Nonclassical symmetries for a family of Cahn–Hilliard equations

Abstract

In this Letter we make a full analysis of the symmetry reductions of the family of Cahn–Hilliard equations by using the classical Lie method of infinitesimals, the functional forms of the diffusion coefficients for which the Cahn–Hilliard equations can be fully reduced to ordinary differential equations by classical Lie symmetries are derived. We prove that by using the nonclassical method, we obtain several solutions which are not invariant under any Lie group admitted by the equation and consequently which are not obtainable through the Lie classical method. For this Cahn–Hilliard equation, we obtain nonclassical symmetries that reduce the original equation to ordinary differential equations with the Painlevé property. We remark that these symmetries have not been derived elsewhere by the singular manifold method.