Nonlinear Evolution Equations Generated from the Backlund Transformation for the Boussinesq Equation

Abstract

A Bäcklund transformation for the Boussinesq equation is given in the bilinear from. It is shown that the Bäcklund transformation generates an important class of nonlinear evolution equations exhibiting N-soliton solutions. They are a modified Boussinesq equation, a higher order water wave equation introduced by Kaup and a coupled equation whose N-soliton solution reduces to that of the nonlinear Schrödinger equation with normal dispersion. The relation between the Bäcklund transformation and the inverse scattering method is also discussed.