Nonlinear relaxation patterns in the Cahn–Hilliard equation: An exact solution

We consider the 1-D Cahn–Hilliard equation with the order parameter v and derive an equation for a modified order parameter g   such that g″=v‘g″=v‘. The new equation allows for separation of variables. This yields exact solutions for v expressed in terms of generalized hypergeometric functions. These solutions have an infinite gradient at their zeros and the first three derivatives of zero at their extrema. The amplitude of these patterns decreases as the inverse square root of time. It is suggested that the phenomenon of compartmentalization of evolving structures typically observed in evolutionary models of the Cahn–Hilliard type is a manifestation of relaxation patterns similar to those derived in this paper.

Graphical abstractFor the Cahn–Hilliard equation, new exact solutions are derived. They have an infinite gradient at their zeros and the first three derivatives of zero at their extrema. Their amplitude decreases as the inverse square root of time.Figure optionsDownload full-size imageDownload as PowerPoint slide