Riccati Equation And The Construction Of The Exact Solutions To The (2+1)-dimensional Nonlinear Evolution Equations

In this paper, based on the Mathematica system, we have generalized the extended tanh-function method and applied it to the (2+1)-dimensional nonlinear evolution equations(NLEEs). By taking advantage of a Riccati equation and a coupled Riccati equations, we get soliton-like solutions, singular solutions and rational solutions to the (2+1)-dimensional NLEEs. In chapter 2, the exact solutions with variable y+nt of the breaking soliton equation and shallow water wave equation are given. The exact solutions to the Burgers equation, Kadomtsev-Petviashvili(KP) equation, Asymmetric Nizhnik-Novikov-Veselov (ANNV) equation are also given by modifying the extended tanh-function method. (2+l)-dimensional dispersive long wave equations is taken as a example of the (2+l)-dimensional coupled NLEEs to show that our method can be used to solve (2+l)-dimensional coupled NLEEs directly. After making a transformation, the modified extended tanh-function method is used to get new solutions of the Broer-Kaup(BK) equations and (2+l)-dimensional dispersive long wave equations. In chapter 3, a coupled Riccati equations is used to solve (2+l)-dimensional NLEEs. We find that if the exact solutions to a nonlinear equation (or a coupled equations) can be constructed by the Riccati equation, then it canalso be constructed by the coupled Riccati equations. There is a relationship between these solutions which are obtained by two kind of methods. Among our solutions, several solutions were previously known, and the others are new.