Swift–Hohenberg equation

Parabolic | Patterns

Swift–Hohenberg equation:

\[\pd{u}{t} = ru - (k_c^2+\nabla^2)^2u+au^2+bu^3+cu^5,\]

with periodic boundary conditions, and we need $c<0$ (or $b<0$ if $c=0$) for stability.

Localised solutions

When $r<0$, $a>0$, and $b<0$, the system can be in a subcritical regime that supports both stable patterned states and the stable homogeneous state $u=0$.

Specific initial conditions can induce localised patterns, which fall off to the background of $u=0$ throughout most of the domain.

This example is based on a 2023 paper by Dan Hill and collaborators which studies symmetric localised solutions of the Swift-Hohenberg equation.