On symmetry group properties of the Benney equations
In the present study, we investigate the symmetry groups of Benney equations that are the system of integro-differential equations. We first investigate the symmetry groups of the Benney equations by using the method that will be explained in the study. Then we obtain all reduced forms of the system of integro-differential equations with fewer variables based on symmetry groups; and lastly, we seek some similarity solutions to the reduced system of the equations. It is shown that one may transform the solution of the reduced forms into nonlinear shallow-water equations and Benney equations derived from two-dimensional Euler equations for inviscid and incompressible fluid.
Keywords: Benney equations; integro-differential equations; Lie groups; similarity solutions.