On Miura transformations among nonlinear partial differential equations

Abstract

In this paper, we study Miura transformations u↦v from partial differential equations uxxx=F(u,ux,ut) to nonlinear partial differential equations G(v,vx,vt,…,∂xlv,…,∂t1v)=0 defined using integrable systems on v. We classify all such Miura transformations under some restrictions, and hence generalize the classical Miura transformation 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 of vt=32vxsin2v+12vx3+vxxx are obtained from constant solutions and soliton solutions of the MKdV equation. As another application of Miura transformations of this paper, we deduce a new Bäcklund transformation for each of vt=32vxsin2v+12vx3+vxxx and vt=−32vxsinh2v−12vx3+vxxx from the known Bäcklund transformations for the MKdV equations.