ON BACKLUND-TRANSFORMATIONS FOR NONLINEAR PARTIAL-DIFFERENTIAL EQUATIONS

In this paper, Backlund 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 u(xxx) = 9(u, u(x), u(t)) that have Backlund transformations whose definition involves only u, u(x), u(xx), and a function determined by u, u(x), and u(xx) via an integrable system of two first-order partial differential equations, and we obtain all such Backlund transformations, In particular, a new nonlinear partial differential equation with a Backlund transformation (i.e., u(xxx) = -3/2 sin 2u u(x) - 1/2 u(x)(3) + u(t)) is found, and the known Backlund 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. (C) 1995 Academic Press, Inc.