158 research outputs found
Numerical continuation for fractional PDEs: sharp teeth and bloated snakes
Partial differential equations (PDEs) involving fractional Laplace operators
have been increasingly used to model non-local diffusion processes and are
actively investigated using both analytical and numerical approaches. The
purpose of this work is to study the effects of the spectral fractional
Laplacian on the bifurcation structure of reaction-diffusion systems on bounded
domains. In order to do this we use advanced numerical continuation techniques
to compute the solution branches. Since current available continuation packages
only support systems involving the standard Laplacian, we first extend the
pde2path software to treat fractional PDEs. The new capabilities are then
applied to the study of the Allen-Cahn equation, the Swift-Hohenberg equation
and the Schnakenberg system (in which the standard Laplacian is each replaced
by the spectral fractional Laplacian). Our study reveals some common effects,
which contributes to a better understanding of fractional diffusion in generic
reaction-diffusion systems. In particular, we investigate the changes in
snaking bifurcation diagrams and also study the spatial structure of
non-trivial steady states upon variation of the order of the fractional
Laplacian. Our results show that the fractional order can induce very
significant qualitative and quantitative changes in global bifurcation
structures
Onset of Patterns in an Ocillated Granular Layer: Continuum and Molecular Dynamics Simulations
We study the onset of patterns in vertically oscillated layers of
frictionless dissipative particles. Using both numerical solutions of continuum
equations to Navier-Stokes order and molecular dynamics (MD) simulations, we
find that standing waves form stripe patterns above a critical acceleration of
the cell. Changing the frequency of oscillation of the cell changes the
wavelength of the resulting pattern; MD and continuum simulations both yield
wavelengths in accord with previous experimental results. The value of the
critical acceleration for ordered standing waves is approximately 10% higher in
molecular dynamics simulations than in the continuum simulations, and the
amplitude of the waves differs significantly between the models. The delay in
the onset of order in molecular dynamics simulations and the amplitude of noise
below this onset are consistent with the presence of fluctuations which are
absent in the continuum theory. The strength of the noise obtained by fit to
Swift-Hohenberg theory is orders of magnitude larger than the thermal noise in
fluid convection experiments, and is comparable to the noise found in
experiments with oscillated granular layers and in recent fluid experiments on
fluids near the critical point. Good agreement is found between the mean field
value of onset from the Swift-Hohenberg fit and the onset in continuum
simulations. Patterns are compared in cells oscillated at two different
frequencies in MD; the layer with larger wavelength patterns has less noise
than the layer with smaller wavelength patterns.Comment: Published in Physical Review
Anyonic Defect Braiding and Spontaneous Chiral Symmetry Breaking in Dihedral Liquid Crystals
Dihedral ('-atic') liquid crystals (DLCs) are assemblies of microscopic
constituent particles that exhibit -fold discrete rotational and reflection
symmetries. Generalizing the half-integer defects in nematic liquid crystals,
two-dimensional -atic DLCs can host point defects of fractional topological
charge . Starting from a generic microscopic model, we derive a
unified hydrodynamic description of DLCs with aligning or anti-aligning
short-range interactions in terms of Ginzburg-Landau and
Landau-Brazovskii-Swift-Hohenberg theories for a universal complex
order-parameter field. Building on this framework, we demonstrate in both
particle and continuum simulations how adiabatic braiding protocols,
implemented through suitable boundary conditions, can emulate anyonic exchange
behavior in a classical system. Analytic solutions and simulations of the
mean-field theory further predict a novel spontaneous chiral symmetry breaking
transition in anti-aligning DLCs, in quantitative agreement with the patterns
observed in particle simulations.Comment: Figs. 3, 6 and S1 added; Analytic solutions added (Sec. V.B.4,
Appendix C), Refs. [8], [14], [26], [53]-[60], [80], [81] adde
Two-dimensional structures in the quintic Ginzburg-Landau equation
By using ZEUS cluster at Embry-Riddle Aeronautical University we perform
extensive numerical simulations based on a two-dimensional Fourier spectral
method Fourier spatial discretization and an explicit scheme for time
differencing) to find the range of existence of the spatiotemporal solitons of
the two-dimensional complex Ginzburg-Landau equation with cubic and quintic
nonlinearities. We start from the parameters used by Akhmediev {\it et. al.}
and slowly vary them one by one to determine the regimes where solitons exist
as stable/unstable structures. We present eight classes of dissipative solitons
from which six are known (stationary, pulsating, vortex spinning, filament,
exploding, creeping) and two are novel (creeping-vortex propellers and spinning
"bean-shaped" solitons). By running lengthy simulations for the different
parameters of the equation, we find ranges of existence of stable structures
(stationary, pulsating, circular vortex spinning, organized exploding), and
unstable structures (elliptic vortex spinning that leads to filament,
disorganized exploding, creeping). Moreover, by varying even the two initial
conditions together with vorticity, we find a richer behavior in the form of
creeping-vortex propellers, and spinning "bean-shaped" solitons. Each class
differentiates from the other by distinctive features of their energy
evolution, shape of initial conditions, as well as domain of existence of
parameters.Comment: 19 pages, 19 figures, 8 tables, updated text and reference
The effect of cusps in time-dependent quantum mechanics
Spatial cusps in initial wavefunctions can lead to non-analytic behavior in
time. We suggest a method for calculating the short-time behavior in such
situations. For these cases, the density does not match its Taylor-expansion in
time, but the Runge-Gross proof of time-dependent density functional theory
still holds, as it requires only the potential to be time-analytic.Comment: 5 pages, 1 figur
Pulses and Snakes in Ginzburg--Landau Equation
Using a variational formulation for partial differential equations (PDEs)
combined with numerical simulations on ordinary differential equations (ODEs),
we find two categories (pulses and snakes) of dissipative solitons, and analyze
the dependence of both their shape and stability on the physical parameters of
the cubic-quintic Ginzburg-Landau equation (CGLE). In contrast to the regular
solitary waves investigated in numerous integrable and non-integrable systems
over the last three decades, these dissipative solitons are not stationary in
time. Rather, they are spatially confined pulse-type structures whose envelopes
exhibit complicated temporal dynamics. Numerical simulations reveal very
interesting bifurcations sequences as the parameters of the CGLE are varied.
Our predictions on the variation of the soliton amplitude, width, position,
speed and phase of the solutions using the variational formulation agree with
simulation results.Comment: 30 pages, 14 figure
Modulational Instability in Equations of KdV Type
It is a matter of experience that nonlinear waves in dispersive media,
propagating primarily in one direction, may appear periodic in small space and
time scales, but their characteristics --- amplitude, phase, wave number, etc.
--- slowly vary in large space and time scales. In the 1970's, Whitham
developed an asymptotic (WKB) method to study the effects of small
"modulations" on nonlinear periodic wave trains. Since then, there has been a
great deal of work aiming at rigorously justifying the predictions from
Whitham's formal theory. We discuss recent advances in the mathematical
understanding of the dynamics, in particular, the instability of slowly
modulated wave trains for nonlinear dispersive equations of KdV type.Comment: 40 pages. To appear in upcoming title in Lecture Notes in Physic
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