468 research outputs found
Nonlinear dynamics in one dimension: On a criterion for coarsening and its temporal law
We develop a general criterion about coarsening for a class of nonlinear
evolution equations describing one dimensional pattern-forming systems. This
criterion allows one to discriminate between the situation where a coarsening
process takes place and the one where the wavelength is fixed in the course of
time. An intermediate scenario may occur, namely `interrupted coarsening'. The
power of the criterion lies in the fact that the statement about the occurrence
of coarsening, or selection of a length scale, can be made by only inspecting
the behavior of the branch of steady state periodic solutions. The criterion
states that coarsening occurs if lambda'(A)>0 while a length scale selection
prevails if lambda'(A)<0, where is the wavelength of the pattern and A
is the amplitude of the profile. This criterion is established thanks to the
analysis of the phase diffusion equation of the pattern. We connect the phase
diffusion coefficient D(lambda) (which carries a kinetic information) to
lambda'(A), which refers to a pure steady state property. The relationship
between kinetics and the behavior of the branch of steady state solutions is
established fully analytically for several classes of equations. Another
important and new result which emerges here is that the exploitation of the
phase diffusion coefficient enables us to determine in a rather straightforward
manner the dynamical coarsening exponent. Our calculation, based on the idea
that |D(lambda)|=lambda^2/t, is exemplified on several nonlinear equations,
showing that the exact exponent is captured. Some speculations about the
extension of the present results to higher dimension are outlined.Comment: 16 pages. Only a few minor changes. Accepted for publication in
Physical Review
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
Self-organization of network dynamics into local quantized states
Self-organization and pattern formation in network-organized systems emerges
from the collective activation and interaction of many interconnected units. A
striking feature of these non-equilibrium structures is that they are often
localized and robust: only a small subset of the nodes, or cell assembly, is
activated. Understanding the role of cell assemblies as basic functional units
in neural networks and socio-technical systems emerges as a fundamental
challenge in network theory. A key open question is how these elementary
building blocks emerge, and how they operate, linking structure and function in
complex networks. Here we show that a network analogue of the Swift-Hohenberg
continuum model---a minimal-ingredients model of nodal activation and
interaction within a complex network---is able to produce a complex suite of
localized patterns. Hence, the spontaneous formation of robust operational cell
assemblies in complex networks can be explained as the result of
self-organization, even in the absence of synaptic reinforcements. Our results
show that these self-organized, local structures can provide robust functional
units to understand natural and socio-technical network-organized processes.Comment: 11 pages, 4 figure
Localised states in an extended Swift-Hohenberg equation
Recent work on the behaviour of localised states in pattern forming partial
differential equations has focused on the traditional model Swift-Hohenberg
equation which, as a result of its simplicity, has additional structure --- it
is variational in time and conservative in space. In this paper we investigate
an extended Swift-Hohenberg equation in which non-variational and
non-conservative effects play a key role. Our work concentrates on aspects of
this much more complicated problem. Firstly we carry out the normal form
analysis of the initial pattern forming instability that leads to
small-amplitude localised states. Next we examine the bifurcation structure of
the large-amplitude localised states. Finally we investigate the temporal
stability of one-peak localised states. Throughout, we compare the localised
states in the extended Swift-Hohenberg equation with the analogous solutions to
the usual Swift-Hohenberg equation
Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation
In this paper we consider the spectral and nonlinear stability of periodic
traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In
particular, we resolve the long-standing question of nonlinear modulational
stability by demonstrating that spectrally stable waves are nonlinearly stable
when subject to small localized (integrable) perturbations. Our analysis is
based upon detailed estimates of the linearized solution operator, which are
complicated by the fact that the (necessarily essential) spectrum of the
associated linearization intersects the imaginary axis at the origin. We carry
out a numerical Evans function study of the spectral problem and find bands of
spectrally stable periodic traveling waves, in close agreement with previous
numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy,
Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also
compare predictions of the associated Whitham modulation equations, which
formally describe the dynamics of weak large scale perturbations of a periodic
wave train, with numerical time evolution studies, demonstrating their
effectiveness at a practical level. For the reader's convenience, we include in
an appendix the corresponding treatment of the Swift-Hohenberg equation, a
nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation
for which the nonlinear stability analysis is considerably simpler, together
with numerical Evans function analyses extending spectral stability analyses of
Mielke and Schneider.Comment: 78 pages, 11 figure
Continuation for thin film hydrodynamics and related scalar problems
This chapter illustrates how to apply continuation techniques in the analysis
of a particular class of nonlinear kinetic equations that describe the time
evolution through transport equations for a single scalar field like a
densities or interface profiles of various types. We first systematically
introduce these equations as gradient dynamics combining mass-conserving and
nonmass-conserving fluxes followed by a discussion of nonvariational amendmends
and a brief introduction to their analysis by numerical continuation. The
approach is first applied to a number of common examples of variational
equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including
certain thin-film equations for partially wetting liquids on homogeneous and
heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal
equations. Second we consider nonvariational examples as the
Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard
equations and thin-film equations describing stationary sliding drops and a
transversal front instability in a dip-coating. Through the different examples
we illustrate how to employ the numerical tools provided by the packages
auto07p and pde2path to determine steady, stationary and time-periodic
solutions in one and two dimensions and the resulting bifurcation diagrams. The
incorporation of boundary conditions and integral side conditions is also
discussed as well as problem-specific implementation issues
Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators
Dozens of exponential integration formulas have been proposed for the
high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries
and Ginzburg-Landau equations. We report the results of extensive comparisons
in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and
higher order methods, and periodic semilinear stiff PDEs with constant
coefficients. Our conclusion is that it is hard to do much better than one of
the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews
Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity
The conserved Swift-Hohenberg equation with cubic nonlinearity provides the
simplest microscopic description of the thermodynamic transition from a fluid
state to a crystalline state. The resulting phase field crystal model describes
a variety of spatially localized structures, in addition to different spatially
extended periodic structures. The location of these structures in the
temperature versus mean order parameter plane is determined using a combination
of numerical continuation in one dimension and direct numerical simulation in
two and three dimensions. Localized states are found in the region of
thermodynamic coexistence between the homogeneous and structured phases, and
may lie outside of the binodal for these states. The results are related to the
phenomenon of slanted snaking but take the form of standard homoclinic snaking
when the mean order parameter is plotted as a function of the chemical
potential, and are expected to carry over to related models with a conserved
order parameter.Comment: 40 pages, 13 figure
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