Homotopy series solutions of perturbed PDEs via approximate symmetry method

We give a general result about the relation between approximate symmetry and approximate homotopy symmetry for perturbed PDEs. We show that the two couple equations derived by approximate symmetry method and approximate homotopy symmetry method are connected by a transformation. Consequently, acting the transformation on the known solutions constructed by approximate symmetry method, we directly obtain approximate homotopy series solutions whose accuracy can be controlled by adjusting the convergence-control parameter. Applications to the Cahn-Hilliard equation illustrate the effectiveness of the transformation.