Application Of The Mapping Method In The Nonlinear Evolution Equation

With the development of science and technology, there are many nonlinear problems in natural and social areas with much concern. These problems are usually characterized by nonlinear evolution equations. So how to construct exact solutions of the associated nonlinear problems plays an important role in understanding the nonlinear problems. This paper will mainly introduce three methods which construct directly the exact solutions of the nonlinear evolution equation --the mapping method,the modified mapping method and the extended mapping method, and we obtain the new exact solutions of the higher order nonlinear Sch?dinger equation and the Zakharov equation through these methods.Chapter 1 is devoted to reviewing the history and development of the soliton theory and the exact solutions of the nonlinear evolution equation, with a introduction of several methods of the nonlinear evolution equation.Chapter 2 considers the main ideas of the mapping method,the modified mapping method and the extended mapping method, and introduces the advantages of these methods.In the chapter 3, by using the mapping method,the modified mapping method and the extended mapping method, we derive some new exact solutions of the higher order nonlinear Sch?dinger equation which are the linear combination of two different Jacobi elliptic functions.In the chapter 4, we take the Zakharov equation as examples, obtain its exact solutions through the mapping method,the modified mapping method and the extended mapping method.