The First Integral Method And Exact Solutions Of Nonlinear Evolution Equations

Through detailed studying and analyzing the basic idea and solving process of thefirst integral method, this paper use the first-integral method to solve some nonlinearevolution equations and nonlinear partial diferential equations with variable coefcients.The paper is divided into the following seven chapters: The first chapter is the intro-duction part, introducing the research status and development trend of nonlinear partialdiferential equations, summarizing the main methods in recent decades for solving exactsolutions of nonlinear partial diferential equations, giving the research background andapplication process of the first integral method, and showing the main content and re-search purposes of this paper. The second chapter to the sixth chapter, through discussingand studying, we respectively use the first integral method to obtain abundant exact so-lutions of the generalized Zakharov-Kuznetsov (ZK) equation, the generalized nonlinearSchro¨dinger equation with parabolic law and dual-power law, Klein-Gordon-Zakharovequation, the generalized long-short wave equations and Zakharov equation. Withoutassuming some form of solution and complicated calculation, we uniformly obtain d-iferent types of abundant explicit exact solutions for these equations, include not onlysome known results of previous references, but also some new explicit exact solutions,such as sech function, csch function, twelve kinds of Jacobi elliptic doubly periodic wavesolutions, Weierstrass elliptic doubly periodic wave solutions, and so on,which modifiesand perfects the known results. In the seventh chapter, the first integral method is ap-plied to obtain the exact solutions of mKdV equation with variable coefcients. we firstlylearning from other literature, we transform the variable coefcient mKdV equation in-to mKdV equation with constant coefcients, and use the first integral method and theextended hyperbolic function method to obtain abundant exact solutions of the variablecoefcient mKdV equation. And we compare and analyze the solutions and the solutionsof existing literature.Finally, we make a summary of this paper and look ahead of research orientation infuture.