Cahn-Hilliard equation

Question

Is there any possibility to write the so-called Cahn-Hilliard equation:

$\frac{\partial c}{\partial t}=D\nabla^2(c^3-c-\gamma\nabla^2c)$,

in terms of two coupled partial differential equations (second order)? (even in $1D$ case, i.e., $c(x,t)$)

Answer 1

One quick suggestion: \begin{align} \frac{\partial c}{\partial t} - D \,\nabla^2 H &= 0,\\ \gamma \nabla^2 c + c - c^3 +H &= 0. \end{align} Which is of course not the only way to write the Cahn-Hilliard equation as two coupled PDE's, but a very straightforward one nonetheless.

Answer 2

I will bring it in a Form like I've learned it

\begin{align*} \frac{\partial c}{\partial t}-D\nabla^2\mu&=0\\ \mu&=-\gamma\nabla^2 c+f(c) \end{align*}

where $f(c)=c^3-c$.