on similarity reductions and conservation laws of the two non-linearity terms benjamin-bona-mahoney equation

In this paper, the lie group of point symmetries for a kind of benjamin-bona-mahoney (bbm) equation is obtained by applying the classical lie symmetry method. an optimal system of sub-algebras of dimension one for the bbm equation is deduced by classifying the adjoint representation orbits of the lie symmetry group. then, for any infinitesimal symmetry generators of the lie group, the related similarity reductions are generated and new group invariant solutions for the bbm equation are obtained. also, new conservation laws for this equation are constructed by the method of scaling. the conservation laws densities are calculated by using the concept of variables weight, scaling symmetry and euler operator and their fluxes are computed by applying the homotopy operator.