On Bäcklund transformations for nonlinear partial differential equations

In this paper, Bäcklund transformations for nonlinear partial differential equations are obtained without using any structure associated with the equations. We classify all the nonlinear partial differential equations of the form uxxx = F(u, ux, ut) that have Bäcklund transformations whose definition involves only u, ux, uxx, and a function determined by u, ux, and uxx via an integrable system of two first-order partial differential equations, and we obtain all such Bäcklund transformations. In particular, a new nonlinear partial differential equation with a Bäcklund transformation (i.e., uxxx = -3/2 sin 2u ux - 1/2 ux3 + ut) is found, and the known Bäcklund transformations for the KdV equation, the MKdV equation, the potential KdV equation, and the potential MKdV equation are recovered without using any knowledge of these equations. Our method is applicable to many other classes of nonlinear partial differential equations.