A note on bifurcation problems in large containers

Abstract

Recent work has shown that the analysis of bifurcation problems on a large, but ÿnite, interval can yield Landau coe cients which are radically di erent from those obtained for the problem on an inÿnite interval. At ÿrst sight, such results seem to challenge the widely held belief that the onset of instability in a large domain should be described by a Ginzburg-Landau-type amplitude equation. In this note, we show that indeed the same phenomenon can occur when the Ginzburg-Landau equation is used as a starting point. An analysis based on neglecting the spatial derivative terms in the Ginzburg-Landau equation is justiÿable only if the amplitude of the solution is large relative to 1=L, where L is the length of the interval. If the amplitude is small relative to 1=L, then the e ect of the boundary conditions is felt throughout the interval.