Abstract
Bäcklund transformations, which are relations among solutions of partial differential equations-usually nonlinear-have been found and applied mainly for systems with two independent variables.A few are known for equations like the Kadomtsev-Petviashvili equation [1], which has three independent variables, but they are rare.Wahlquist and Estabrook [2] discovered a systematic method for searching for Bäcklund transformations, using an auxiliary linear system called a prolongation structure.The integrability conditions for the prolongation structure are to be the original differential equation system, most of which systems have just two independent variables.This paper discusses how the Wahlquist-Estabrook method might be applied to systems with larger numbers of variables, with the Kadomtsev-Petviashvili equation as an example.The Zakharov-Shabat method is also discussed.Applications to other equations, such as the Davey-Stewartson and Einstein equation systems, are presented.