Symmetry analysis and solutions for a family of Cahn-Hilliard equations

In this paper we find some new classes of solutions for a family of Cahn-Hilliard equations. For some equations of this family several solutions have already been obtained by using several methods: the Lie method, the direct method and the singular manifold method. We make full analysis of the symmetry reductions of the family of Cahn-Hilliard equations by using the classical Lie method of infinitesimals and the nonclassical method. New classes of nonlocal symmetries for the family of Cahn-Hilliard equations are obtained. These nonclassical potential symmetries are realized as local nonclassical symmetries of a related integrated equation. For an equation of the Cahn-Hilliard family with the conditional Painlevé condition, we also compare symmetry reductions by using the nonclassical method with those obtained elsewhere by the singular manifold method. For this equation, we obtain nonclassical symmetries that reduce the original equation to ordinary differential equations with the Painlevé property. Such symmetries have not been derived elsewhere neither by the direct method nor by the singular manifold method.