Research About Two Methods To Obtain Solutions Of Variable-Coefficient Korteweg-de Vries Equations (vcKdV)

In recent years, as an important branch of the nonlinear science soliton theory was developed rapidly and widely used in fluid dynamics, plasma physics, nonlinear optics, condensed matter physics, biology and other fields. Many of these models can be described properly by nonlinear partial differential equations. Therefore, it's worth studying about how to get the analytical solution (exact solutions) of these equations. The existing methods include Inverse Scattering Method, Backhand transformation method, Homogeneous balance method, Hirota method, Wronskian method and Exp-function method. On the basis of introducing several of them, two kinds of variable-coeffcient Kdv equations were studied by using Hirota method and Exp-function method.The paper is organized as follows:In chapter one we first introduce the history and development of the soliton theory and several corresponding research methods. By solving some of the classical wave equations and nonlinear partial differential equations, we illustrate some kinds of traditional methods such as travelling wave method, Homogeneous balance method, Painleve analysis method and Backlund transformation method.Chapter two is about Hirota method. It's basic idea is to change a nonlinear equation into homogeneous form through a suitable transformation. It is often bilinear form for those integrable systems. We'll take the Kdv equation and MKdv equation as examples to introduce the Hirota method, bilinear form, seek soliton solution, backlund transformation and Wronskian technique. In the end through the Hirota method we obtain the soliton-type solutions and backlund transformation of a kind of variable-coeffcient Kdv equation then verify them with Wronskian technique.Chapter three is about Exp-function method. Because most soliton solutions of the nonlinear evolution equations have the Exp-function form, we can assume the solution initially then solve the overdetermined equations with powerful symbolic computation. Next by modifying Exp-function method we derive another kind of variable-coeffcient Kdv equation's soliton-type solution and periodic solution and analysis the solutions with figures.