Studies On The Exp-function Method For Some Nonlinear Evolution Equations

With the rapid development of science and technology, nonlinear science hasachieved an deep research and wide applications in all fields of natural sciences. Inthis dissertation, soliton theory which is one of the important branch of nonlinearscience is considered as the research background, multi-wave solutions of theSine-Gordon equation and exact solutions of the (4+1)-dimensional Fokas equationand variable-coefficient BBM equation and Burgers equation has been constructedthrough the generalization of the exponential function method. In the concretesolution process, the powerful symbolic operation function of the mathematicalsoftware Mathematica is utilized to describe the characteristic of evolutionarybehavior of the obtained solutions and the figures of solitary wave solutions aregiven. The main work of this dissertation is summarized as follows:1. The universality of applications, research status and future trends ofnonlinear science have been summarized. The origin, theoretical demonstration,physical properties and applications of soliton have been introduced.2. A generalized exponential function method is proposed for constructingmulti-wave solutions of Sine-Gordon equation. As a result, single, double and triplewave solution of Sine-Gordon equation have been obtained. Also, the evolutionaryfigures of wave propagation have been shown.3. The exponential function method has been extended to high-dimensionalnonlinear partial differential equations and new solutions of (4+1)-dimensionalFokas equation have been obtained.4. The exponential function method has been extended to variable-coefficientnonlinear partial differential equations and some exact solutions of BBM equationand Burgers equation with variable coefficients have been obtained.