A New Auxiliary Equation Method And Its Application In Solving Nonlinear Partial Differential Equations

Nonlinear science research not only has important scientific significance, but also has important practical significance by using of human survival environment. So it is in the international leading position. Nonlinear partial differential equations are widely applied to describe the complicated physics phenomena in nonlinear science. Soliton as analytical solution of nonlinear partial differential equation has important physical meaning in the nonlinear science. And it is a solitary wave which has elastic collision. Through in-depth understanding and researching of solitary wave motion, we will better understand the nonlinear field. So how to obtain exact solitary solution is a vital task in nonlinear science research.There is no unified method of solving nonlinear partial differential equations. These exact solutions including singular soliton solution, periodic solution, double periodic solution and multi-soliton solution are usually obtained. Few interaction solutions which include rational function, trigonometric function, exponential func-tion, hyperbolic function and Jacobi elliptic function at the same time are obtained successfully. It is very meaningful for understanding the world better to research the interaction solutions of nonlinear partial differential equations. Hence, the im-portant work of this paper is to research interaction solution.Chapter one, we describe the background and research status of the soliton theory. It is focused on the traditional and new methods of solving nonlinear partial differential equations. Then the main work in this paper is introduced.Chapter two, a novel auxiliary equation method is presented. New interaction solutions of the (2+1) dimensional KdV equation and Hirota-Satsuma equation are obtained successfully by applying this method.Chapter three, the novel auxiliary equation method is improved. And we obtain new interaction solutions of the fifth-order KdV equation with variable coefficients and the generalized Hirota-Satsuma coupled KdV equation.Chapter four, four types of functions solutions of this novel auxiliary equa-tion are gained. We solve the generalized dispersion equation and the nonlinear Schrodinger equation with perturbed terms successfully.