Solitons and nonlinear waves along quantum vortex filaments under the low-temperature two-dimensional local induction approximation.

A vector equation describing the motion of a vortex filament in an external flow under the localized induction approximation is considered. The initial-boundary value problem for it is proved to have at least a solution when both ends of the filament are fixed. Unlike the case without an external flow, the stretch and shrink of the filament generally occur. But, for our solution, they are finite at any finite time.

We review two formulations of the fully nonlinear local induction equation approximating the self-induced motion of the vortex filament (in the local induction approximation), corresponding to the Cartesian and arc-length coordinate systems. The arc-length representation put forth by Umeki [Theor. Comput. Fluid Dyn. 24, 383 (2010)] results in a type of 1+1 derivative nonlinear Schrödinger (NLS) equation describing the motion of such a vortex filament. We obtain exact stationary solutions to this derivative NLS equation; such exact solutions are a rarity. These solutions are periodic in space and we determine the nonlinear dependence of the period on the amplitude.

In R. A. Van Gorder, “General rotating quantum vortex filaments in the low-temperature Svistunov model of the local induction approximation,” Phys. Fluids 26, 065105 (2014) I discussed properties of generalized vortex filaments exhibiting purely rotational motion under the low-temperature Svistunov model of the local induction approximation. Such solutions are stationary in terms of translational motion. In the Comment [N. Hietala, “Comment on ‘General rotating quantum vortex filaments in the low-temperature Svistunov model of the local induction approximation’ [Phys. Fluids 26, 065105 (2014)],” Phys. Fluids 26, 119101 (2014)], the author criticizes my paper for not including translational motion (although it was clearly stated that the filament motion was assumed rotational). As it turns out, if one is interested in studying the geometric structure of solutions (which was the point of my paper), one obtains the needed qualitative results on the structure of such solutions by studying the purely rotational case. Nevertheless, in this Response I shall discuss the vortex filaments that have both rotational and translational motions. I then briefly discuss why one might want to study such generalized rotating filament solutions, in contrast to simple the standard helical or planar examples (which are really special cases). I also discuss how one can study the time evolution of filaments which exhibit more complicated dynamics than pure translation and rotation. Doing this, one can study non-stationary solutions which initially appear purely rotational and gradually display other dynamics as the filaments evolve.

In this paper, we take into account the dynamical equations of a vortex filament in superfluid helium at finite temperature (1 K < T < 2.17 K) and at very low temperature, which is called Biot–Savart law. The last equation is also valid for a vortex tube in a frictionless, unbounded, and incompressible fluid. Both the equations are approximated by the Local Induction Approximation (LIA) and Fukumoto’s approximation. The obtained equations are then considered in the extrinsic frame of reference, where exact solutions (Kelvin waves) are shown. These waves are then compared one to each other in terms of their dispersion relations in the frictionless case. The same equations are then investigated for a quantized vortex line in superfluid helium at higher temperature, where friction terms are needed for a full description of the motion.

Reconnections between quantum vortex filaments in presence of trapped particles are investigated using numerical simulations of the Gross--Pitaevskii equation. Particles are described with classical degrees of freedom and modeled as highly repulsive potentials which deplete the superfluid. First, the case of a vortex dipole with a single particle trapped inside one of the vortices is studied. It is shown that the reconnection takes place at the position of the particle as a consequence of the symmetry breaking induced by it. The separation rate between the reconnecting points is compatible with the known dynamics of quantum vortex reconnections and it is independent of the particle mass and size. After the reconnection, the particle is pushed away with a constant velocity and its trajectory is deflected because of the transverse momentum exchange with the vortex filaments. The momentum exchanges between the particle, the vortex, and a density pulse are characterized. Finally, the reconnection of two linked rings, each of them with several initially randomly distributed particles is studied. It is observed that generically, reconnections take place at the location of trapped particles. It is shown that reconnection dynamics is unaffected for light particles.

