Exact Solutions Of Two Nonlinear Evolution Equations

This paper studies mainly the Darboux transformation and Hirota bilinear method, which can obtain explicit solutions of nonlinear partial differential equations. In the first section of the paper, the basic theories of Darboux transformation are introduced. Based on this method, we construct a DT of a coupled equation which is related to a 3×3 spectral problem, then we obtain the explicit solutions of the coupled equation. Moreover, several interesting figures of the solutions are plotted. In the second section, the new N-soliton solution to the Mel'nikov equation is obtained by using the Hirota bilinear method. At last, the figures of the soliton solution are obtained by choosing the suitable parameters.