Numerical analysis of slender vortex motion

Abstract

It is found that the boundaries between stable, recurrent, and chaotic regimes in the parameter space of the model problem depend on the method used. The boundaries of these domains in the parameter space for the Klein-Majda equation and for the Klein-Knio equation are closely related to the core size. When the core size is large enough, the Klein-Majda equation always exhibits stable solutions for our model problem.Various conclusions are drawn; in particular, the behavior of turbulent vortices cannot be captured by these local approximations, and probably cannot be captured by any slender vortex model with constant vortex cross-section. Speculations about the differences between classical and superfluid hydrodynamics are also offered.

The phase space of the focused cubic Schroedinger equation: A numerical study

“…We also discover that in terms of vortex approximations, the phase transition type behavior for the cubic Schriidinger equation results in the transition from an almost straight filament with a sharp kink for low temperatures to a filament of random shape at high temperatures. We remark that in the limit the temperature at which the .transition occurs increases to infinity, which is in agreement with the results of Chorin [12] [57]. If the phase transition in the cubic Schrodinger equation were to exist, this might lead to a reconsideration of the role of these equations and their modifications in the study of classical and superfluid turbulence.…”

On the motion of slender vortex filaments

“…Also in the stability analysis, we deal with the filament function instead of the filament curve. We have shown 33 that a small sideband perturbation to the filament function is, to the leading order, equivalent to a small sideband perturbation to the filament curve. Thus the small oscillation in the graphs, caused by other sidebands and the nonlinear Hasimoto transformation, does not contradict with the stability analysis.…”