= Swift–Hohenberg equation
{c}
{title2=$w_t=[r-(1+\partial_x^2)^2]w+sw^3-w^5$}
The <Swift–Hohenberg equation> is a <partial differential equation> for <pattern formation>. In one spatial dimension, a cubic-quintic version is
$$
w_t=[r-(1+\partial_x^2)^2]w+sw^3-w^5.
$$
Linearizing at $w=0$ and substituting a <Fourier mode> $e^{\sigma t+ikx}$ gives $\sigma(k)=r-(1-k^2)^2$. The first unstable modes therefore have <wavenumber> near one when $r$ becomes positive. Replacing one by $q_0^2$ inside the differential operator selects wavenumbers near $q_0$.
There is also a useful <Lyapunov functional>. On a periodic interval, or with sufficient decay at infinity, define
$$
E[w]=\int\left\{\frac12(w+w_{xx})^2-\frac r2w^2-\frac s4w^4+\frac16w^6\right\}\,dx.
$$
Twice using <integration by parts> gives the <functional derivative> $\delta E/\delta w=(1+\partial_x^2)^2w-rw-sw^3+w^5$. The equation is therefore a <gradient flow>, and
$$
\frac{dE}{dt}=\int\frac{\delta E}{\delta w}w_t\,dx=-\int w_t^2\,dx\le0.
$$
This dissipative structure constrains the evolution without requiring the final spatial pattern to be uniform. For $s>0$, the cubic term enhances finite amplitudes while the quintic term limits growth. Near onset, a slowly varying envelope can satisfy a <quintic real Ginzburg-Landau equation>.
Back to article page
