On symmetry group properties and general similarity forms of the Benney equations in the Lagrangian variables

In the present study, we investigate the symmetry groups of Benney equations in Lagrangian variables in the form of the system of the nonlinear integro-differential equations. We obtain the Lie point symmetries by using the invariance criterion for a specific type of integro-differential equation and find some reduced forms that have fewer independent variables by using the symmetry groups.