Comparison of approximate symmetry and approximate homotopy symmetry to the Cahn-Hilliard equation

Complete infinite order approximate symmetry and approximate homotopy symmetry classifications of the Cahn-Hilliard equation are performed and the reductions are constructed by an optimal system of one-dimensional subalgebras. Zero order similarity reduced equations are nonlinear ordinary differential equations while higher order similarity solutions can be obtained by solving linear variable coefficient ordinary differential equations. The relationship between two methods for different order are studied and the results show that the approximate homotopy symmetry method is more effective to control the convergence of series solutions than the approximate symmetry one.