678 research outputs found
The fundamental solution of the unidirectional pulse propagation equation
The fundamental solution of a variant of the three-dimensional wave equation
known as "unidirectional pulse propagation equation" (UPPE) and its paraxial
approximation is obtained. It is shown that the fundamental solution can be
presented as a projection of a fundamental solution of the wave equation to
some functional subspace. We discuss the degree of equivalence of the UPPE and
the wave equation in this respect. In particular, we show that the UPPE, in
contrast to the common belief, describes wave propagation in both longitudinal
and temporal directions, and, thereby, its fundamental solution possesses a
non-causal character.Comment: accepted to J. Math. Phy
Dynamic Modes of Microcapsules in Steady Shear Flow: Effects of Bending and Shear Elasticities
The dynamics of microcapsules in steady shear flow was studied using a
theoretical approach based on three variables: The Taylor deformation parameter
, the inclination angle , and the phase angle of
the membrane rotation. It is found that the dynamic phase diagram shows a
remarkable change with an increase in the ratio of the membrane shear and
bending elasticities. A fluid vesicle (no shear elasticity) exhibits three
dynamic modes: (i) Tank-treading (TT) at low viscosity of
internal fluid ( and relaxes to constant values), (ii)
Tumbling (TB) at high ( rotates), and (iii) Swinging
(SW) at middle and high shear rate (
oscillates). All of three modes are accompanied by a membrane ()
rotation. For microcapsules with low shear elasticity, the TB phase with no
rotation and the coexistence phase of SW and TB motions are induced by
the energy barrier of rotation. Synchronization of rotation with
TB rotation or SW oscillation occurs with integer ratios of rotational
frequencies. At high shear elasticity, where a saddle point in the energy
potential disappears, intermediate phases vanish, and either or
rotation occurs. This phase behavior agrees with recent simulation results of
microcapsules with low bending elasticity.Comment: 11 pages, 14 figure
Stable spatiotemporal solitons in Bessel optical lattices
We investigate the existence and stability of three-dimensional (3D) solitons
supported by cylindrical Bessel lattices (BLs) in self-focusing media. If the
lattice strength exceeds a threshold value, we show numerically, and using the
variational approximation, that the solitons are stable within one or two
intervals of values of their norm. In the latter case, the Hamiltonian-vs.-norm
diagram has a "swallowtail" shape, with three cuspidal points. The model
applies to Bose-Einstein condensates (BECs) and to optical media with saturable
nonlinearity, suggesting new ways of making stable 3D BEC solitons and "light
bullets" of an arbitrary size.Comment: 9 pages, 4 figures, Phys. Rev. Lett., in pres
Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length
We consider, by means of the variational approximation (VA) and direct
numerical simulations of the Gross-Pitaevskii (GP) equation, the dynamics of 2D
and 3D condensates with a scattering length containing constant and
harmonically varying parts, which can be achieved with an ac magnetic field
tuned to the Feshbach resonance. For a rapid time modulation, we develop an
approach based on the direct averaging of the GP equation,without using the VA.
In the 2D case, both VA and direct simulations, as well as the averaging
method, reveal the existence of stable self-confined condensates without an
external trap, in agreement with qualitatively similar results recently
reported for spatial solitons in nonlinear optics. In the 3D case, the VA again
predicts the existence of a stable self-confined condensate without a trap. In
this case, direct simulations demonstrate that the stability is limited in
time, eventually switching into collapse, even though the constant part of the
scattering length is positive (but not too large). Thus a spatially uniform ac
magnetic field, resonantly tuned to control the scattering length, may play the
role of an effective trap confining the condensate, and sometimes causing its
collapse.Comment: 7 figure
Direct transition to high-dimensional chaos through a global bifurcation
In the present work we report on a genuine route by which a high-dimensional
(with d>4) chaotic attractor is created directly, i.e., without a
low-dimensional chaotic attractor as an intermediate step. The high-dimensional
chaotic set is created in a heteroclinic global bifurcation that yields an
infinite number of unstable tori.The mechanism is illustrated using a system
constructed by coupling three Lorenz oscillators. So, the route presented here
can be considered a prototype for high-dimensional chaotic behavior just as the
Lorenz model is for low-dimensional chaos.Comment: 7 page
Logarithmic periodicities in the bifurcations of type-I intermittent chaos
The critical relations for statistical properties on saddle-node bifurcations
are shown to display undulating fine structure, in addition to their known
smooth dependence on the control parameter. A piecewise linear map with the
