Swift–Hohenberg equation

Swift–Hohenberg equation:

ut=ru(kc2+2)2u+au2+bu3+cu5,

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.