266 research outputs found
Steady Stokes flow with long-range correlations, fractal Fourier spectrum, and anomalous transport
We consider viscous two-dimensional steady flows of incompressible fluids
past doubly periodic arrays of solid obstacles. In a class of such flows, the
autocorrelations for the Lagrangian observables decay in accordance with the
power law, and the Fourier spectrum is neither discrete nor absolutely
continuous. We demonstrate that spreading of the droplet of tracers in such
flows is anomalously fast. Since the flow is equivalent to the integrable
Hamiltonian system with 1 degree of freedom, this provides an example of
integrable dynamics with long-range correlations, fractal power spectrum, and
anomalous transport properties.Comment: 4 pages, 4 figures, published in Physical Review Letter
A Kelvin-wave cascade on a vortex in superfluid He at a very low temperature
A study by computer simulation is reported of the behaviour of a quantized
vortex line at a very low temperature when there is continuous excitation of
low-frequency Kelvin waves. There is no dissipation except by phonon radiation
at a very high frequency. It is shown that non-linear coupling leads to a net
flow of energy to higher wavenumbers and to the development of a simple
spectrum of Kelvin waves that is insensitive to the strength and frequency of
the exciting drive. The results are likely to be relevant to the decay of
turbulence in superfluid He at very low temperatures
Localized induction equation and pseudospherical surfaces
We describe a close connection between the localized induction equation
hierarchy of integrable evolution equations on space curves, and surfaces of
constant negative Gauss curvature.Comment: 21 pages, AMSTeX file. To appear in Journal of Physics A:
Mathematical and Genera
Hamiltonians for curves
We examine the equilibrium conditions of a curve in space when a local energy
penalty is associated with its extrinsic geometrical state characterized by its
curvature and torsion. To do this we tailor the theory of deformations to the
Frenet-Serret frame of the curve. The Euler-Lagrange equations describing
equilibrium are obtained; Noether's theorem is exploited to identify the
constants of integration of these equations as the Casimirs of the euclidean
group in three dimensions. While this system appears not to be integrable in
general, it {\it is} in various limits of interest. Let the energy density be
given as some function of the curvature and torsion, . If
is a linear function of either of its arguments but otherwise arbitrary, we
claim that the first integral associated with rotational invariance permits the
torsion to be expressed as the solution of an algebraic equation in
terms of the bending curvature, . The first integral associated with
translational invariance can then be cast as a quadrature for or for
.Comment: 17 page
Slow flows of an relativistic perfect fluid in a static gravitational field
Relativistic hydrodynamics of an isentropic fluid in a gravitational field is
considered as the particular example from the family of Lagrangian
hydrodynamic-type systems which possess an infinite set of integrals of motion
due to the symmetry of Lagrangian with respect to relabeling of fluid particle
labels. Flows with fixed topology of the vorticity are investigated in
quasi-static regime, when deviations of the space-time metric and the density
of fluid from the corresponding equilibrium configuration are negligibly small.
On the base of the variational principle for frozen-in vortex lines dynamics,
the equation of motion for a thin relativistic vortex filament is derived in
the local induction approximation.Comment: 4 pages, revtex, no figur
The hodograph method applicability in the problem of long-scale nonlinear dynamics of a thin vortex filament near a flat boundary
Hamiltonian dynamics of a thin vortex filament in ideal incompressible fluid
near a flat fixed boundary is considered at the conditions that at any point of
the curve determining shape of the filament the angle between tangent vector
and the boundary plane is small, also the distance from a point on the curve to
the plane is small in comparison with the curvature radius. The dynamics is
shown to be effectively described by a nonlinear system of two
(1+1)-dimensional partial differential equations. The hodograph transformation
reduces that system to a single linear differential equation of the second
order with separable variables. Simple solutions of the linear equation are
investigated at real values of spectral parameter when the filament
projection on the boundary plane has shape of a two-branch spiral or a smoothed
angle, depending on the sign of .Comment: 9 pages, revtex4, 6 eps-figure
Interaction of Nonlinear Schr\"odinger Solitons with an External Potential
Employing a particularly suitable higher order symplectic integration
algorithm, we integrate the 1- nonlinear Schr\"odinger equation numerically
for solitons moving in external potentials. In particular, we study the
scattering off an interface separating two regions of constant potential. We
find that the soliton can break up into two solitons, eventually accompanied by
radiation of non-solitary waves. Reflection coefficients and inelasticities are
computed as functions of the height of the potential step and of its steepness.Comment: 14 pages, uuencoded PS-file including 10 figure
Tunable tunneling: An application of stationary states of Bose-Einstein condensates in traps of finite depth
The fundamental question of how Bose-Einstein condensates tunnel into a
barrier is addressed. The cubic nonlinear Schrodinger equation with a finite
square well potential, which models a Bose-Einstein condensate in a
quasi-one-dimensional trap of finite depth, is solved for the complete set of
localized and partially localized stationary states, which the former evolve
into when the nonlinearity is increased. An immediate application of these
different solution types is tunable tunneling. Magnetically tunable Feshbach
resonances can change the scattering length of certain Bose-condensed atoms,
such as Rb, by several orders of magnitude, including the sign, and
thereby also change the mean field nonlinearity term of the equation and the
tunneling of the wavefunction. We find both linear-type localized solutions and
uniquely nonlinear partially localized solutions where the tails of the
wavefunction become nonzero at infinity when the nonlinearity increases. The
tunneling of the wavefunction into the non-classical regime and thus its
localization therefore becomes an external experimentally controllable
parameter.Comment: 11 pages, 5 figure
Stationary solutions of the one-dimensional nonlinear Schroedinger equation: II. Case of attractive nonlinearity
All stationary solutions to the one-dimensional nonlinear Schroedinger
equation under box or periodic boundary conditions are presented in analytic
form for the case of attractive nonlinearity. A companion paper has treated the
repulsive case. Our solutions take the form of bounded, quantized, stationary
trains of bright solitons. Among them are two uniquely nonlinear classes of
nodeless solutions, whose properties and physical meaning are discussed in
detail. The full set of symmetry-breaking stationary states are described by
the character tables from the theory of point groups. We make
experimental predictions for the Bose-Einstein condensate and show that, though
these are the analog of some of the simplest problems in linear quantum
mechanics, nonlinearity introduces new and surprising phenomena.Comment: 11 pages, 9 figures -- revised versio
Finite-gap Solutions of the Vortex Filament Equation: Isoperiodic Deformations
We study the topology of quasiperiodic solutions of the vortex filament
equation in a neighborhood of multiply covered circles. We construct these
solutions by means of a sequence of isoperiodic deformations, at each step of
which a real double point is "unpinched" to produce a new pair of branch points
and therefore a solution of higher genus. We prove that every step in this
process corresponds to a cabling operation on the previous curve, and we
provide a labelling scheme that matches the deformation data with the knot type
of the resulting filament.Comment: 33 pages, 5 figures; submitted to Journal of Nonlinear Scienc
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