We now study the Cahn–Hilliard equation with an extra reaction term,

\[\pd{u}{t} = r\nabla^2 (u^3-u-g\nabla^2u)+u-u^3,\]

with periodic boundary conditions.

  • Load the interactive simulation

  • The initial condition is taken to be random noise at the level of the discretised system, and the initial timescale, given by $r$, is small. Increase $r$ by one or two orders of magnitude to speed up the simulation, and observe the coarsening process as described in this 2001 article.

There are lots of things that you can do with this equation (with or without the reaction term). For instance, you can extend it to a ‘non-reciprocal’ Cahn–Hilliard system as in Brauns and Marchetti, which exhibits pretty patterns that you can play with in this interactive simulation. Thanks to Lloyd Fung for pointing out this example!