The Exact Solutions Of Some Nonlinear Equations

The soliton theory is an important part of the nonlinear science. There are many nonlinear partial differential equations which appear in the area of mathematical physics have soliton solutions. Therefore, the problem of solving soliton equations is important in theory and applications.This dissertation is organized by four parts. The first part is due to the introduction.In the second part, the Li spectral problem is studied. The Lax pair of the Li equations and its explicit N-fold Darboux transformation is derived from the zero curvature equation .In the third part, the N-fold Darboux transformation of Jaulent-Miodek equations is obtained by a direct way. In fact, it is a special gauge transformation as following with of the Lax pairφx = Uφ,φt = Vφ.The new solutions of Jaulent-Miodek equations is obtained by using the Darboux transformation.In the fourth part, the N-fold Darboux transformation of Generalized MKdV equations is obtained as followingThe figures of the solutions are also presented.