Related papers: Vortex Filament Equation and Non-Lightlike Space C…
This paper presents a new representation of curve dynamics, with applications to vortex filaments in fluid dynamics. Instead of representing these filaments with explicit curve geometry and Lagrangian equations of motion, we represent…
The binormal (or vortex filament) equation provides the localized induction approximation of the 3D incompressible Euler equation. We present explicit solutions of the binormal equation in higher-dimensions that collapse in finite time. The…
In this paper we explore the nature of self-similar solutions of the Curve Shortening Flow and the Vortex Filament Equation, also known as the Binormal Flow. We explore some of their fundamental conservation properties and describe the…
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…
It is well known that the Lorentzian length of a timelike curve in Minkowski spacetime is smaller than the Lorentzian length of the geodesic connecting its initial and final endpoints. The difference is known as the 'differential aging' and…
We develop a new interpretation of the geometric phase in evolution with a non-Hermitian real value Hamiltonian by relating it to the angle developed during the parallel transport along a closed curve by a unit vector triad in the…
Using techniques of integrable systems, we study a Weierstrass representation formula for timelike surfaces with prescribed mean curvature in Minkowski 3-space. It is shown that timelike minimal surfaces are obtained by integrating a pair…
In this paper we study curves in Lorentz-Minkowski space $\mathbb{L}^2$ that are critical points of the moment of inertia with respect to the origin. This extends a problem posed by Euler in the Lorentzian setting. We obtain explicit…
In two previous papers two evolution equations for the vortex line density $L$, proposed by Vinen, were generalized to rotating superfluid turbulence and compared with each other. Here, the already generalized alternative Vinen equation is…
The motion of a vortex filament in superfluid 4He is considered by using the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) phenomenological model for the scattering process between the vortex and thermal excitations in liquid 4He. The HVBK…
Modelling incompressible ideal fluids as a finite collection of vortex filaments is important in physics (super-fluidity, models for the onset of turbulence) as well as for numerical algorithms used in computer graphics for the real time…
For a regular curve on a spacelike surface in Lorentz-Minkowski $3$-space, we have a moving frame along the curve which is called a Lorentzian Darboux frame. We introduce five special vector fields along the curve associated to the…
A one-dimensional space curve in $\mathbb{R}^{3}$ is a useful nonlinear medium for modeling vortex filaments and biological soft-matter capable of supporting a variety of wave motions. The Hasimoto transformation defines a mapping between…
By considering a lattice model of extended phase space, and using techniques of noncommutative differential geometry, we are led to: (a) the conception of vector fields as generators of motion and transition probability distributions on the…
Here we present first results simulating plasma filaments in non-axisymmetric geometries, using a fluid turbulence extension of the \boutxx~framework. This is made possible by the implementation of the Flux Coordinate Independent scheme for…
The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow…
In [1], we gave a method for constructing Bertrand curves from the spherical curves in 3 dimensional Minkowski space. In this work, we construct the Bertrand curves corresponding to a spacelike geodesic and a null helix in Minkowski 4…
We deal with solitons of the mean curvature flow. The definition of \textit{translating solitons on a light-like direction} in Minkowski 3-space is introduced. Firstly, we classify those which are graphical, \textit{translation surfaces},…
The motion of noncircular two-dimensional vortices is shown to depend on a form of coupling between vortex ellipticity and the gradient of fluid density. The approach is based on the perspective that an elliptic vortex can be described as…
In this paper, we consider the concentration property of solutions to the dispersive Ginzburg-Landau (or Gross-Pitaevskii) equation in three dimensions. On a spatial domain, it has long been conjectured that such a solution concentrates…