New similarity reductions of the Boussinesq equation

Abstract

Some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering, are presented. These new similarity reductions, including some new reductions to the first, second, and fourth Painlevé equations, cannot be obtained using the standard Lie group method for finding group-invariant solutions of partial differential equations; they are determined using a new and direct method that involves no group theoretical techniques.