100 research outputs found
Numerical study of pattern formation following a convective instability in non-Boussinesq fluids
We present a numerical study of a model of pattern formation following a
convective instability in a non-Boussinesq fluid. It is shown that many of the
features observed in convection experiments conducted on gas can be
reproduced by using a generalized two-dimensional Swift-Hohenberg equation. The
formation of hexagonal patterns, rolls and spirals is studied, as well as the
transitions and competition among them. We also study nucleation and growth of
hexagonal patterns and find that the front velocity in this two dimensional
model is consistent with the prediction of marginal stability theory for one
dimensional fronts.Comment: 9 pages, report FSU-SCRI-92-6
Transverse Patterns in Nonlinear Optical Resonators
The book is devoted to the formation and dynamics of localized structures
(vortices, solitons) and extended patterns (stripes, hexagons, tilted waves) in
nonlinear optical resonators such as lasers, optical parametric oscillators,
and photorefractive oscillators. The theoretical analysis is performed by
deriving order parameter equations, and also through numerical integration of
microscopic models of the systems under investigation. Experimental
observations, and possible technological implementations of transverse optical
patterns are also discussed. A comparison with patterns found in other
nonlinear systems, i.e. chemical, biological, and hydrodynamical systems, is
given. This article contains the table of contents and the introductory chapter
of the book.Comment: 37 pages, 14 figures. Table of contents and introductory chapter of
the boo
Pattern formation for the Swift-Hohenberg equation on the hyperbolic plane
We present an overview of pattern formation analysis for an analogue of the
Swift-Hohenberg equation posed on the real hyperbolic space of dimension two,
which we identify with the Poincar\'e disc D. Different types of patterns are
considered: spatially periodic stationary solutions, radial solutions and
traveling waves, however there are significant differences in the results with
the Euclidean case. We apply equivariant bifurcation theory to the study of
spatially periodic solutions on a given lattice of D also called H-planforms in
reference with the "planforms" introduced for pattern formation in Euclidean
space. We consider in details the case of the regular octagonal lattice and
give a complete descriptions of all H-planforms bifurcating in this case. For
radial solutions (in geodesic polar coordinates), we present a result of
existence for stationary localized radial solutions, which we have adapted from
techniques on the Euclidean plane. Finally, we show that unlike the Euclidean
case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf
bifurcation to traveling waves which are invariant along horocycles of D and
periodic in the "transverse" direction. We highlight our theoretical results
with a selection of numerical simulations.Comment: Dedicated to Klaus Kirchg\"assne
Localized radial roll patterns in higher space dimensions
Localized roll patterns are structures that exhibit a spatially periodic profile in their center. When following such patterns in a system parameter in one space dimension, the length of the spatial interval over which these patterns resemble a periodic profile stays either bounded, in which case branches form closed bounded curves (“isolas”), or the length increases to infinity so that branches are unbounded in function space (“snaking”). In two space dimensions, numerical computations show that branches of localized rolls exhibit a more complicated structure in which both isolas and snaking occur. In this paper, we analyse the structure of branches of localized radial roll solutions in dimension 1+ε, with 0 < ε 1, through a perturbation analysis. Our analysis sheds light on some of the features visible in the planar case.http://math.bu.edu/people/mabeck/Bramburgeretal18.pdfFirst author draf
Snakes and ladders in an inhomogeneous neural field model
Continuous neural field models with inhomogeneous synaptic connectivities are
known to support traveling fronts as well as stable bumps of localized
activity. We analyze stationary localized structures in a neural field model
with periodic modulation of the synaptic connectivity kernel and find that they
are arranged in a snakes-and-ladders bifurcation structure. In the case of
Heaviside firing rates, we construct analytically symmetric and asymmetric
states and hence derive closed-form expressions for the corresponding
bifurcation diagrams. We show that the ideas proposed by Beck and co-workers to
analyze snaking solutions to the Swift-Hohenberg equation remain valid for the
neural field model, even though the corresponding spatial-dynamical formulation
is non-autonomous. We investigate how the modulation amplitude affects the
bifurcation structure and compare numerical calculations for steep sigmoidal
firing rates with analytic predictions valid in the Heaviside limit
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