The Application Of The Modified Simple Equation Method And The Study Of A Variable-coefficient Lie Equations

It is known that a large number of nonlinear phenomena in many fields of science and technology can be described by the nonlinear evolution equations. Therefore, the nonlinear model has become a widely used tool for describing the nature of the real world in nonlinear science.Because of this, the study of seeking exact solutions of nonlinear evolution equations is regarded as a routine to explain the physical properties of the nonlinear model. Hence, it is important to study how to construct new methods to solve nonlinear evolution equations and how to improve those previously known methods so as to find some properties of physical problems and new methods to solve nonlinear models.In addition, the integrability of nonlinear model is an important problem in soliton theory. The Lax pair and auto-Backlund transformation is always connected with the integrable property. The former is regarded as a bridge for linearilizing the nonlinear equations.The latter is an effective tool to construct. the exact solution of the nonlinear equations. Therefore, it is important to get the Lax pair and auto-Backlund transformation for nonlinear equations. It not only can provide the foundation for studying various kinds of properties of integrable systems, but also can give useful tools to construct the solution of nonlinear equations.This work will focus on the above mentioned subject and mainly consider how to obtain the solutions of nonlinear evolution equations, and the exact solution of the sine-Gordon equation, the generalized variable-coefficient KdV-mKdV equation, the (2+1)-dimensional dispersive long wave equations are obtained as well. Moreover, we shall study the variable-coefficient Lie equations and give its Painleve property,Lax pair, Backlund transformation and exact solutions. This dissertation is arranged as followsThe first chapter is the introduction, which gives a brief introduction to Painleve analysis, research status for nonlinear evolution equations with variable coefficients, the modified simple equation method and the auxiliary equation method,etc. Our work in this paper is also briefly introduced.In the second chapter, the modified simple equation method is used to construct the exact solutions of the sine-Gordon equation, the generalized variable-coefficient KdV-mKdV equation and the (2+1)-dimensional dispersive long wave equation. And the exact solitary wave solutions of these equations are obtained from their exact solutions.In the third chapter, a variable-coeff-icient Lie equations is investigated. The variable-coefficients Lie equations is proved to be integrable in the sence of Painleve property,at the same time its Lax pair,auto-Backlund transformation and exact solutions are also presented.The fourth chapter, the exact solutions of the (2+1)-dimensional AKNS equation and the higher order dispersive NLS equation are obtained by utilizing the Raccati auxiliary equation method and the improved G’\G-expansion method respectively. The obtained solutions will include the periodic solutions, the solitary wave solutions and the rational solutions.The fifth chapter is the summary of this paper and an outline of our future work.