The Swift-Hohenberg equation

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Partial differential equations and boundary-value problems

Many Parts, One Behaviour, Ch Pattern formation: instabilities that choose a wavelength

Problem

The Swift-Hohenberg equation, in dimensionless units, is

\partial_t u = ru - (1+\partial_x^2)^2 u,

where

u(x,t)

is the pattern field and

r

is the control parameter. (a) Substitute a Fourier mode

u = e^{\sigma t + iqx}

and derive the growth rate

\sigma(q)

. State the threshold value of

r

and the critical wavenumber

q_{\rm c}

at which the first mode destabilises. (b) Find the unstable band of wavenumbers

q

when

r = 0.05

Browse the problem library