42 research outputs found
Stabilized Kuramoto-Sivashinsky system
A model consisting of a mixed Kuramoto - Sivashinsky - KdV equation, linearly
coupled to an extra linear dissipative equation, is proposed. The model applies
to the description of surface waves on multilayered liquid films. The extra
equation makes its possible to stabilize the zero solution in the model,
opening way to the existence of stable solitary pulses (SPs). Treating the
dissipation and instability-generating gain in the model as small
perturbations, we demonstrate that balance between them selects two
steady-state solitons from their continuous family existing in the absence of
the dissipation and gain. The may be stable, provided that the zero solution is
stable. The prediction is completely confirmed by direct simulations. If the
integration domain is not very large, some pulses are stable even when the zero
background is unstable. Stable bound states of two and three pulses are found
too. The work was supported, in a part, by a joint grant from the Israeli
Minsitry of Science and Technology and Japan Society for Promotion of Science.Comment: A text file in the latex format and 20 eps files with figures.
Physical Review E, in pres
Potential model of a 2D Bunsen flame
The Michelson Sivashinsky equation, which models the non linear dynamics of
premixed flames, has been recently extended to describe oblique flames. This
approach was extremely successful to describe the behavior on one side of the
flame, but some qualitative effects involving the interaction of both sides of
the front were left unexplained. We use here a potential flow model, first
introduced by Frankel, to study numerically this configuration. Furthermore,
this approach allows us to provide a physical explanation of the phenomena
occuring in this geometry by means of an electrostatic analogy
Flame front propagation V: Stability Analysis of Flame Fronts: Dynamical Systems Approach in the Complex Plane
We consider flame front propagation in channel geometries. The steady state
solution in this problem is space dependent, and therefore the linear stability
analysis is described by a partial integro-differential equation with a space
dependent coefficient. Accordingly it involves complicated eigenfunctions. We
show that the analysis can be performed to required detail using a finite order
dynamical system in terms of the dynamics of singularities in the complex
plane, yielding detailed understanding of the physics of the eigenfunctions and
eigenvalues.Comment: 17 pages 7 figure
Studies of Phase Turbulence in the One Dimensional Complex Ginzburg-Landau Equation
The phase-turbulent (PT) regime for the one dimensional complex
Ginzburg-Landau equation (CGLE) is carefully studied, in the limit of large
systems and long integration times, using an efficient new integration scheme.
Particular attention is paid to solutions with a non-zero phase gradient. For
fixed control parameters, solutions with conserved average phase gradient
exist only for less than some upper limit. The transition from phase to
defect-turbulence happens when this limit becomes zero. A Lyapunov analysis
shows that the system becomes less and less chaotic for increasing values of
the phase gradient. For high values of the phase gradient a family of
non-chaotic solutions of the CGLE is found. These solutions consist of
spatially periodic or aperiodic waves travelling with constant velocity. They
typically have incommensurate velocities for phase and amplitude propagation,
showing thereby a novel type of quasiperiodic behavior. The main features of
these travelling wave solutions can be explained through a modified
Kuramoto-Sivashinsky equation that rules the phase dynamics of the CGLE in the
PT phase. The latter explains also the behavior of the maximal Lyapunov
exponents of chaotic solutions.Comment: 16 pages, LaTeX (Version 2.09), 10 Postscript-figures included,
submitted to Phys. Rev.
Biharmonic pattern selection
A new model to describe fractal growth is discussed which includes effects
due to long-range coupling between displacements . The model is based on the
biharmonic equation in two-dimensional isotropic defect-free
media as follows from the Kuramoto-Sivashinsky equation for pattern formation
-or, alternatively, from the theory of elasticity. As a difference with
Laplacian and Poisson growth models, in the new model the Laplacian of is
neither zero nor proportional to . Its discretization allows to reproduce a
transition from dense to multibranched growth at a point in which the growth
velocity exhibits a minimum similarly to what occurs within Poisson growth in
planar geometry. Furthermore, in circular geometry the transition point is
estimated for the simplest case from the relation
such that the trajectories become stable at the growing surfaces in a
continuous limit. Hence, within the biharmonic growth model, this transition
depends only on the system size and occurs approximately at a distance far from a central seed particle. The influence of biharmonic patterns on
the growth probability for each lattice site is also analysed.Comment: To appear in Phys. Rev. E. Copies upon request to
[email protected]
High-dimensional interior crisis in the Kuramoto-Sivashinsky equation
An investigation of interior crisis of high dimensions in an extended spatiotemporal system exemplified by the Kuramoto-Sivashinsky equation is reported. It is shown that unstable periodic orbits and their associated invariant manifolds in the Poincaré hyperplane can effectively characterize the global bifurcation dynamics of high-dimensional systems.A. C.-L. Chian, E. L. Rempel, E. E. Macau, R. R. Rosa, and F. Christianse
Stochastic Model for Surface Erosion Via Ion-Sputtering: Dynamical Evolution from Ripple Morphology to Rough Morphology
Surfaces eroded by ion-sputtering are sometimes observed to develop
morphologies which are either ripple (periodic), or rough (non-periodic). We
introduce a discrete stochastic model that allows us to interpret these
experimental observations within a unified framework. We find that a periodic
ripple morphology characterizes the initial stages of the evolution, whereas
the surface displays self-affine scaling in the later time regime. Further, we
argue that the stochastic continuum equation describing the surface height is a
noisy version of the Kuramoto-Sivashinsky equation.Comment: 4 pages, 7 postscript figs., Revtex, to appear in Phys. Rev. Let
Stable two-dimensional solitary pulses in linearly coupled dissipative Kadomtsev-Petviashvili equations
A two-dimensional (2D) generalization of the stabilized Kuramoto -
Sivashinsky (KS) system is presented. It is based on the Kadomtsev-Petviashvili
(KP) equation including dissipation of the generic (Newell -- Whitehead --
Segel, NWS) type and gain. The system directly applies to the description of
gravity-capillary waves on the surface of a liquid layer flowing down an
inclined plane, with a surfactant diffusing along the layer's surface.
Actually, the model is quite general, offering a simple way to stabilize
nonlinear waves in media combining the weakly-2D dispersion of the KP type with
gain and NWS dissipation. Parallel to this, another model is introduced, whose
dissipative terms are isotropic, rather than of the NWS type. Both models
include an additional linear equation of the advection-diffusion type, linearly
coupled to the main KP-NWS equation. The extra equation provides for stability
of the zero background in the system, opening a way to the existence of stable
localized pulses. The consideration is focused on the case when the dispersive
part of the system of the KP-I type, admitting the existence of 2D localized
pulses. Treating the dissipation and gain as small perturbations and making use
of the balance equation for the field momentum, we find that the equilibrium
between the gain and losses may select two 2D solitons, from their continuous
family existing in the conservative counterpart of the model (the latter family
is found in an exact analytical form). The selected soliton with the larger
amplitude is expected to be stable. Direct simulations completely corroborate
the analytical predictions.Comment: a latex text file and 16 eps files with figures; Physical Review E,
in pres
Solitary and periodic solutions of the generalized Kuramoto-Sivashinsky equation
The generalized Kuramoto-Sivashinsky equation in the case of the power
nonlinearity with arbitrary degree is considered. New exact solutions of this
equation are presented