309 research outputs found
Normalizations with exponentially small remainders for nonautonomous analytic periodic vector fields
In this paper we deal with analytic nonautonomous vector fields with a
periodic time-dependancy, that we study near an equilibrium point. In a first
part, we assume that the linearized system is split in two invariant subspaces
E0 and E1. Under light diophantine conditions on the eigenvalues of the linear
part, we prove that there is a polynomial change of coordinates in E1 allowing
to eliminate up to a finite polynomial order all terms depending only on the
coordinate u0 of E0 in the E1 component of the vector field. We moreover show
that, optimizing the choice of the degree of the polynomial change of
coordinates, we get an exponentially small remainder. In the second part, we
prove a normal form theorem with exponentially small remainder. Similar
theorems have been proved before in the autonomous case : this paper
generalizes those results to the nonautonomous periodic case
Symmetry Breaking in Symmetric and Asymmetric Double-Well Potentials
Motivated by recent experimental studies of matter-waves and optical beams in
double well potentials, we study the solutions of the nonlinear Schr\"{o}dinger
equation in such a context. Using a Galerkin-type approach, we obtain a
detailed handle on the nonlinear solution branches of the problem, starting
from the corresponding linear ones and predict the relevant bifurcations of
solutions for both attractive and repulsive nonlinearities. The results
illustrate the nontrivial differences that arise between the steady
states/bifurcations emerging in symmetric and asymmetric double wells
Stable periodic density waves in dipolar Bose-Einstein condensates trapped in optical lattices
Density-wave patterns in (quasi-) discrete media with local interactions are
known to be unstable. We demonstrate that \emph{stable} double- and triple-
period patterns (DPPs and TPPs), with respect to the period of the underlying
lattice, exist in media with nonlocal nonlinearity. This is shown in detail for
dipolar Bose-Einstein condensates (BECs), loaded into a deep one-dimensional
(1D) optical lattice (OL), by means of analytical and numerical methods in the
tight-binding limit. The patterns featuring multiple periodicities are
generated by the modulational instability of the continuous-wave (CW) state,
whose period is identical to that of the OL. The DPP and TPP emerge via phase
transitions of the second and first kind, respectively. The emerging patterns
may be stable provided that the dipole-dipole (DD) interactions are repulsive
and sufficiently strong, in comparison with the local repulsive nonlinearity.
Within the set of the considered states, the TPPs realize a minimum of the free
energy. Accordingly, a vast stability region for the TPPs is found in the
parameter space, while the DPP\ stability region is relatively narrow. The same
mechanism may create stable density-wave patterns in other physical media
featuring nonlocal interactions, such as arrayed optical waveguides with
thermal nonlinearity.Comment: 7 pages, 4 figures, Phys. Rev. Lett., in pres
Gravity travelling waves for two superposed fluid layers, one being of infinite depth: a new type of bifurcation
International audienceIn this paper, we study the travelling gravity waves in a system of two layers of perfect fluids, the bottom one being infinitely deep, the upper one having a finite thickness h. We assume that the flow is potential, and the dimensionless parameters are the ratio between densities Ï = Ï 2 /Ï 1 and λ = gh/c^2. We study special values of the parameters such that λ(1 â Ï) is near 1 â , where a bifurcation of a new type occurs. We formulate the problem as a spatial reversible dynamical system, where U = 0 corresponds to a uniform state (velocity c in a moving reference frame), and we consider the linearized operator around 0. We show that its spectrum contains the entire real axis (essential spectrum), with in addition a double eigenvalue in 0, a pair of simple imaginary eigenvalues ±iλ at a distance O(1) from 0, and for λ(1 â Ï) above 1, another pair of simple imaginary eigenvalues tending towards 0 as λ(1 â Ï) â 1 +. When λ(1 â Ï) †1 this pair disappears into the essential spectrum. The rest of the spectrum lies at a distance at least O(1) from the imaginary axis. We show in this paper that for λ(1 â Ï) close to 1 â , there is a family of periodic solutions like in the Lyapunov-Devaney theorem (despite the resonance due to the point 0 in the spectrum). Moreover, showing that the full system can be seen as a perturbation of the Benjamin-Ono equation, coupled with a nonlinear oscillation, we also prove the existence of a family of homoclinic connections to these periodic orbits, provided that these ones are not too small
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
A model for the orientational ordering of the plant microtubule cortical array
The plant microtubule cortical array is a striking feature of all growing
plant cells. It consists of a more or less homogeneously distributed array of
highly aligned microtubules connected to the inner side of the plasma membrane
and oriented transversely to the cell growth axis. Here we formulate a
continuum model to describe the origin of orientational order in such confined
arrays of dynamical microtubules. The model is based on recent experimental
observations that show that a growing cortical microtubule can interact through
angle dependent collisions with pre-existing microtubules that can lead either
to co-alignment of the growth, retraction through catastrophe induction or
crossing over the encountered microtubule. We identify a single control
parameter, which is fully determined by the nucleation rate and intrinsic
dynamics of individual microtubules. We solve the model analytically in the
stationary isotropic phase, discuss the limits of stability of this isotropic
phase, and explicitly solve for the ordered stationary states in a simplified
version of the model.Comment: 15 pages, 5 figure
Separatrix splitting at a Hamiltonian bifurcation
We discuss the splitting of a separatrix in a generic unfolding of a
degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We
assume that the unperturbed fixed point has two purely imaginary eigenvalues
and a double zero one. It is well known that an one-parametric unfolding of the
corresponding Hamiltonian can be described by an integrable normal form. The
normal form has a normally elliptic invariant manifold of dimension two. On
this manifold, the truncated normal form has a separatrix loop. This loop
shrinks to a point when the unfolding parameter vanishes. Unlike the normal
form, in the original system the stable and unstable trajectories of the
equilibrium do not coincide in general. The splitting of this loop is
exponentially small compared to the small parameter. This phenomenon implies
non-existence of single-round homoclinic orbits and divergence of series in the
normal form theory. We derive an asymptotic expression for the separatrix
splitting. We also discuss relations with behaviour of analytic continuation of
the system in a complex neighbourhood of the equilibrium
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]
Action minimizing fronts in general FPU-type chains
We study atomic chains with nonlinear nearest neighbour interactions and
prove the existence of fronts (heteroclinic travelling waves with constant
asymptotic states). Generalizing recent results of Herrmann and Rademacher we
allow for non-convex interaction potentials and find fronts with non-monotone
profile. These fronts minimize an action integral and can only exists if the
asymptotic states fulfil the macroscopic constraints and if the interaction
potential satisfies a geometric graph condition. Finally, we illustrate our
findings by numerical simulations.Comment: 19 pages, several figure
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