Filament Equation Research Articles

Steady solutions for the Schrödinger map equation

In this paper we use bifurcation methods to construct a new family of solutions of the binormal flow, also known as the vortex filament equation, which do not change their form. Our examples are complementary to those obtain by S. Kida in 1981, and therefore they are also related, thanks to the so-called Hasimoto transformation, to traveling wave solutions of the 1d cubic non-linear Schrödinger equation.

On the harmonic evolute of time-like Hasimoto surfaces in Lorentz–Minkowski space

The movement of a thin vortex in a thin viscous fluid by the motion of a curve propagating in Lorentz–Minkowski space [Formula: see text] is described by the vortex filament or smoke ring equation and can be viewed as a dynamical system on the space curves in [Formula: see text]. This paper investigates the harmonic evolute surfaces of time-like Hasimoto surfaces in [Formula: see text]. Also, we discuss the geometric properties of these surfaces, namely, we obtain the Gaussian and mean curvatures of the first and second fundamental forms. As a verification, we construct a concrete example for the meant surfaces to demonstrate our theoretical results.

Revisiting surfaces with a common null geodesic curve in Minkowski 3-space 𝔼13

In this paper, we study the problem of constructing a surface family from a given null geodesic curve in Minkowski 3-space [Formula: see text]. Using the Cartan frame of the given null curve, we present the surface tangent plane is coincident with the curve rectifying plane, and analyze the necessary and sufficient condition for that curve to be null geodesic. Then, the extension to ruled, and developable surfaces are also outlined. As an application, we derive an explicit solution of the Da Rios vortex filament equation lying in the B-scroll.

A different modelling of complex Hasimoto map for pseudo-null curves via Bishop frame

In this paper, we construct some links between visco model of motion and equations attained by taking geometric differential description of Hasimoto map of pseudo-null curves framed by the Bishop frame and some physical consequences. First, we find that the equations of Hasimoto transformation are occurred by adapting Hasimoto map for pseudo-null curves. Then, we investigate the correspondence of both the Landau–Lifshitz system and the vortex filament equation by means of the Hasimoto map of pseudo-null curves. Lastly, we portray the links of findings obtained by geometric physical issues. Finally, we simulate optical solutions with graphically physical constructions.

Pseudo null growth model and its classifications based on generalized vortex filament equation

The pseudo null growth model in Minkowski 3-space is shaped by evolving a pseudo null curve as denoted direction, growth velocity, and stretch in this paper. Meanwhile, the idea of the generalized Hasimoto surfaces is proposed based on the vortex filament equation. The conditions and geometric structures of the generalized Hasimoto pseudo null growth surfaces are investigated accompanied by some typical examples.

The Trefoil Soliton

The Kiepert trefoil is an algebraic curve with remarkable geometric and number theoretic properties. Ludwig Kiepert, generalizing ideas due to Serret and Liouville, determined that it could be parametrized by arc length in terms of elliptic functions. In this note, we observe some other properties of the curve. In particular, the curve is a special example of a buckled ring, and thus a solitary wave solution to the planar filament equation, evolving by rotation. It is also a solitary wave solution to a flow in the (three-dimensional) filament hierarchy, evolving by translation.

Optical recursion systems for the Hasimoto map and optical applications with spherical frame

In this research, we analyze the Hasimoto map and some flow equations for spherical frame (s-frame) in S2 spherical space. Here, we first obtain Hasimoto map, which we call the s-Hasimoto map, for the s-frame in S2. Then, we get some soliton equations with the help of s-Hasimoto map and vortex filament (Da Rios) equation. Later, by examining the effects of visco-Da Rios equation and visco-Landau Lifshitz equation on s-Hasimoto map, we obtain some flow equations. Finally, we present some of the contributions that all our results will make to magnetic fluxes, fluid dynamics and, especially, optical motion.

