Miura Transformation Between The Partial Differential Equation

In this paper, we study Miura transformations uâ†'v from partial differential equations u_xxx = F{u,u_x,u_t) to nonlinear partial differential equationsdefined using integrable systems on v. We classify allsuch Miura transformations under some restrictions, and hence generalize the classical Miura transformations to a large class of nonlinear partial differential equations. For some examples, by applying Miura transformations found in this paper, we derive exact solutions v from known solutions u. In particular, kink and soliton -kink solutions ofare obtained from constant solutions and soliton solutions of the MKdV equation. As another application of Miura transformations of this paper, we deduce a new Backhund transformation for each of andfrom the known Backhand transformation for the MKdV equation.