4,736 research outputs found
Relative Periodic Solutions of the Complex Ginzburg-Landau Equation
A method of finding relative periodic orbits for differential equations with
continuous symmetries is described and its utility demonstrated by computing
relative periodic solutions for the one-dimensional complex Ginzburg-Landau
equation (CGLE) with periodic boundary conditions. A relative periodic solution
is a solution that is periodic in time, up to a transformation by an element of
the equation's symmetry group. With the method used, relative periodic
solutions are represented by a space-time Fourier series modified to include
the symmetry group element and are sought as solutions to a system of nonlinear
algebraic equations for the Fourier coefficients, group element, and time
period. The 77 relative periodic solutions found for the CGLE exhibit a wide
variety of temporal dynamics, with the sum of their positive Lyapunov exponents
varying from 5.19 to 60.35 and their unstable dimensions from 3 to 8.
Preliminary work indicates that weighted averages over the collection of
relative periodic solutions accurately approximate the value of several
functionals on typical trajectories.Comment: 32 pages, 12 figure
Coupled complex Ginzburg-Landau systems with saturable nonlinearity and asymmetric cross-phase modulation
We formulate and study dynamics from a complex Ginzburg-Landau system with
saturable nonlinearity, including asymmetric cross-phase modulation (XPM)
parameters. Such equations can model phenomena described by complex
Ginzburg-Landau systems under the added assumption of saturable media. When the
saturation parameter is set to zero, we recover a general complex cubic
Ginzburg-Landau system with XPM. We first derive conditions for the existence
of bounded dynamics, approximating the absorbing set for solutions. We use this
to then determine conditions for amplitude death of a single wavefunction. We
also construct exact plane wave solutions, and determine conditions for their
modulational instability. In a degenerate limit where dispersion and
nonlinearity balance, we reduce our system to a saturable nonlinear
Schr\"odinger system with XPM parameters, and we demonstrate the existence and
behavior of spatially heterogeneous stationary solutions in this limit. Using
numerical simulations we verify the aforementioned analytical results, while
also demonstrating other interesting emergent features of the dynamics, such as
spatiotemporal chaos in the presence of modulational instability. In other
regimes, coherent patterns including uniform states or banded structures arise,
corresponding to certain stable stationary states. For sufficiently large yet
equal XPM parameters, we observe a segregation of wavefunctions into different
regions of the spatial domain, while when XPM parameters are large and take
different values, one wavefunction may decay to zero in finite time over the
spatial domain (in agreement with the amplitude death predicted analytically).
While saturation will often regularize the dynamics, such transient dynamics
can still be observed - and in some cases even prolonged - as the saturability
of the media is increased, as the saturation may act to slow the timescale.Comment: 36 page
Analytical Proof of Space-Time Chaos in Ginzburg-Landau Equations
We prove that the attractor of the 1D quintic complex Ginzburg-Landau
equation with a broken phase symmetry has strictly positive space-time entropy
for an open set of parameter values. The result is obtained by studying chaotic
oscillations in grids of weakly interacting solitons in a class of
Ginzburg-Landau type equations. We provide an analytic proof for the existence
of two-soliton configurations with chaotic temporal behavior, and construct
solutions which are closed to a grid of such chaotic soliton pairs, with every
pair in the grid well spatially separated from the neighboring ones for all
time. The temporal evolution of the well-separated multi-soliton structures is
described by a weakly coupled lattice dynamical system (LDS) for the
coordinates and phases of the solitons. We develop a version of normal
hyperbolicity theory for the weakly coupled LDSs with continuous time and
establish for them the existence of space-time chaotic patterns similar to the
Sinai-Bunimovich chaos in discrete-time LDSs. While the LDS part of the theory
may be of independent interest, the main difficulty addressed in the paper
concerns with lifting the space-time chaotic solutions of the LDS back to the
initial PDE. The equations we consider here are space-time autonomous, i.e. we
impose no spatial or temporal modulation which could prevent the individual
solitons in the grid from drifting towards each other and destroying the
well-separated grid structure in a finite time. We however manage to show that
the set of space-time chaotic solutions for which the random soliton drift is
arrested is large enough, so the corresponding space-time entropy is strictly
positive
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
Disordered Regimes of the one-dimensional complex Ginzburg-Landau equation
I review recent work on the ``phase diagram'' of the one-dimensional complex
Ginzburg-Landau equation for system sizes at which chaos is extensive.
Particular attention is paid to a detailed description of the spatiotemporally
disordered regimes encountered. The nature of the transition lines separating
these phases is discussed, and preliminary results are presented which aim at
evaluating the phase diagram in the infinite-size, infinite-time, thermodynamic
limit.Comment: 14 pages, LaTeX, 9 figures available by anonymous ftp to
amoco.saclay.cea.fr in directory pub/chate, or by requesting them to
[email protected]
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