Pattern formation with a conservation law
Matthews, P.; Cox, Stephen M.
Stephen M. Cox
Pattern formation in systems with a conserved quantity is considered by studying the appropriate amplitude equations. The conservation law leads to a
large-scale neutral mode that must be included in the asymptotic analysis for pattern formation near onset. Near a stationary bifurcation, the usual
Ginzburg--Landau equation for the amplitude of the pattern is then coupled to an equation for the large-scale mode. These amplitude equations show
that for certain parameters all roll-type solutions are unstable. This new instability differs from the Eckhaus instability in that it is amplitude-driven and is
supercritical. Beyond the stability boundary, there exist stable stationary solutions in the form of strongly modulated patterns. The envelope of these
modulations is calculated in terms of Jacobi elliptic functions and, away from the onset of modulation, is closely approximated by a sech profile.
Numerical simulations indicate that as the modulation becomes more pronounced, the envelope broadens. A number of applications are considered,
including convection with fixed-flux boundaries and convection in a magnetic field, resulting in new instabilities for these systems.
Matthews, P., & Cox, S. M. (2000). Pattern formation with a conservation law. Nonlinearity, 13,
|Journal Article Type||Article|
|Publication Date||Jan 1, 2000|
|Deposit Date||Nov 30, 2001|
|Publicly Available Date||Oct 9, 2007|
|Peer Reviewed||Peer Reviewed|