type-I intermittency is studied and a log-periodic dependence is numerically
obtained for the average time between laminar events, the Lyapunov exponent and
attractor moments. The origin of the oscillations is built in the natural
probabilistic measure of the map and can be traced back to the existence of
logarithmically distributed discrete values of the control parameter giving
Markov partition. Reinjection and noise effect dependences are discussed and
indications are given on how the oscillations are potentially applicable to
complement predictions made with the usual critical exponents, taken from data
in critical phenomena.Comment: 4 pages, 6 figures, accepted for publication in PRL (2004
Nonlinearity Management in Higher Dimensions
In the present short communication, we revisit nonlinearity management of the
time-periodic nonlinear Schrodinger equation and the related averaging
procedure. We prove that the averaged nonlinear Schrodinger equation does not
support the blow-up of solutions in higher dimensions, independently of the
strength in the nonlinearity coefficient variance. This conclusion agrees with
earlier works in the case of strong nonlinearity management but contradicts
those in the case of weak nonlinearity management. The apparent discrepancy is
explained by the divergence of the averaging procedure in the limit of weak
nonlinearity management.Comment: 9 pages, 1 figure
Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia
The nonlinear stability of immiscible twoâfluid Couette flows in the presence of inertia is considered. The interface between the two viscous fluids can support insoluble surfactants and the interplay between the underlying hydrodynamic instabilities and Marangoni ef- fects is explored analytically and computationally in both two and three dimensions. Asymptotic analysis when one of the layers is thin relative to the other yields a coupled system of nonlinear equations describing the spatiotemporal evolution of the interface and its local surfactant concentration. The system is nonlocal and arises by appropri- ately matching solutions of the linearised NavierâStokes equations in the thicker layer to the solution in the thin layer. The scaled models are used to study different physical mechanisms by varying the Reynolds number, the viscosity ratio between the two layers, the total amount of surfactant present initially and a scaled P Ìeclet number measuring diffusion of surfactant along the interface. The linear stability of the underlying flow to twoâ and threeâdimensional disturbances is investigated and a Squireâs type theorem is found to hold when inertia is absent. When inertia is present, threeâdimensional distur- bances can be more unstable than twoâdimensional ones and so Squireâs theorem does not hold. The linear instabilities are followed into the nonlinear regime by solving the evo- lution equations numerically; this is achieved by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for finite Reynolds numbers indicate that for twoâdimensional flows the solutions are mostly nonlinear travelling waves of permanent form, even though these can lose stabil- ity via Hopf bifurcations to timeâperiodic travelling waves. As the length of the system (that is the wavelength of periodic waves) increases, the dynamics become more complex and include timeâperiodic, quasiâperiodic as well as chaotic fluctuations. It is also found that oneâdimensional interfacial travelling waves of permanent form can become unstable to spanwise perturbations for a wide range of parameters, producing threeâdimensional flows with interfacial profiles that are twoâdimensional and travel in the direction of the underlying shear. Nonlinear flows are also computed for parameters which predict linear instability to threeâdimensional disturbances but not twoâdimensional ones. These are found to have a oneâdimensional interface in a rotated frame with respect to the direction of the underlying shear and travel obliquely without changing form
Discrete Solitons and Vortices on Anisotropic Lattices
We consider effects of anisotropy on solitons of various types in
two-dimensional nonlinear lattices, using the discrete nonlinear
Schr{\"{o}}dinger equation as a paradigm model. For fundamental solitons, we
develop a variational approximation, which predicts that broad quasi-continuum
solitons are unstable, while their strongly anisotropic counterparts are
stable. By means of numerical methods, it is found that, in the general case,
the fundamental solitons and simplest on-site-centered vortex solitons ("vortex
crosses") feature enhanced or reduced stability areas, depending on the
strength of the anisotropy. More surprising is the effect of anisotropy on the
so-called "super-symmetric" intersite-centered vortices ("vortex squares"),
with the topological charge equal to the square's size : we predict in
an analytical form by means of the Lyapunov-Schmidt theory, and confirm by
numerical results, that arbitrarily weak anisotropy results in dramatic changes
in the stability and dynamics in comparison with the \emph{degenerate}, in this
case, isotropic limit.Comment: 10 pages + 7 figure
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