The Cahn-Hilliard Equation – the application of the exponential transform method leading to a new insight for interfaces

Abstract

The Cahn-Hilliard Equation was introduced to study phase separation in binary alloy glasses and polymers and is a good approach to spinodal decomposition. General families of solutions are of basic interest. Therefore a complete characterization of the group properties is given. We determine the Lie point symmetry vector fields and calculate similarity solutions. Depending upon the approach, here, in this case, the traveling-wave reduction leads to new nonlinear ordinary differential equations which only can be solved by numerical methods. We show that these ordinary differential equations are not of Painlevé type and can therefore not fully be integrated. The crucial point of the paper is the fact that we are able to calculate new classes of solutions by a new algebraic method derived recently by the author. These new solutions, important in relevant applications, are discussed whereby we support the theoretical results by illustrative figures.