A Kelvin-wave cascade on a vortex in superfluid 4^4He 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 $^4$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

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 $\lambda$ when the filament projection on the boundary plane has shape of a two-branch spiral or a smoothed angle, depending on the sign of $\lambda$.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-$d$ 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

Stability of the selfsimilar dynamics of a vortex filament

In this paper we continue our investigation about selfsimilar solutions of the vortex filament equation, also known as the binormal flow (BF) or the localized induction equation (LIE). Our main result is the stability of the selfsimilar dynamics of small pertubations of a given selfsimilar solution. The proof relies on finding precise asymptotics in space and time for the tangent and the normal vectors of the perturbations. A main ingredient in the proof is the control of the evolution of weighted norms for a cubic 1-D Schr\"odinger equation, connected to the binormal flow by Hasimoto's transform.Comment: revised version, 36 page

Simulating (electro)hydrodynamic effects in colloidal dispersions: smoothed profile method

Previously, we have proposed a direct simulation scheme for colloidal dispersions in a Newtonian solvent [Phys.Rev.E 71,036707 (2005)]. An improved formulation called the ``Smoothed Profile (SP) method'' is presented here in which simultaneous time-marching is used for the host fluid and colloids. The SP method is a direct numerical simulation of particulate flows and provides a coupling scheme between the continuum fluid dynamics and rigid-body dynamics through utilization of a smoothed profile for the colloidal particles. Moreover, the improved formulation includes an extension to incorporate multi-component fluids, allowing systems such as charged colloids in electrolyte solutions to be studied. The dynamics of the colloidal dispersions are solved with the same computational cost as required for solving non-particulate flows. Numerical results which assess the hydrodynamic interactions of colloidal dispersions are presented to validate the SP method. The SP method is not restricted to particular constitutive models of the host fluids and can hence be applied to colloidal dispersions in complex fluids

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 $^{85}$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

A Dissipative-Particle-Dynamics Model for Simulating Dynamics of Charged Colloid

A mesoscopic colloid model is developed in which a spherical colloid is represented by many interacting sites on its surface. The hydrodynamic interactions with thermal fluctuations are taken accounts in full using Dissipative Particle Dynamics, and the electrostatic interactions are simulated using Particle-Particle-Particle Mesh method. This new model is applied to investigate the electrophoretic mobility of a charged colloid under an external electric field, and the influence of salt concentration and colloid charge are systematically studied. The simulation results show good agreement with predictions from the electrokinetic theory.Comment: 17 pages, 8 figures, submitted to the proceedings of High Performance Computing in Science & Engineering '1

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

The Viscous Nonlinear Dynamics of Twist and Writhe

Exploiting the "natural" frame of space curves, we formulate an intrinsic dynamics of twisted elastic filaments in viscous fluids. A pair of coupled nonlinear equations describing the temporal evolution of the filament's complex curvature and twist density embodies the dynamic interplay of twist and writhe. These are used to illustrate a novel nonlinear phenomenon: ``geometric untwisting" of open filaments, whereby twisting strains relax through a transient writhing instability without performing axial rotation. This may explain certain experimentally observed motions of fibers of the bacterium B. subtilis [N.H. Mendelson, et al., J. Bacteriol. 177, 7060 (1995)].Comment: 9 pages, 4 figure