Spots in the Swift-Hohenberg Equation

The existence of stationary localized spots for the planar and the three-dimensional Swift–Hohenberg equation is proved using geometric blow-up techniques. The spots found in this paper have a much larger amplitude than that expected from a formal scaling in the far field. One advantage of the geometric blow-up methods used here is that the anticipated amplitude scaling does not enter as an assumption into the analysis but emerges naturally during the construction. Thus, the approach used here may also be useful in other contexts where the scaling is not known a priori.