potential equations and Maxwell field equations. However, because of the complication of the nonlinear theory, it is not easy to solve these equations. The principle of superposition, the fourier series expand and the Laplace transform method are not suitable for solving nonlinear equations. So we usually use the numerical method to solve these equations. Generally speaking, it's difficult to find the exact solutions and there are few researches on this subject can be found, so it's very attractive to study on how to solve the nonlinear equations.To get the traveling solutions of the nonlinear evolution equations, we study some powerful methods for solving the nonlinear evolution equations. Finally, we obtain many new solitary solutions and exact solutions. Firstly, we compare the advantage and the disadvantage between the sine-cosine method and the extend sine-cosine method. Then we apply the extend sine-cosine method to the following nonlinear evolution equations: the Klein-Gordon-type equation,the RLW-type equation,the Boussinesq-type equation,a variant of the KdV equation,a second variant of the KdV equation,a third variant of the KdV equation and a fourth variant of the KdV equation. By using the extend sine-cosine method, we gain their compacton solutions successfully. The results include not only some known results, but also some brand-new exact solutions. Secondly, we compare the advantage and the disadvantage between the sine-cosine method and the tanh method. The tanh method is applied to the following nonlinear evolution equations: the Boussinesq equation, the modified Boussinesq equation, the fifth order Boussinesq equation,Porus medium equation and the Fisher's equations. It is very meaningful we study these equations, because researchers pained no or very few attentions on them. By using some necessary transforming technique and the tanh method, we deal with the tedious algebraic calculation with the aid of mathematic software. As a result, we not only obtain many exact solutions, but also extend the method to solve equations with higher order.