Translation of waves along quantum vortex filaments in the low-temperature two-dimensional local induction approximation

2016

Phys. Rev. E

Self Cite

Very recent experimental work has demonstrated the existence of Kelvin waves along quantized vortex filaments in superfluid helium. The possible configurations and motions of such filaments is of great physical interest, and Svistunov previously obtained a Hamiltonian formulation for the dynamics of quantum vortex filaments in the low-temperature limit under the assumption that the vortex filament is essentially aligned along one axis, resulting in a two-dimensional (2D) problem. It is standard to approximate the dynamics of thin filaments by employing the local induction approximation (LIA), and we show that by putting the two-dimensional LIA into correspondence with the first equation in the integrable Wadati-Konno-Ichikawa-Schimizu (WKIS) hierarchy, we immediately obtain solutions to the two-dimensional LIA, such as helix, planar, and self-similar solutions. These solutions are obtained in a rather direct manner from the WKIS equation and then mapped into the 2D-LIA framework. Furthermore, the approach can be coupled to existing inverse scattering transform results from the literature in order to obtain solitary wave solutions including the analog of the Hasimoto one-soliton for the 2D-LIA. One large benefit of the approach is that the correspondence between the 2D-LIA and the WKIS allows us to systematically obtain vortex filament solutions directly in the Cartesian coordinate frame without the need to solve back from curvature and torsion. Implications of the results for the physics of experimentally studied solitary waves, Kelvin waves, and postvortex reconnection events are mentioned.

Section: Introductionsupporting

confidence: 92%

“…III, we give the most general form of stationary solutions for the WKIS model, before transforming these into solutions of the 2D-LIA. These solutions are related back to those of [21,22] which were obtained directly from the 2D-LIA. We find that the solutions more readily come out of the WKIS framework.…”

Section: Introductionmentioning

confidence: 99%

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

2016

Phys. Rev. E

Self Cite

“…III of Ref. 17, one may similarly show that for the LNE, the only pure traveling wave solutions are helical filaments.…”

Section: A Traveling Waves Along Vortex Filaments Under the Lnementioning

confidence: 68%

“…While the nonlinearity was omitted when calculating the approximate solution, note that the influence of the nonlinearity does show up in the particular type of similarity solution (t 1/3 ) obtained. One may compare this result with the t 1/2 similarity scaling obtained for the 2D-LIA directly 17 or for the transformed solutions governed by the first equation in the WKIS hierarchy. 18 The t 1/2 type scalings also appear in numerical studies for the LIA or Biot-Savart model, in addition to the fully non-local Biot-Savart asymptotics which were recently studied in Ref.…”

Section: Self-similar Vortex Filament Structuresmentioning

confidence: 84%

See 1 more Smart Citation

Motion of isolated open vortex filaments evolving under the truncated local induction approximation

2017

Physics of Fluids

Self Cite

The study of nonlinear waves along open vortex filaments continues to be an area of active research. While the local induction approximation (LIA) is attractive due to locality compared with the nonlocal Biot-Savart formulation, it has been argued that LIA appears too simple to model some relevant features of Kelvin wave dynamics, such as Kelvin wave energy transfer. Such transfer of energy is not feasible under the LIA due to integrability, so in order to obtain a non-integrable model, a truncated LIA, which breaks the integrability of the classical LIA, has been proposed as a candidate model with which to study such dynamics. Recently Laurie et al. ["Interaction of Kelvin waves and nonlocality of energy transfer in superfluids," Phys. Rev. B 81, 104526 (2010)] derived truncated LIA systematically from Biot-Savart dynamics. The focus of the present paper is to study the dynamics of a section of common open vortex filaments under the truncated LIA dynamics. We obtain the analog of helical, planar, and more general filaments which rotate without a change in form in the classical LIA, demonstrating that while quantitative differences do exist, qualitatively such solutions still exist under the truncated LIA. Conversely, solitons and breather solutions found under the LIA should not be expected under the truncated LIA, as the existence of such solutions relies on the existence of an infinite number of conservation laws which is violated due to loss of integrability. On the other hand, similarity solutions under the truncated LIA can be quite different to their counterparts found for the classical LIA, as they must obey a t 1/3 type scaling rather than the t 1/2 type scaling commonly found in the LIA and Biot-Savart dynamics. This change in similarity scaling means that Kelvin waves are radiated at a slower rate from vortex kinks formed after reconnection events. The loss of soliton solutions and the difference in similarity scaling indicate that dynamics emergent under the truncated LIA can indeed differ a great deal from those previously studied under the classical LIA.

Self-similar vortex filament motion under the non-local Biot–Savart model

2016

J. Fluid Mech.

Self Cite

One type of thin vortex filament structure that has attracted interest in recent years is that which obeys self-similar scaling. Among various applications, these filaments have been used to model the motion of quantized vortex filaments in superfluid helium after reconnection events. While similarity solutions have been described analytically and numerically using the local induction approximation (LIA), they have not been studied (or even shown to exist) under the non-local Biot–Savart model. In this present paper, we show not only that self-similar vortex filament solutions exist for the non-local Biot–Savart model, but that such solutions are qualitatively similar to their LIA counterparts. This suggests that the various LIA solutions found previously should be valid physically (at least in the small amplitude regime), since they agree well with the more accurate Biot–Savart model.