The Gross-Pitaevskii (GP) equation admits a two-dimensional solitary wave solution representing two mutually self-propelled, anti-parallel straight line vortices. The complete sequence of such solitary wave solutions has been computed by Jones and Roberts (J. Phys. A, 15, 2599, 1982). These solutions are unstable with respect to three-dimensional perturbations (the Crow instability). The most unstable mode has a wavelength along the direction of the vortices of the same order as their separation. The growth rate associated with this mode is evaluated here, and it is found to increase very rapidly with decreasing separation. It is shown, through numerical integrations of the GP equation that, as the perturbations grow to finite amplitude, the lines reconnect to produce a sequence of almost circular vortex rings. 1
We analyse the dynamical properties of three-dimensional solitary waves in cylindrically trapped Bose–Einstein condensates. Families of solitary waves bifurcate from the planar dark soliton and include the solitonic vortex, the vortex ring and more complex structures of intersecting vortex lines known collectively as Chladni solitons. The particle-like dynamics of these guided solitary waves provides potentially profitable features for their implementation in atomtronic circuits, and play a key role in the generation of metastable loop currents. Based on the time-dependent Gross–Pitaevskii equation we calculate the dispersion relations of moving solitary waves and their modes of dynamical instability. The dispersion relations reveal a complex crossing and bifurcation scenario. For stationary structures we find that for the solitonic vortex is the only stable solitary wave. More complex Chladni solitons still have weaker instabilities than planar dark solitons and may be seen as transient structures in experiments. Fully time-dependent simulations illustrate typical decay scenarios, which may result in the generation of multiple separated solitonic vortices.
We show theoretically that elongated vortex-antivortex dipoles can be created controllably in trapped Bose-Einstein condensates, using known experimental techniques. Vortex dipoles of sufficient length are unstable and cascade into slow vortex rings which ultimately decay via sound emission. This instability of antiparallel vortex line elements, which self-generates Kelvin waves on vortex loops in trapped atomic gases, may play a role in bridging the Kelvin-wave and Kolmogorov-Richardson cascades of quantum turbulence.
In classical inviscid fluids, antiparallel vortices perturbed by Kelvin waves exhibit the Crow instability, where the mutual interaction of the Kelvin modes renders them dynamically unstable. This results in the approach and reconnection of the vortices, leading to a cascaded decay into ever-smaller vortex loops. Through mean-field simulations we study the Crow instability of quantum vortex lines in a superfluid whose atoms are subject to the anisotropic, long-ranged dipole-dipole interaction. We observe that the direction of dipole polarization plays a crucial role in determining the dynamically favored Kelvin modes. The subsequent rate of the instability is linked to the mediation of the vortex curvature by the effective dipole-dipole interaction between the vortices themselves. The vortex curvature is strongly suppressed and modes of lower wavenumber are preferred when the dipole polarization is parallel to the vortices, whereas the curvature is maximized for polarizations along the vortices' separation axis. For polarizations along the binormal axis, modes of higher wavenumber are favorable but the instability rate is considerably inhibited. This paves the way to a deeper understanding of vortex reconnections, vortex loop cascades and turbulence in dipolar superfluids.
Quantum vortex dynamics in Bose-Einstein condensates or superfluid helium can be informatively described by the Gross-Pitaevskii (GP) equation. Various approximate analytical formulas for a single stationary vortex are recalled and their shortcomings demonstrated. Significantly more accurate two-point [2/2] and [3/3] Pad\'e approximants for stationary vortex profiles are presented. Two straight, singly quantized, antiparallel vortices, located at a distance
d0 apart, form a vortex dipole, which, in the GP model, can either annihilate
or propagate indefinitely as a ``solitary wave.'' We show, through calculations performed in a periodic domain, that the details and types of behavior displayed by vortex dipoles depend strongly on the initial conditions rather than only on the separation distance
d0 (as has been previously claimed). It is found, indeed, that the choice of the initial two-vortex profile (i.e., the modulus of the ``effective wave function''), strongly affects the vortex trajectories and the time scale of the process: annihilation proceeds more rapidly when low-energy (or ``relaxed'') initial profiles are imposed. The initial ``circular'' phase distribution contours, customarily obtained by multiplying an effective wave function for each individual vortex, can be generalized to explicit elliptical forms specified by two parameters; then by ``tuning'' the elliptical shape at fixed
d0, a sharp transition between solitary-wave propagation and annihilation is captured. Thereby, a ``phase diagram'' for this ``AnSol'' transition is constructed in the space of ellipticity and separation and various limiting forms of the boundary are discussed.
Axisymmetric disturbances that preserve their form as they move along the vortex lines in uniform Bose-Einstein condensates are obtained numerically by the solution of the Gross-Pitaevskii equation. A continuous family of such solitary waves is shown in the momentum (
p)-substitution energy ($\stackrel{^}{\mathcal{E}}$) plane with $p\ensuremath{\rightarrow}0.09\ensuremath{\rho}{\ensuremath{\kappa}}^{3}/{c}^{2}$, $\stackrel{^}{\mathcal{E}}\ensuremath{\rightarrow}0.091\ensuremath{\rho}{\ensuremath{\kappa}}^{3}/c$ as $U\ensuremath{\rightarrow}c$, where $\ensuremath{\rho}$ is the density,
c is the speed of sound, $\ensuremath{\kappa}$ is the quantum of circulation, and
U is the solitary wave velocity. It is shown that collapse of a bubble captured by a vortex line leads to the generation of such solitary waves in condensates. The various stages of collapse are elucidated. In particular, it is shown that during collapse the vortex core becomes significantly compressed, and after collapse two solitary wave trains moving in opposite directions are formed on the vortex line.