Pattern formation with a conservation law

Pattern formation in systems with a conserved quantity is considered by stu dying the appropriate amplitude equations. The conservation law leads to a large-scale neutral mode that must be included in the asymptotic analysis f or pattern formation near onset. Near a stationary bifurcation, the usual G inzburg-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 insta bility differs from the Eckhaus instability in that it is amplitude-driven and is supercritical. Beyond the stability boundary, there exist stable sta tionary solutions in the form of strongly modulated patterns. The envelope of these modulations is calculated in terms of Jacobi elliptic functions an d, away from the onset of modulation, is closely approximated by a sech pro file. Numerical simulations indicate that as the modulation becomes more pr onounced, the envelope broadens. A number of applications are considered, i ncluding convection with fixed-flux boundaries and convection in a magnetic held, resulting in new instabilities for these systems.