Vortex Filament in Three-manifold and the Duistermaat-Heckman Formula

Symplectic geometry of the vortex filament in a curved three-manifold is investigated. There appears an infinite sequence of constants of motion in involution in the case of constant curvature. The Duistermaat-Heckman formula is examined perturbatively for the classical partition function in our model and verified up to the 3-loop order.Comment: Revised to include Figure2 (a PostScript file). 15 pages, LaTex2e, 2 figure

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

Multiple Breathers on a Vortex Filament

In this paper we investigate the correspondence between the Da Rios-Betchov equation, which appears in the three-dimensional motion of a vortex filament, and the nonlinear Schrödinger equation. Using this correspondence we map a set of solutions corresponding to breathers in the nonlinear Schrödinger equation to waves propagating along a vortex filament. The work presented generalizes the recently derived family of vortex configurations associated with these breather solutions to a wider class of configurations that are associated with combination homoclinic/heteroclinic orbits of the 1D self-focussing nonlinear Schrödinger equation. We show that by considering these solutions of the governing nonlinear Schrödinger equation, highly nontrivial vortex filament configurations can be obtained that are associated with a pair of breather excitations. These configurations can lead to loop-like excitations emerging from an otherwise weakly perturbed helical vortex. The results presented further demonstrate the rich class of solutions that are supported by the Da Rios-Betchov equation that is recovered within the local induction approximation for the motion of a vortex filament

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

Motion of a Vortex Filament in the Local Induction Approximation: Reformulation of the Da Rios-Betchov Equations in the Extrinsic Filament Coordinate Space

In recognition of the highly non-trivial task of computation of the inverse Hasimoto transformation mapping the intrinsic geometric parameter space onto the extrinsic vortex filament coordinate space a reformulation of the Da Rios-Betchov equations in the latter space is given. The nonlinear localized vortex filament structure solution given by the present formulation is in detailed agreement with the Betchov-Hasimoto solution in the small-amplitude limit and is also in qualitative agreement with laboratory experiment observations of helical-twist solitary waves propagating on concentrated vortices in rotating fluids. The present formulation also provides for a discernible effect of the slipping motion of a vortex filament on the vortex evolution

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

Darboux transformation for the modified Veselov-Novikov equation

A Darboux transformation is constructed for the modified Veselov-Novikov equation.Comment: Latex file,8 pages, 0 figure

Differential Geometry of the Vortex Filament Equation

Differential calculus on the space of asymptotically linear curves is developed. The calculus is applied to the vortex filament equation in its Hamiltonian description. The recursion operator generating the infinite sequence of commuting flows is shown to be hereditary. The system is shown to have a description with a Hamiltonian pair. Master symmetries are found and are applied to deriving an expression of the constants of motion in involution. The expression agrees with the inspection of Langer and Perline.Comment: 20 pages, LaTeX, no figure

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

Vortex String Dynamics in an External Antisymmetric Tensor Field

We study the Lund-Regge equation that governs the motion of strings in a constant background antisymmetric tensor field by using the duality between the Lund-Regge equation and the complex sine-Gordon equation. Similar to the cases of vortex filament configurations in fluid dynamics, we find various exact solitonic string configurations which are the analogue of the Kelvin wave, the Hasimoto soliton and the smoke ring. In particular, using the duality relation, we obtain a completely new type of configuration which corresponds to the breather of the complex sine-Gordon equation.Comment: 20 pages, 9 figure