209 research outputs found
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
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
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
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
Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces
A moving frame formulation of non-stretching geometric curve flows in
Euclidean space is used to derive a 1+1 dimensional hierarchy of integrable
SO(3)-invariant vector models containing the Heisenberg ferromagnetic spin
model as well as a model given by a spin-vector version of the mKdV equation.
These models describe a geometric realization of the NLS hierarchy of soliton
equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet
equations of the moving frame. This derivation yields an explicit
bi-Hamiltonian structure, recursion operator, and constants of motion for each
model in the hierarchy. A generalization of these results to geometric surface
flows is presented, where the surfaces are non-stretching in one direction
while stretching in all transverse directions. Through the Frenet equations of
a moving frame, such surface flows are shown to encode a hierarchy of 2+1
dimensional integrable SO(3)-invariant vector models, along with their
bi-Hamiltonian structure, recursion operator, and constants of motion,
describing a geometric realization of 2+1 dimensional bi-Hamiltonian NLS and
mKdV soliton equations. Based on the well-known equivalence between the
Heisenberg model and the Schrodinger map equation in 1+1 dimensions, a
geometrical formulation of these hierarchies of 1+1 and 2+1 vector models is
given in terms of dynamical maps into the 2-sphere. In particular, this
formulation yields a new integrable generalization of the Schrodinger map
equation in 2+1 dimensions as well as a mKdV analog of this map equation
corresponding to the mKdV spin model in 1+1 and 2+1 dimensions.Comment: Published version with typos corrected. Significantly expanded
version of a talk given by the first author at the 2008 BIRS workshop on
"Geometric Flows in Mathematics and Physics
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
Electromigration-Induced Propagation of Nonlinear Surface Waves
Due to the effects of surface electromigration, waves can propagate over the
free surface of a current-carrying metallic or semiconducting film of thickness
h_0. In this paper, waves of finite amplitude, and slow modulations of these
waves, are studied. Periodic wave trains of finite amplitude are found, as well
as their dispersion relation. If the film material is isotropic, a wave train
with wavelength lambda is unstable if lambda/h_0 < 3.9027..., and is otherwise
marginally stable. The equation of motion for slow modulations of a finite
amplitude, periodic wave train is shown to be the nonlinear Schrodinger
equation. As a result, envelope solitons can travel over the film's surface.Comment: 13 pages, 2 figures. To appear in Phys. Rev.
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
Superfluid Flow Past an Array of Scatterers
We consider a model of nonlinear superfluid flow past a periodic array of
point-like scatterers in one dimension. An application of this model is the
determination of the critical current of a Josephson array in a regime
appropriate to a Ginzburg-Landau formulation. Here, the array consists of short
normal-metal regions, in the presence of a Hartree electron-electron
interaction, and embedded within a one-dimensional superconducting wire near
its critical temperature, . We predict the critical current to depend
linearly as , while the coefficient depends sensitively on the
sizes of the superconducting and normal-metal regions and the strength and sign
of the Hartree interaction. In the case of an attractive interaction, we find a
further feature: the critical current vanishes linearly at some temperature
less than , as well as at itself. We rule out a simple
explanation for the zero value of the critical current, at this temperature
, in terms of order parameter fluctuations at low frequencies.Comment: 23 pages, REVTEX, six eps-figures included; submitted to PR
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