189 research outputs found
Moving Embedded Solitons
The first theoretical results are reported predicting {\em moving} solitons
residing inside ({\it embedded} into) the continuous spectrum of radiation
modes. The model taken is a Bragg-grating medium with Kerr nonlinearity and
additional second-derivative (wave) terms. The moving embedded solitons (ESs)
are doubly isolated (of codimension 2), but, nevertheless, structurally stable.
Like quiescent ESs, moving ESs are argued to be stable to linear approximation,
and {\it semi}-stable nonlinearly. Estimates show that moving ESs may be
experimentally observed as 10 fs pulses with velocity th that
of light.Comment: 9 pages 2 figure
Embedded Solitons in a Three-Wave System
We report a rich spectrum of isolated solitons residing inside ({\it embedded
} into) the continuous radiation spectrum in a simple model of three-wave
spatial interaction in a second-harmonic-generating planar optical waveguide
equipped with a quasi-one-dimensional Bragg grating. An infinite sequence of
fundamental embedded solitons are found, which differ by the number of internal
oscillations. Branches of these zero-walkoff spatial solitons give rise,
through bifurcations, to several secondary branches of walking solitons. The
structure of the bifurcating branches suggests a multistable configuration of
spatial optical solitons, which may find straightforward applications for
all-optical switching.Comment: 5 pages 5 figures. To appear in Phys Rev
Thirring Solitons in the presence of dispersion
The effect of dispersion or diffraction on zero-velocity solitons is studied
for the generalized massive Thirring model describing a nonlinear optical fiber
with grating or parallel-coupled planar waveguides with misaligned axes. The
Thirring solitons existing at zero dispersion/diffraction are shown numerically
to be separated by a finite gap from three isolated soliton branches. Inside
the gap, there is an infinity of multi-soliton branches. Thus, the Thirring
solitons are structurally unstable. In another parameter region (far from the
Thirring limit), solitons exist everywhere.Comment: 12 pages, Latex. To appear in Phys. Rev. Let
Radiationless Travelling Waves In Saturable Nonlinear Schr\"odinger Lattices
The longstanding problem of moving discrete solitary waves in nonlinear
Schr{\"o}dinger lattices is revisited. The context is photorefractive crystal
lattices with saturable nonlinearity whose grand-canonical energy barrier
vanishes for isolated coupling strength values. {\em Genuinely localised
travelling waves} are computed as a function of the system parameters {\it for
the first time}. The relevant solutions exist only for finite velocities.Comment: 5 pages, 4 figure
Stripe to spot transition in a plant root hair initiation model
A generalised Schnakenberg reaction-diffusion system with source and loss
terms and a spatially dependent coefficient of the nonlinear term is studied
both numerically and analytically in two spatial dimensions. The system has
been proposed as a model of hair initiation in the epidermal cells of plant
roots. Specifically the model captures the kinetics of a small G-protein ROP,
which can occur in active and inactive forms, and whose activation is believed
to be mediated by a gradient of the plant hormone auxin. Here the model is made
more realistic with the inclusion of a transverse co-ordinate. Localised
stripe-like solutions of active ROP occur for high enough total auxin
concentration and lie on a complex bifurcation diagram of single and
multi-pulse solutions. Transverse stability computations, confirmed by
numerical simulation show that, apart from a boundary stripe, these 1D
solutions typically undergo a transverse instability into spots. The spots so
formed typically drift and undergo secondary instabilities such as spot
replication. A novel 2D numerical continuation analysis is performed that shows
the various stable hybrid spot-like states can coexist. The parameter values
studied lead to a natural singularly perturbed, so-called semi-strong
interaction regime. This scaling enables an analytical explanation of the
initial instability, by describing the dispersion relation of a certain
non-local eigenvalue problem. The analytical results are found to agree
favourably with the numerics. Possible biological implications of the results
are discussed.Comment: 28 pages, 44 figure
Discrete embedded solitons
We address the existence and properties of discrete embedded solitons (ESs),
i.e., localized waves existing inside the phonon band in a nonlinear
dynamical-lattice model. The model describes a one-dimensional array of optical
waveguides with both the quadratic (second-harmonic generation) and cubic
nonlinearities. A rich family of ESs was previously known in the continuum
limit of the model. First, a simple motivating problem is considered, in which
