9,634 research outputs found
Small oscillations and the Heisenberg Lie algebra
The Adler Kostant Symes [A-K-S] scheme is used to describe mechanical systems
for quadratic Hamiltonians of on coadjoint orbits of the
Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie
algebra that admits an ad-invariant metric. Its quadratic induces
the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that
one on . This system is a Lax pair equation whose solution can
be computed with help of the Adjoint representation. For a certain class of
functions, the Poisson commutativity on the coadjoint orbits in
is related to the commutativity of a family of derivations of the
2n+1-dimensional Heisenberg Lie algebra . Therefore the complete
integrability is related to the existence of an n-dimensional abelian
subalgebra of certain derivations in . For instance, the motion
of n-uncoupled harmonic oscillators near an equilibrium position can be
described with this setting.Comment: 17 pages, it contains a theory about small oscillations in terms of
the AKS schem
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
Hodograph solutions of the dispersionless coupled KdV hierarchies, critical points and the Euler-Poisson-Darboux equation
It is shown that the hodograph solutions of the dispersionless coupled KdV
(dcKdV) hierarchies describe critical and degenerate critical points of a
scalar function which obeys the Euler-Poisson-Darboux equation. Singular
sectors of each dcKdV hierarchy are found to be described by solutions of
higher genus dcKdV hierarchies. Concrete solutions exhibiting shock type
singularities are presented.Comment: 19 page
Langevin equation with scale-dependent noise
A new wavelet based technique for the perturbative solution of the Langevin
equation is proposed. It is shown that for the random force acting in a limited
band of scales the proposed method directly leads to a finite result with no
renormalization required. The one-loop contribution to the Kardar-Parisi-Zhang
equation Green function for the interface growth is calculated as an example.Comment: LaTeX, 5 page
Magneto-Conductance Anisotropy and Interference Effects in Variable Range Hopping
We investigate the magneto-conductance (MC) anisotropy in the variable range
hopping regime, caused by quantum interference effects in three dimensions.
When no spin-orbit scattering is included, there is an increase in the
localization length (as in two dimensions), producing a large positive MC. By
contrast, with spin-orbit scattering present, there is no change in the
localization length, and only a small increase in the overall tunneling
amplitude. The numerical data for small magnetic fields , and hopping
lengths , can be collapsed by using scaling variables , and
in the perpendicular and parallel field orientations
respectively. This is in agreement with the flux through a `cigar'--shaped
region with a diffusive transverse dimension proportional to . If a
single hop dominates the conductivity of the sample, this leads to a
characteristic orientational `finger print' for the MC anisotropy. However, we
estimate that many hops contribute to conductivity of typical samples, and thus
averaging over critical hop orientations renders the bulk sample isotropic, as
seen experimentally. Anisotropy appears for thin films, when the length of the
hop is comparable to the thickness. The hops are then restricted to align with
the sample plane, leading to different MC behaviors parallel and perpendicular
to it, even after averaging over many hops. We predict the variations of such
anisotropy with both the hop size and the magnetic field strength. An
orientational bias produced by strong electric fields will also lead to MC
anisotropy.Comment: 24 pages, RevTex, 9 postscript figures uuencoded Submitted to PR
Nonadiabatic charged spherical evolution in the postquasistatic approximation
We apply the postquasistatic approximation, an iterative method for the
evolution of self-gravitating spheres of matter, to study the evolution of
dissipative and electrically charged distributions in General Relativity. We
evolve nonadiabatic distributions assuming an equation of state that accounts
for the anisotropy induced by the electric charge. Dissipation is described by
streaming out or diffusion approximations. We match the interior solution, in
noncomoving coordinates, with the Vaidya-Reissner-Nordstr\"om exterior
solution. Two models are considered: i) a Schwarzschild-like shell in the
diffusion limit; ii) a Schwarzschild-like interior in the free streaming limit.
These toy models tell us something about the nature of the dissipative and
electrically charged collapse. Diffusion stabilizes the gravitational collapse
producing a spherical shell whose contraction is halted in a short
characteristic hydrodynamic time. The streaming out radiation provides a more
efficient mechanism for emission of energy, redistributing the electric charge
on the whole sphere, while the distribution collapses indefinitely with a
longer hydrodynamic time scale.Comment: 11 pages, 16 Figures. Accepted for publication in Phys Rev
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