One-dimensional optimal system of subalgebras for Lie symmetry group of the Benjamin-Bona-Mahoney equation

In this paper, by using the classical Lie symmetry method, Lie symmetry group of Benjamin- Bona-Mahoney equation is determined. Th en by classifying the orbits of the adjoint representation of the symmetry group on its Lie algebra, the optimal system of one-dimensional subalgebras of the BBM equation is obtained. For each class, we will find the similarity reduction of the equation and by solving them we will obtain new sets of group invariant solutions.