The authors are interested in localized roll solutions, i.e., in understanding how localized roll patterns and their plateau lengths depend on parameters. For a scalar spatial variable \(x\in\mathbb{R}\) these structures are spatially periodic for \(x\) in a bounded region, and they decay exponentially fast to zero as \(x\rightarrow \pm \infty\). In planar systems with \(x\in\mathbb{R}^{2}\), localized roll solutions may take the form of radial patterns, which are often referred to as spots and rings depending on whether the roll structures extend into the center of the pattern (spots) or not (rings)
The emphasis of the authors is on the Swift-Hohenberg equation \[U_{t}=-(1+\Delta)^{2}U-\mu U+\nu U^{2}-U^{3}, \quad x\in\mathbb{R}^{n}, \quad U\in \mathbb{R},\] where \(\Delta\) denotes the Laplace operator, \(\nu\) is held fixed, and \(\mu\) is a parameter which varies.
The authors analyze the structure of branches of localized radial roll solutions in dimension \(1+\epsilon\), with \(0 < \epsilon \ll 1\), through a perturbation analysis. Many detailed results are provided and discussed.
For the planar and three-dimensional Swift-Hohenberg equations, the authors show, using analytical and numerical methods, that snaking branches need to collapse onto the Maxwell point.
They also provide results for: (i) dynamics near the boundary layer; (ii) dynamics near the family of periodic orbits; (iii) dynamics near the stable manifold of the homogeneous state, which is based on radial solutions of the Swift-Hohenberg equation satisfy the partial differential equation \[0=-\bigg(1+\frac{n-1}{n}\partial_{r}+\partial_{r}^{2}\bigg)^{2}U-\mu U+\nu U^{2}-U^{3},\] where \(r=|x|\) denotes the radial direction in \(\mathbb{R}^{n}\).
The construction of radial pulses is also given. It is very rigorously constructed by using the following steps: boundary-layer solution, plateau solution, matching solutions at \(x = r_{0}\), matching solutions at \(x = L\) and matching the phase at \(x = r_{0}\).