Exact Solutions, Symmetry Reductions, Painleve Test and Backlund Transformations of A Coupled KdV Equation

A coupled KdV equation is studied in this manuscript. The exact solutions, such as the periodic wave solutions and solitary wave solutions by means of the deformation and mapping approach from the solutions of the nonlinear phi(4) model are given. Using the symmetry theory, the Lie point symmetries and symmetry reductions of the coupled KdV equation are presented. The results show that the coupled KdV equation possesses in finitely many symmetries and may be considered as an integrable system. Also, the Painleve test shows the coupled KdV equation possesses Painleve property. The Backlund transformations of the coupled KdV equation related to Painleve property and residual symmetry are shown.