The motion of a vortex filament in superfluid 4He is considered by using the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) [H.E. Hall, W.F. Vinen, Proc. Roy. Soc. Lond. A 238, 204 (1956); H.E. Hall, W.F. Vinen, Proc. Roy. Soc. Lond. A 238, 215 (1956); I.L. Bekarevich, I.M. Khalatnikov, Sov. Phys. J. Exp. Theor. Phys. 13, 643 (1961)] phenomenological model for the scattering process between the vortex and thermal excitations in liquid 4He. The HVBK equations are analytically formulated first in the intrinsic geometric parameter space to obtain insights into the physical implications of the friction terms, associated with the friction coefficients α and α′ (in the Hall-Vinen notation) as well as the previous neglect of the friction term associated with the friction coefficient α′. The normal fluid velocity components both along and transverse to the vortex filament are included. This analytical development also serves to highlight the difficulties arising in making further progress on this route. A reformulation of the HVBK equation in the extrinsic vortex filament coordinate space is then given which is known [B.K. Shivamoggi, Phys. Rev. B 84, 012506 (2011)] to provide a useful alternative analytical approach in this regard. A nonlinear Schrodinger equation for the propagation of nonlinear Kelvin waves on a vortex filament in a superfluid is given taking into account the generalized normal fluid flow. The friction term associated with α′, even in the presence of the normal fluid velocity components transverse to the vortex filament, is shown to produce merely an algebraic growth of the Kelvin waves hence providing further justification for the neglect of this term. On the other hand, the instability produced by the friction term associated with α via the normal fluid velocity component along the vortex filament is shown to manifest itself as a parametric amplification on considering the problem of a rotating planar vortex filament in a superfluid.

Three-dimensional motion of a slender vortex tube, embedded in an inviscid incompressible fluid, is investigated based on the Euler equations. Using the method of matched asymptotic expansions in a small parameter ∈, the ratio of core radius to curvature radius, the velocity of a vortex filament is derived to O(∈ 3), whereby the influence of elliptical deformation of the core due to the self-induced strain is taken into account. In the localized induction approximation, this is reducible to a completely integrable evolution equation among the localized induction hierarchy.KeywordsEuler EquationVortex RingAxial FlowCore RadiusVortex FilamentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Effects of dust charges on the nonlinear ion acoustic solitary waves in a fully relativistic dusty plasma for both cases of negative and positive dusts are numerically studied based on the pseudopotential method. In the presence of dusty particles, it is found that various types of nonlinear acoustic waves exist in forms which can be viewed as sequential combinations of three kinds of elementary solitary waves: bump, dip, and kink-type solitary waves. The number and the sequence of the constituent elementary solitary waves in a given nonlinear waves depend more sensitively on dust particle density than any other parameters. For negatively charged dust particles of low density, the nonlinear wave is in the shape of bumpy solitary wave. For a somewhat higher density, the wave changes into a form which can be viewed as a combination of bump and dip-type solitary waves. As the density is increased further, a more complex nonlinear wave composed of bump, kink, and dip-type solitary waves emerges. For a much higher density of dust particle, the nonlinear wave can have a shape that can be considered as a combination of bump and kink-type solitary waves. For the case of positively charged dust particles, two kinds of nonlinear waves can exist: bump-type solitary wave and a combination of bump and kink solitary waves. For both cases of negative and positive dust particles, it is found that single dip-type solitary wave does not exist. It is also found that as dust particle density increases, the signature of the elementary waves becomes less prominent.

The turbulent flow of helium II is complicated by the fact that one must actually consider two different, but interacting, turbulent velocity fields: that of the normal fluid component and that of the superfluid component. Aside from the forcing due to the interaction with the superfluid, the normal fluid obeys a Navier-Stokes equation, and the turbulence of this component is expected to be classical. The superfluid, however, is an Eulerian fluid with the additional restriction that the vorticity in the superfluid is confined to quantized vortex filaments. These superfluid vortex filaments have an extremely small core radius (approximately one Angstrom) and are thus true vortex filaments. The circulation of these filaments is quantized, and due to energy considerations only vortices with one quantum of circulation need to be considered. At high flow rates these quantized vortex filaments can form dense tangles which are referred to as quantum turbulence.

