310 research outputs found
Contour dynamics model for electric discharges
A contour dynamics model for electrical discharges is obtained and analyzed.
The model is deduced as the asymptotic limit of the minimal streamer model for
the propagation of electric discharges, in the limit of small electron
diffusion. The dispersion relation for a non planar 2-D discharge is
calculated. The development and propagation of finger-like patterns are studied
and their main features quantified.Comment: 4 pages, 2 fi
Constrained Reductions of 2D dispersionless Toda Hierarchy, Hamiltonian Structure and Interface Dynamics
Finite-dimensional reductions of the 2D dispersionless Toda hierarchy,
constrained by the ``string equation'' are studied. These include solutions
determined by polynomial, rational or logarithmic functions, which are of
interest in relation to the ``Laplacian growth'' problem governing interface
dynamics. The consistency of such reductions is proved, and the Hamiltonian
structure of the reduced dynamics is derived. The Poisson structure of the
rationally reduced dispersionless Toda hierarchies is also derivedComment: 18 pages LaTex, accepted to J.Math.Phys, Significantly updated
version of the previous submissio
Electric discharge contour dynamics model: the effects of curvature and finite conductivity
In this paper we present the complete derivation of the effective contour
model for electrical discharges which appears as the asymptotic limit of the
minimal streamer model for the propagation of electric discharges, when the
electron diffusion is small. It consists of two integro-differential equations
defined at the boundary of the plasma region: one for the motion and a second
equation for the net charge density at the interface. We have computed explicit
solutions with cylindrical symmetry and found the dispersion relation for small
symmetry-breaking perturbations in the case of finite resistivity. We implement
a numerical procedure to solve our model in general situations. As a result we
compute the dispersion relation for the cylindrical case and compare it with
the analytical predictions. Comparisons with experimental data for a 2-D
positive streamers discharge are provided and predictions confirmed.Comment: 23 pages, 3 figure
Diffusion-Limited Aggregation on Curved Surfaces
We develop a general theory of transport-limited aggregation phenomena
occurring on curved surfaces, based on stochastic iterated conformal maps and
conformal projections to the complex plane. To illustrate the theory, we use
stereographic projections to simulate diffusion-limited-aggregation (DLA) on
surfaces of constant Gaussian curvature, including the sphere () and
pseudo-sphere (), which approximate "bumps" and "saddles" in smooth
surfaces, respectively. Although curvature affects the global morphology of the
aggregates, the fractal dimension (in the curved metric) is remarkably
insensitive to curvature, as long as the particle size is much smaller than the
radius of curvature. We conjecture that all aggregates grown by conformally
invariant transport on curved surfaces have the same fractal dimension as DLA
in the plane. Our simulations suggest, however, that the multifractal
dimensions increase from hyperbolic () geometry, which
we attribute to curvature-dependent screening of tip branching.Comment: 4 pages, 3 fig
Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the KdV hierarchy
We show that unstable fingering patterns of two dimensional flows of viscous
fluids with open boundary are described by a dispersionless limit of the KdV
hierarchy. In this framework, the fingering instability is linked to a known
instability leading to regularized shock solutions for nonlinear waves, in
dispersive media. The integrable structure of the flow suggests a dispersive
regularization of the finite-time singularities.Comment: Published versio
Anomalous diffusion and anisotropic nonlinear Fokker-Planck equation
We analyse a bidimensional nonlinear Fokker-Planck equation by considering an
anisotropic case, whose diffusion coefficients are
and with . In this
context, we also investigate two situations with the drift force
. The first one is characterized by
and the second is given by and
. In these cases, we can verify an anomalous behavior induced in
different directions by the drift force applied. The found results are exact
and exhibit, in terms of the -exponentials, functions which emerge from the
Tsallis formalism. The generalization for the -dimensional case is
discussed.Comment: 6 pages, tex fil
Logarithmic diffusion and porous media equations: a unified description
In this work we present the logarithmic diffusion equation as a limit case
when the index that characterizes a nonlinear Fokker-Planck equation, in its
diffusive term, goes to zero. A linear drift and a source term are considered
in this equation. Its solution has a lorentzian form, consequently this
equation characterizes a super diffusion like a L\'evy kind. In addition is
obtained an equation that unifies the porous media and the logarithmic
diffusion equations, including a generalized diffusion equation in fractal
dimension. This unification is performed in the nonextensive thermostatistics
context and increases the possibilities about the description of anomalous
diffusive processes.Comment: 5 pages. To appear in Phys. Rev.
Bubble break-off in Hele-Shaw flows : Singularities and integrable structures
Bubbles of inviscid fluid surrounded by a viscous fluid in a Hele-Shaw cell
can merge and break-off. During the process of break-off, a thinning neck
pinches off to a universal self-similar singularity. We describe this process
and reveal its integrable structure: it is a solution of the dispersionless
limit of the AKNS hierarchy. The singular break-off patterns are universal, not
sensitive to details of the process and can be seen experimentally. We briefly
discuss the dispersive regularization of the Hele-Shaw problem and the
emergence of the Painlev\'e II equation at the break-off.Comment: 27 pages, 9 figures; typo correcte
Laplacian Growth and Whitham Equations of Soliton Theory
The Laplacian growth (the Hele-Shaw problem) of multi-connected domains in
the case of zero surface tension is proven to be equivalent to an integrable
systems of Whitham equations known in soliton theory. The Whitham equations
describe slowly modulated periodic solutions of integrable hierarchies of
nonlinear differential equations. Through this connection the Laplacian growth
is understood as a flow in the moduli space of Riemann surfaces.Comment: 33 pages, 7 figures, typos corrected, new references adde
Solutions For A Generalized Fractional Anomalous Diffusion Equation
In this paper, we investigate the solutions for a generalized fractional
diffusion equation that extends some known diffusion equations by taking a
spatial time-dependent diffusion coefficient and an external force into
account, which subjects to the natural boundaries and the generic initial
condition. We obtain explicit analytical expressions for the probability
distribution and study the relation between our solutions and those obtained
within the maximum entropy principle by using the Tsallis entropy.Comment: 10 pages, LaTeX, 3 figure
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