On classification of Bäcklund transformations

In this paper, we study Bäcklund transformations u ↦ v from nonlinear partial differential equations uxx = F(u, ux, ut) to Gv,vx,vt,…,∂xlv,…,∂tlv=0 defined via integrable systems on v. We classify all such Bäcklund transformations under some restrictions. Our results show that there exist many such Bäcklund transformations.