Constructing Exact Solutions Of Some Nonlinear Evolution Equations With Variable Coefficient

With the development of science and technology, nonlinear evolution equations are play a important role in field of Social science and natural world. and it serves as a kind of mathematical model, it is used to describe the nonlinear problem which appeares in the field of physics、computer、chemical、biological、environmental etc. At the same time, it is also an important subject for the study of soliton theory.At present, with the development and broad apply of computer symbol computation softwares, these nonlinear problems which have not been solved are actively explored by scientist using computer methods, find rules of systemic and features, explore the mystery. And from their generality and universality discusses all kinds of nonlinear behavior of the system, now has achieved fruitful results, has a good development future. lots of important results have been achieved and having a better development prospect.But the study about nonlinear evolution equations with variable coefficient is so rare, and thereis also lack of a unified solution, so the the study about nonlinear evolution equations with variable coefficient has taken more people’s attention, this is also the key research content of this paper. In this paper, a number of classic variable coefficient equations have been solved by using improved (G’/G)-expansion method, and good effect been obtained too. Then a number of equations also have been solved by the change of wave transformation and using improved (G’/G)-expansion method, and successfully proved in the previous equations. Moreover, variable coefficient Gardner equations also have been tried by using improved algebraic method, and ideal results are obtained.This whole work is presented in5chapters. In chapter1, the theoretical background, research status and development trend of nonlinear evolution equations are introduced; In chapter2, some new exact solutions of the variable coefficient Burgers equation and the first kind of KdV equation of variable coefficient; In chapter3, the variable coefficient combi-nation of KdV equation and KP equation have been solved by change the wave transforma-tion ξ=k(x)+c(t)+1to ξ=k(t)x+c(t)y+l(t) using improved (G’/G)-expansion methodand ideal results are obtained; In chapter4, variable coefficient Gardner equation has been tried by using improved algebraic method, and obtained more exact solutions; In chapter5, the whole research is summarized and also given an outlook in the direction of this research.