Abstract
Backlund transformations, which are relations among solutions of partial dierential 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 Backlund transforma- tions, using an auxiliary linear system called a prolongation structure. The integrabil- ity conditions for the prolongation structure are to be the original dierential equation system, most of which systems have just two independent variables. This paper dis- cusses 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.
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