The quantum turbulence is a collection of the interacting quantum vortex loops and filaments. The main goal of this work is to systematize the information on the current state of the research on the reconnecting quantized vortex loops in the superfluid helium: the geometry, the dynamics, the properties of the vortex loops, the energy spectrum before and after the reconnections. The paper discusses the possible role of the reconnections in the formation of the turbulent spectrum. In addition, in this paper the main methods and approaches to the study of the reconnecting vortex loops and quantum turbulence are discussed.

Through the Hasimoto map, various dynamical systems can be mapped to different integrodifferential generalizations of Nonlinear Schrodinger (NLS) family of equations some of which are known to be integrable. Two such continuum limits, corresponding to the inhomogeneous XXX Heisenberg spin chain [Balakrishnan, J. Phys. C 15, L1305 (1982)] and that of a thin vortex filament moving in a superfluid with drag [Shivamoggi, Eur. Phys. J. B 86, 275 (2013) 86; Van Gorder, Phys. Rev. E 91, 053201 (2015)], are shown to be particular non-holonomic deformations (NHDs) of the standard NLS system involving generalized parameterizations. Crucially, such NHDs of the NLS system are restricted to specific spectral orders that exactly complements NHDs of the original physical systems. The specific non-holonomic constraints associated with these integrodifferential generalizations additionally posses distinct semi-classical signature.

Motion of a vortex filament with axial flow in the core, i.e. a vortex-jet filament, is governed, in the localized induction approximation, by the first two terms of the localized induction hierarchy. Permanent form of a steadily translating vortexjet filament is found to be identical with equilibrium form of a thin inextensible elastic rod of circular cross section, and, as a dynamical extension, traveling waves on an extensible rod is sought. We establish a variational principle for motion of an elastic rod with respect to material frames.

Interest in the study of invariant quantities is generally motivated by the need to interpret and to understand their meaning and their fundamental role in the theory. The invariants we shall consider in this paper emerge in two contexts. In the context of the localized induction approximation (LIA) for the motion of an inextensible vortex filament in a perfect fluid flow, we shall deal with certain conserved quantities that emerge from the BetchovDa Rios equations; the polynomial invariants for the related nonlinear Schrodinger equation (NLSE) are calculated employing the recurrence formula of Zakharov and Shabat (1974); these quantities are constants of the motion for the vortex as long as self-intersection does not occur and long-distance effects are neglected. Some of them are then interpreted in terms of kinetic energy, momentum, angular momentum, “pseudo-helicity,” and some inequalities between physical and global geometrical properties are stated. In the context of the Euler equations, given that the topology of the vortex structure is conserved and the helicity invariant is the natural measure of the topological complexity of the field (Moffatt, 1969; Arnol’d, 1974), we comment further on this role of helicity; moreover, by associating an “energy spectrum” with any knot or link present in the fluid, the lowest ground-state enstrophy is interpreted as an even more powerful invariant for evaluating the knot or link complexity of the field structure.

We demonstrate that a single reconnection of two quantum vortices can lead to the creation of a cascade of vortex rings. Our analysis, motivated by the analytical results in localized induction approximation, involved high resolution Biot-Savart and Gross-Pitaevskii simulations. The latter showed that the rings cascade starts on the atomic scale, with rings diameters orders of magnitude smaller than the characteristic line spacing in the tangle, and thus capable of penetrating the tangle and annihilating on the boundaries. That way such process can be an efficient decay mechanism for sparse or moderately dense vortex tangle at very low temperatures.

The equation of motion of a quantized vortex filament in a trapped Bose-Einstein condensate [A. A. Svidzinsky and A. L. Fetter, Phys. Rev. A {\bf 62}, 063617 (2000)] has been generalized to the case of an arbitrary anharmonic anisotropic rotating trap and presented in a variational form. For condensate density profiles of the form $\rho=f(x^2+y^2+\mbox{Re\,}\Psi(x+iy))$ in the presence of the plane of symmetry $y=0$, the solutions $x(z)$ describing stationary vortices of U and S types coming to the surface and solitary waves have been found in quadratures. Analogous three-dimensional configurations of the vortex filament uniformly moving along the $z$ axis have also been found in strictly cylindrical geometry. The dependence of solutions on the form of the function $f(q)$ has been analyzed.