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Localized hexagon patterns of the planar SwiftHohenberg equation
 SIAM J. Appl. Dyn. Syst
"... We investigate stationary spatially localized hexagon patterns of the twodimensional Swift–Hohenberg equation in the parameter region where the trivial state and regular hexagon patterns are both stable. Using numerical continuation techniques, we trace out the existence regions of fully localized ..."
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Cited by 26 (8 self)
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We investigate stationary spatially localized hexagon patterns of the twodimensional Swift–Hohenberg equation in the parameter region where the trivial state and regular hexagon patterns are both stable. Using numerical continuation techniques, we trace out the existence regions of fully localized hexagon patches and of planar pulses which consist of a strip filled with hexagons that is embedded in the trivial state. We find that these patterns exhibit snaking: for each parameter value in the snaking region, an infinite number of patterns exist that are connected in parameter space and whose width increases without bound. Our computations also indicate a relation between the limits of the snaking regions of planar hexagon pulses with different orientations and of the fully localized hexagon patches. To investigate which hexagons among the oneparameter family of hexagons are selected in a hexagon pulse or front, we derive a conserved quantity of the spatial dynamical system that describes planar patterns which are periodic in the transverse direction and use it to calculate the Maxwell curves along which the selected hexagons have the same energy as the trivial state. We find that the Maxwell curve lies within the snaking region as expected from heuristic arguments.
Numerical Results for Snaking of Patterns over Patterns in Some 2D SelkovSchnakenberg ReactionDiffusion Systems
 SIAM J. Appl. Dyn. Syst
, 2014
"... For a Selkov–Schnakenberg model as a prototype reactiondiffusion system on two dimensional domains we use the continuation and bifurcation software pde2path to numerically calculate branches of patterns embedded in patterns, for instance hexagons embedded in stripes and vice versa, with a planar in ..."
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Cited by 4 (3 self)
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For a Selkov–Schnakenberg model as a prototype reactiondiffusion system on two dimensional domains we use the continuation and bifurcation software pde2path to numerically calculate branches of patterns embedded in patterns, for instance hexagons embedded in stripes and vice versa, with a planar interface between the two patterns. We use the GinzburgLandau reduction to approximate the locations of these branches by Maxwell points for the associated Ginzburg–Landau system. For our basic model, some but not all of these branches show a snaking behaviour in parameter space, over the given computational domains. The (numerical) non–snaking behaviour appears to be related to too narrow bistable ranges with rather small GinzburgLandau energy differences. This claim is illustrated by a suitable generalized model. Besides the localized patterns with planar interfaces we also give a number of examples of fully localized patterns over patterns, for instance hexagon patches embedded in radial stripes, and
Spinodal decomposition and coarsening fronts in the CahnHilliard equation
, 2014
"... We study spinodal decomposition and coarsening when initiated by localized disturbances in the CahnHilliard equation. Spatiotemporal dynamics are governed by multistage invasion fronts. The first front invades a spinodal unstable equilibrium and creates a spatially periodic unstable pattern. Seco ..."
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Cited by 2 (2 self)
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We study spinodal decomposition and coarsening when initiated by localized disturbances in the CahnHilliard equation. Spatiotemporal dynamics are governed by multistage invasion fronts. The first front invades a spinodal unstable equilibrium and creates a spatially periodic unstable pattern. Secondary fronts invade this unstable pattern and create a coarser pattern in the wake. We give linear predictions for speeds and wavenumbers in this process and show existence of corresponding nonlinear fronts. The existence proof is based on Conley index theory, a priori estimates, and Galerkin approximations. We also compare our results and predictions with direct numerical simulations and report on some interesting bifurcations. Running head: Spinodal decomposition and coarsening fronts
Stationary Coexistence of Hexagons and Rolls via Rigorous Computations *
, 2015
"... Abstract. In this work we introduce a rigorous computational method for finding heteroclinic solutions of a system of two second order differential equations. These solutions correspond to standing waves between rolls and hexagonal patterns of a twodimensional pattern formation PDE model. After re ..."
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Abstract. In this work we introduce a rigorous computational method for finding heteroclinic solutions of a system of two second order differential equations. These solutions correspond to standing waves between rolls and hexagonal patterns of a twodimensional pattern formation PDE model. After reformulating the problem as a projected boundary value problem (BVP) with boundaries in the stable/unstable manifolds, we compute the local manifolds using the parameterization method and solve the BVP using Chebyshev series and the radii polynomial approach. Our results settle a conjecture by Doelman et al. [European J. Appl. Math., 14 (2003), pp. 85110] about the coexistence of hexagons and rolls.
Spinodal decomposition and coarsening fronts in the
, 2012
"... CahnHilliard equation ..."
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Abstract Available online at www.sciencedirect.com Physics Reports 386 (2003) 29–222
, 2003
"... Front propagation into unstable states ..."
Modulated traveling fronts for a nonlocal FisherKPP equation: a dynamical systems approach
, 2014
"... We consider a nonlocal generalization of the FisherKPP equation in one spatial dimension. As a parameter is varied the system undergoes a Turing bifurcation. We study the dynamics near this Turing bifurcation. Our results are twofold. First, we prove the existence of a twoparameter family of bifu ..."
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We consider a nonlocal generalization of the FisherKPP equation in one spatial dimension. As a parameter is varied the system undergoes a Turing bifurcation. We study the dynamics near this Turing bifurcation. Our results are twofold. First, we prove the existence of a twoparameter family of bifurcating stationary periodic solutions and derive a rigorous asymptotic approximation of these solutions. We also study the spectral stability of the bifurcating stationary periodic solutions with respect to almost coperiodic perturbations. Secondly, we restrict to a specific class of exponential kernels for which the nonlocal problem is transformed into a higher order partial differential equation. In this context, we prove the existence of modulated traveling fronts near the Turing bifurcation that describe the invasion of the Turing unstable homogeneous state by the periodic pattern established in the first part. Both results rely on a center manifold reduction to a finite dimensional ordinary differential equation.