180 research outputs found
Takens-Bogdanov bifurcation of travelling wave solutions in pipe flow
The appearance of travelling-wave-type solutions in pipe Poiseuille flow that
are disconnected from the basic parabolic profile is numerically studied in
detail. We focus on solutions in the 2-fold azimuthally-periodic subspace
because of their special stability properties, but relate our findings to other
solutions as well. Using time-stepping, an adapted Krylov-Newton method and
Arnoldi iteration for the computation and stability analysis of relative
equilibria, and a robust pseudo-arclength continuation scheme we unfold a
double-zero (Takens-Bogdanov) bifurcating scenario as a function of Reynolds
number (Re) and wavenumber (k). This scenario is extended, by the inclusion of
higher order terms in the normal form, to account for the appearance of
supercritical modulated waves emanating from the upper branch of solutions at a
degenerate Hopf bifurcation. These waves are expected to disappear in
saddle-loop bifurcations upon collision with lower-branch solutions, thereby
leaving stable upper-branch solutions whose subsequent secondary bifurcations
could contribute to the formation of the phase space structures that are
required for turbulent dynamics at higher Re.Comment: 26 pages, 15 figures (pdf and png). Submitted to J. Fluid Mec
Nonlinear modes and symmetry breaking in rotating double-well potentials
We study modes trapped in a rotating ring carrying the self-focusing (SF) or
defocusing (SDF) cubic nonlinearity and double-well potential , where is the angular coordinate. The model, based on the nonlinear
Schr\"{o}dinger (NLS) equation in the rotating reference frame, describes the
light propagation in a twisted pipe waveguide, as well as in other optical
settings, and also a Bose-Einstein condensate (BEC)trapped in a torus and
dragged by the rotating potential. In the SF and SDF regimes, five and four
trapped modes of different symmetries are found, respectively. The shapes and
stability of the modes, and transitions between them are studied in the first
rotational Brillouin zone. In the SF regime, two symmetry-breaking transitions
are found, of subcritical and supercritical types. In the SDF regime, an
antisymmetry-breaking transition occurs. Ground-states are identified in both
the SF and SDF systems.Comment: Physical Review A, in pres
Thermodynamic Limit Of The Ginzburg-Landau Equations
We investigate the existence of a global semiflow for the complex
Ginzburg-Landau equation on the space of bounded functions in unbounded domain.
This semiflow is proven to exist in dimension 1 and 2 for any parameter values
of the standard cubic Ginzburg-Landau equation. In dimension 3 we need some
restrictions on the parameters but cover nevertheless some part of the
Benjamin-Feijer unstable domain.Comment: uuencoded dvi file (email: [email protected]
Three-dimensional gravity-capillary solitary waves in water of finite depth and related problems
A dimension-breaking phenomenon for water waves with weak surface tension
It is well known that the water-wave problem with weak surface tension has
small-amplitude line solitary-wave solutions which to leading order are
described by the nonlinear Schr\"odinger equation. The present paper contains
an existence theory for three-dimensional periodically modulated solitary-wave
solutions which have a solitary-wave profile in the direction of propagation
and are periodic in the transverse direction; they emanate from the line
solitary waves in a dimension-breaking bifurcation. In addition, it is shown
that the line solitary waves are linearly unstable to long-wavelength
transverse perturbations. The key to these results is a formulation of the
water wave problem as an evolutionary system in which the transverse horizontal
variable plays the role of time, a careful study of the purely imaginary
spectrum of the operator obtained by linearising the evolutionary system at a
line solitary wave, and an application of an infinite-dimensional version of
the classical Lyapunov centre theorem.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s00205-015-0941-
Symmetric and asymmetric localized modes in linear lattices with an embedded pair of -nonlinear sites
We construct families of symmetric, antisymmetric, and asymmetric solitary
modes in one-dimensional bichromatic lattices with the
second-harmonic-generating () nonlinearity concentrated at a pair
of sites placed at distance . The lattice can be built as an array of
optical waveguides. Solutions are obtained in an implicit analytical form,
which is made explicit in the case of adjacent nonlinear sites, . The
stability is analyzed through the computation of eigenvalues for small
perturbations, and verified by direct simulations. In the cascading limit,
which corresponds to large mismatch , the system becomes tantamount to the
recently studied single-component lattice with two embedded sites carrying the
cubic nonlinearity. The modes undergo qualitative changes with the variation of
. In particular, at , the symmetry-breaking bifurcation (SBB),
which creates asymmetric states from symmetric ones, is supercritical and
subcritical for small and large values of , respectively, while the
bifurcation is always supercritical at . In the experiment, the
corresponding change of the phase transition between the second and first kinds
may be implemented by varying the mismatch, via the wavelength of the input
beam. The existence threshold (minimum total power) for the symmetric modes
vanishes exactly at , which suggests a possibility to create the solitary
mode using low-power beams. The stability of solution families also changes
with
Hierarchical model for the scale-dependent velocity of seismic waves
Elastic waves of short wavelength propagating through the upper layer of the
Earth appear to move faster at large separations of source and receiver than at
short separations. This scale dependent velocity is a manifestation of Fermat's
principle of least time in a medium with random velocity fluctuations. Existing
perturbation theories predict a linear increase of the velocity shift with
increasing separation, and cannot describe the saturation of the velocity shift
at large separations that is seen in computer simulations. Here we show that
this long-standing problem in seismology can be solved using a model developed
originally in the context of polymer physics. We find that the saturation
velocity scales with the four-third power of the root-mean-square amplitude of
the velocity fluctuations, in good agreement with the computer simulations.Comment: 7 pages including 3 figure
Open TURNS: An industrial software for uncertainty quantification in simulation
The needs to assess robust performances for complex systems and to answer
tighter regulatory processes (security, safety, environmental control, and
health impacts, etc.) have led to the emergence of a new industrial simulation
challenge: to take uncertainties into account when dealing with complex
numerical simulation frameworks. Therefore, a generic methodology has emerged
from the joint effort of several industrial companies and academic
institutions. EDF R&D, Airbus Group and Phimeca Engineering started a
collaboration at the beginning of 2005, joined by IMACS in 2014, for the
development of an Open Source software platform dedicated to uncertainty
propagation by probabilistic methods, named OpenTURNS for Open source Treatment
of Uncertainty, Risk 'N Statistics. OpenTURNS addresses the specific industrial
challenges attached to uncertainties, which are transparency, genericity,
modularity and multi-accessibility. This paper focuses on OpenTURNS and
presents its main features: openTURNS is an open source software under the LGPL
license, that presents itself as a C++ library and a Python TUI, and which
works under Linux and Windows environment. All the methodological tools are
described in the different sections of this paper: uncertainty quantification,
uncertainty propagation, sensitivity analysis and metamodeling. A section also
explains the generic wrappers way to link openTURNS to any external code. The
paper illustrates as much as possible the methodological tools on an
educational example that simulates the height of a river and compares it to the
height of a dyke that protects industrial facilities. At last, it gives an
overview of the main developments planned for the next few years
Onset of Surface-Tension-Driven Benard Convection
Experiments with shadowgraph visualization reveal a subcritical transition to
a hexagonal convection pattern in thin liquid layers that have a free upper
surface and are heated from below. The measured critical Marangoni number (84)
and observation of hysteresis (3%) agree with theory. In some experiments,
imperfect bifurcation is observed and is attributed to deterministic forcing
caused in part by the lateral boundaries in the experiment.Comment: 4 pages. The RevTeX file has a macro allowing various styles. The
appropriate style is "mypprint" which is the defaul
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