Darboux Transformation And Explicit Solutions For A Hierarchy Of Soliton Equation

In section one, begaining with the known spectral problem, a hierarchy of soliton equation is derived with the help of zero curvature equation. Forthemore, we aslo obtain the famous mKP equations. 4qt= qxxx-6q2qx-6qx(?)x-1qy+3(?)x-1qyyIn section two, the Darboux transformation for the hierarchy of soliton equation is studied. We found the concrete form of Darboux transformation. And prove that it is a Darboux matrix exactly.In section three, taking a trivial soution (u, v)=(0,0) as the explicit solution of the equation (2.29) are obatained by using of the derived Darboux transformation.Moreover the figures are drawn through the Mathematica.In section four, while the case m= 2, we discusse an infinite conservation law of soliton equation.