Localized radial solutions of the Swift–Hohenberg equation

Abstract

Stationary localized solutions of the planar Swift-Hohenberg equation are investigated in the parameter region where the trivial solution is stable. In the parameter region where rolls bifurcate subcritically, localized radial ring-like pulses are shown to bifurcate from the trivial solution. Furthermore, radial spot-like pulses are shown to bifurcate from the trivial state, regardless of the criticality of roll patterns. These theoretical results apply also to general reaction-diffusion systems near Turing instabilities. Numerical computations show that planar radial pulses 'snake' near the Maxwell point where, by definition, the one-dimensional roll patterns have the same energy as the trivial state. These computations also reveal that spots, which are stable in a certain parameter region, become unstable with respect to hexagonal perturbations, leading to fully localized hexagon patterns.

Three-dimensional spatially localized binary-fluid convection in a porous medium

“…Thus some of the small oscillations correspond to the nucleation of new rolls while others reflect roll destruction. In addition, while the large excursions governing the growth of the core are more or less vertically aligned as expected of homoclinic snaking of target-like structures in simpler systems (Lloyd & Sandstede 2009) the small superposed oscillations are not ( figure 3). This is indicative of the fact that the nucleus and the arms tend to grow simultaneously.…”

“…Of these the hexagons, worms and targets are all found in the subcritical regime where a periodic pattern, be it hexagons or rolls, coexists with the trivial solution. As in one dimension, all these patterns 'snake', at least initially, as the structure is followed in parameter space and grows by nucleating additional cells or rings along its periphery (Lloyd et al 2008;Lloyd & Sandstede 2009). Spot-like states differ in that they are present even in the supercritical regime (Lloyd & Sandstede 2009), and related states have been observed in non-variational systems, such as reaction-diffusion equations (Coullet, Riera & Tresser 2000) or the equations arising in nonlinear optics Vladimirov et al 2002), as well as in ferrofluids (Richter & Barashenkov 2005) and an optical light valve experiment (Bortolozzo, Clerc & Residori 2009).…”

Localized Turing patterns in nonlinear optical cavities

“…By examining the topology of their phase space trajectory, a lot of useful informations have been deduced [4,[29][30][31], notably the fact that they follow a specific bifurcation sequence, called the snaking bifurcation diagram [29]. Further structures in the bifurcation diagram were later revealed through a combination of numerical and asymptotic studies [32][33][34][35][36][37][38][39][40][41][42][43].…”

Homoclinic snaking in bounded domains

“…Recent years have seen rapid developments in the theory of spatially localized structures in reversible systems in both one [7,26,11] and two [8,28,27] spatial dimensions. These structures occur in a great variety of physical systems, including nonlinear optics where they are usually referred to as dissipative solitons, reaction-diffusion systems where they take the form of spots or ensembles of spots, and convection where they are called convectons.…”