the cubic nonlinearity acts in a single waveguide. An explicit solution is
constructed asymptotically in the large-wavenumber limit. The general problem
is then shown to be equivalent to the existence of a homoclinic orbit in a
four-dimensional reversible map. From properties of such maps, it is shown that
(unlike ordinary gap solitons), discrete ESs have the same codimension as their
continuum counterparts. A specific numerical method is developed to compute
homoclinic solutions of the map, that are symmetric under a specific reversing
transformation. Existence is then studied in the full parameter space of the
problem. Numerical results agree with the asymptotic results in the appropriate
limit and suggest that the discrete ESs may be semi-stable as in the continuous
case.Comment: A revtex4 text file and 51 eps figure files. To appear in
Nonlinearit
Relativistic solitary waves modulating long laser pulses in plasmas
This article discusses the existence of solitary electromagnetic waves
trapped in a self-generated Langmuir wave and embedded in an infinitely long
circularly polarized electromagnetic wave propagating through a plasma. From
the mathematical point of view they are exact solutions of the 1-dimensional
relativistic cold fluid plasma model with nonvanishing boundary conditions.
Under the assumption of traveling wave solutions with velocity and vector
potential frequency , the fluid model is reduced to a Hamiltonian
system. The solitary waves are homoclinic (grey solitons) or heteroclinic (dark
solitons) orbits to fixed points. By using a dynamical systems description of
the Hamiltonian system and a spectral method, we identify a great variety of
solitary waves, including asymmetric ones, discuss their disappearance for
certain parameter values, and classify them according to: (i) grey or dark
character, (ii) the number of humps of the vector potential envelope and (iii)
their symmetries. The solutions come in continuous families in the parametric
plane and extend up to velocities that approach the speed of light.
The stability of certain types of grey solitary waves is investigated with the
aid of particle-in-cell simulations that demonstrate their propagation for a
few tens of the inverse of the plasma frequency.Comment: 20 pages, 10 figure
Asynchronous partial contact motion due to internal resonance in multiple degree-of-freedom rotordynamics
Sudden onset of violent chattering or whirling rotorstator contact motion in rotational machines can cause significant damage in many industrial applications. It is shown that internal resonance can lead to the onset of bouncing-type partial contact motion away from primary resonances. These partial contact limit cycles can involve any two modes of an arbitrarily high degree-of-freedom system, and can be seen as an extension of a synchronisation condition previously reported for a single disc system. The synchronisation formula predicts multiple drivespeeds, corresponding to different forms of mode-locked bouncing orbits. These results are backed up by a brute-force bifurcation analysis which reveals numerical existence of the corresponding family of bouncing orbits at supercritical drivespeeds, provided the dampingis sufficiently low. The numerics reveal many overlapping families of solutions, which leads to significant multi-stability of the response at given drive speeds. Further secondary bifurcations can also occur within each family, altering the nature of the response, and ultimately leading to chaos. It is illustrated how stiffness and damping of the stator have a large effect on the number and nature of the partial contact solutions, illustrating the extreme sensitivity that would be observed in practice
Predictions from a stochastic polymer model for the MinDE dynamics in E.coli
The spatiotemporal oscillations of the Min proteins in the bacterium
Escherichia coli play an important role in cell division. A number of different
models have been proposed to explain the dynamics from the underlying
biochemistry. Here, we extend a previously described discrete polymer model
from a deterministic to a stochastic formulation. We express the stochastic
evolution of the oscillatory system as a map from the probability distribution
of maximum polymer length in one period of the oscillation to the probability
distribution of maximum polymer length half a period later and solve for the
fixed point of the map with a combined analytical and numerical technique. This
solution gives a theoretical prediction of the distributions of both lengths of
the polar MinD zones and periods of oscillations -- both of which are
experimentally measurable. The model provides an interesting example of a
stochastic hybrid system that is, in some limits, analytically tractable.Comment: 16 page
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