An Application of Prolongation Algebras to Determine Bäcklund Transformations for Nonlinear Equations

Abstract

Prolongation algebras which are determined by applying a version of the Wahlquist-Estabrook method to three different nonlinear partial differential equations can be employed to obtain not only Lax pairs but Bäcklund transformations as well. By solving Maurer-Cartan equations for the related group specified by the prolongation algebra, a set of differential forms is obtained which can lead directly to these kinds of results. Although specific equations are studied, the approach should be applicable to large classes of partial differential equations.