Pattern formation with a conservation law

Abstract

Abstract. 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.

A Mathematical Model of Liver Cell Aggregation In Vitro

“…In this case, variation of the dependent variables on the long lengthscale (i.e. much longer than the critical wavelength) must be included, together with the potential for mode interaction (Matthews and Cox, 2000).…”

Swift-Hohenberg model for magnetoconvection

“…This process is resisted only by the diffusion of magnetic field, which is weak over large horizontal scales. A quantitative calculation of the circumstances under which flux separation may develop was given by Matthews and Cox [7,13], by considering the stability of small-amplitude two-dimensional convection rolls near the onset of convection, under "ideal" boundary conditions. They began by noting a crucial feature of magnetoconvection, which is that the total flux of the magnetic field through the layer is a conserved quantity, and this leads to a large-scale neutral mode representing rearrangement of the magnetic field.…”

“…Near the onset of convection, the usual Ginzburg-Landau equation for the amplitude of convection rolls must be coupled to this large-scale mode: analysis of this coupled pair of equations then leads to the conclusion that all convection rolls can be made unstable near onset to an amplitude modulation on large horizontal scales. This instability occurs for small values of the magnetic diffusivity and moderate values of the imposed magnetic field [13]. Of course, such an idealized analysis of convection near onset cannot be applied directly to the numerical simulations of strongly nonlinear, compressible magnetoconvection [5,10], but it may help to suggest parameter regimes for future numerical studies.…”

Stability of patterns with arbitrary period for a Ginzburg-Landau equation with a mean field

“…Both µ 1 (L) and µ 2 (L) are functions of L -the domain size -only. Thus we have rigorously established the following bifurcation picture (see also Figure 3 of [7]). Recall that we do not have any condition on the minimal period L. Our results show that nontrival stable patterns exist for a Ginzburg-Landau equation coupled to an equation for a mean field, even when the coefficients of the equations are real and when the minimal period is finite.…”

“…Previous numerical and analytical studies of these amplitude equation include [7] (numerical simulation, asymptotic expansion and bifurcation theory, in particular the use of Jacobi elliptic integrals to describe the shape of solutions and a numerical study of stability) and [8] (rigorous study of the limit when the minimal period is large enough). In [8] the resulting steady-states are pulses or spikes and nonlocal eigenvalue problems are used, but no Jacobi elliptic integrals.…”