Darboux Transformations Of Two Soliton Equations

The paper studies mainly the N times Darboux transformations of two important soliton equations, that is mKP equation and vcKdV equation. There are five sections in this article. In the first section, the development of soliton theory and the fundamental theory of the Darboux matrix and Darboux transformations are mainly introduced. In the second section, It gives two soliton equations.Their compatible solution w=v2 is the solution of mKP equation,then construct one time Darboux transformation. In the third section, we get the N times Darboux transformations and provide the proof. In the forth section,we obtain the N-soliton solution of the soliton equation through the Darboux transformations and the seed solutions u=0, v=1.Then we get the explicit solutions when N=1 and N=2. In the fifth section,we construct N times Darboux transformations of vcKdV equation.