Exact Solutions For Two Classes Of Partial Differential Equation And Its Bifurcation

It is well known that construction of exact solutions of nonlinear partial differ-ential equations is a significant problem of differential equations and computer algebra. In this paper, we firstly introduce the theoretial knowledge of exact solution and obtain exact solutions of a mixed-KdV equation by using homogeneous balance method, (^)expansion-method. A series of new exact solutions which include rational function solutions, trigonometric functon solutons, soliton solutions and hyperbolic function solutions are obtained. Secondly, we use method of dynamical system to gain several kinds of exact solutions, at the same time, we draw the phase portraits of system. At the last, we use bifurcation theory to discuss the exact solutions of a more complex equation——ac-driven complex Ginzburg-Landau equation.This paper is organized as follows:In the first chapter, the evolution of method to construct the exact solutions of nonlinear partial differential equations, the development of rogue wave and bifurcation theroy of dynam-ical system are introduced.In the second chapter, homogeneous balance method and (G’/G)-expansion method are intro-duced firstly, and then using these methods, we obtain the exact solutions of (1+1)-dimensional KdV equation which conclude rational function solutions, trigonometric functon solutons, soli-ton solutions and hyperbolic function solutions.In the third chapter, we use travelling wave variable and the method of dynamical system to gain many travelling wave solutions, such as some periodical wave solutions solitary wave solutions, break wave solutions, kink wave solutions, anti-kink wave solutions. At the same time, we obtain periodic homoclinic orbits, bounded open orbits, heteroclinic orbits according-iy-In the fourth chapter, we study ac-driven complex Ginzburg—landau equation by applying bifurcation theroy to obtain various types of exact solutions with different parameters c.