Uniqueness of the Kadomtsev-Petviashvili and Boussinesq Equations

The Kadomtsev-Petviashvili and Boussinesq equations (u xxx - 6uu x)x - ut x ± uyy = 0, (u xxx - 6uu x)x + u xx ± u tt = 0, are completely integrable, and in particular, they possess the three-soliton solution. This article aims to expose a uniqueness property of the Kadomtsev-Petviashvili (KP) and Boussinesq equations in the integrability theory. It is shown that the Kadomtsev-Petviashvili and Boussinesq equations and their dimensional reductions are the only integrable equations among a class of generalized Kadomtsev-Petviashvili and Boussinesq equations (u x1x1x1 - 6uu x1) x1 + σ M i, j=1 a iju xixj = 0, where the a i j's are arbitrary constants and M is an arbitrary natural number, if the existence of the three-soliton solution is required. © 2011 Verlag der Zeitschrift für Naturforschung, Tübingen.