Uniqueness of the Kadomtsev-Petviashvili and Boussinesq Equations

The Kadomtsev-Petviashvili and Boussinesq equations (u(xxx) - 6uu(x))(x) - u(tx) +/- u(yy) = 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) + Sigma(M)(i,j)=1a(ij)u(xixj) = 0, where the aij's are arbitrary constants and M is an arbitrary natural number, if the existence of the three-soliton solution is required.