172 research outputs found
Normal form for travelling kinks in discrete Klein-Gordon lattices
We study travelling kinks in the spatial discretizations of the nonlinear
Klein--Gordon equation, which include the discrete lattice and the
discrete sine--Gordon lattice. The differential advance-delay equation for
travelling kinks is reduced to the normal form, a scalar fourth-order
differential equation, near the quadruple zero eigenvalue. We show numerically
non-existence of monotonic kinks (heteroclinic orbits between adjacent
equilibrium points) in the fourth-order equation. Making generic assumptions on
the reduced fourth-order equation, we prove the persistence of bounded
solutions (heteroclinic connections between periodic solutions near adjacent
equilibrium points) in the full differential advanced-delay equation with the
technique of center manifold reduction. Existence and persistence of multiple
kinks in the discrete sine--Gordon equation are discussed in connection to
recent numerical results of \cite{ACR03} and results of our normal form
analysis
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 ρ = ρ 2 /ρ 1 and λ = gh/c^2. We study special values of the parameters such that λ(1 − ρ) is near 1 − , 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 − ρ) above 1, another pair of simple imaginary eigenvalues tending towards 0 as λ(1 − ρ) → 1 +. When λ(1 − ρ) ≤ 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 − ρ) close to 1 − , 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
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
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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