Geometry of Solutions of the Quasi-Vortex Filament Equation in Euclidean 3-Space E3

This work aims at investigating the geometry of surfaces corresponding to the geometry of solutions of the vortex filament equation in Euclidean 3-space E3 using the quasi-frame. In particular, we discuss some geometric properties and some characterizations of parameter curves of these surfaces in E3.

Vortex Filament Equation for a Regular Polygon in the Hyperbolic Plane

The aim of this paper 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 up exponentially, which makes the problem more challenging from a numerical point of view. However, using a finite difference scheme in space combined with a fourth-order Runge–Kutta method in time and fixed boundary conditions, we show that the numerical solution is in complete agreement with the one obtained by means of algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with the evolution in the Euclidean case.

Optical modeling of Hasimoto map for antiferromagnetic timelike optical fiber

In this study, we study optical hasimoto map corresponds to under the quasi-frame for timelike curves in 3D Minkowski space. We investigate the relationship of this transformation with some flow equations, which are known examples for nonlinear optical systems. Here, we first obtain the q-Hasimoto map as equivalent of the Hasimoto map in the q-frame for timelike space curve in 3D Minkowski Space. Then, we examine the effect on the q-Hasimoto map of the some flow equations such as vortex filament equation, Heisenberg antiferromagnetic flow. Finally, we present comments on some of the physical consequences of these effects that we have examined.

Differential geometric aspects of nonlinear Schrödinger equation

The main scope of this paper is to examine the smoke ring (or vortex filament) equation which can be viewed as a dynamical system on the space curve in E³. The differential geometric properties the soliton surface accociated with Nonlinear Schrödinger (NLS) equation, which is called NLS surface or Hasimoto surface, are investigated by using Darboux frame. Moreover, Gaussian and mean curvature of Hasimoto surface are found in terms of Darboux curvatures k_{n}, k_{g} and τ_{g.}. Then, we give a different proof of that the s- parameter curves of NLS surface are the geodesics of the soliton surface. As applications we examine two NLS surfaces with Darboux Frame.

Local existence and uniqueness of skew mean curvature flow

Abstract The Skew Mean Curvature Flow (SMCF) is a Schrödinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general-dimensional SMCF in Euclidean spaces.

Higher-dimensional Euler fluids and Hasimoto transform: counterexamples and generalizations

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 local nature of this phenomenon suggests a possibility of the singularity appearance in nearby vortex blob solutions of the Euler equation in 5D and higher. Furthermore, the Hasimoto transform takes the binormal equation to the NLS and barotropic fluid equations. We show that in higher dimensions the existence of such a transform would imply the conservation of the Willmore energy in skew-mean-curvature flows and present counterexamples for vortex membranes based on products of spheres. These (counter)examples imply that there is no straightforward generalization to higher dimensions of the 1D Hasimoto transform. We derive its replacement, the evolution equations for the mean curvature and torsion form for membranes, thus generalizing the barotropic fluid and Da Rios equations.

Geometric differentiability of Riemann's non-differentiable function

Riemann's non-differentiable function is a classic example of a continuous function which is almost nowhere differentiable, and many results concerning its analytic regularity have been shown so far. However, it can also be given a geometric interpretation, so questions on its geometric regularity arise. This point of view is developed in the context of the evolution of vortex filaments, modelled by the Vortex Filament Equation or the binormal flow, in which a generalisation of Riemann's function to the complex plane can be regarded as the trajectory of a particle. The objective of this document is to show that the trajectory represented by its image does not have a tangent anywhere. For that, we discuss several concepts of tangent vectors in view of the set's irregularity.

On the Evolution of the Vortex Filament Equation for Regular $M$-Polygons with Nonzero Torsion

In this paper, we consider the evolution of the vortex filament equation (VFE): $X_t = X_s \wedge X_{ss},$ taking $M$-sided regular polygons with nonzero torsion as initial data. Using algebraic te...

Some geometric properties of Riemann's non-differentiable function

Riemann's non-differentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the binormal flow or the vortex filament equation. In this setting, we analyse certain geometric properties of its image in C. The objective of this note is to assert that the Hausdorff dimension of its image is no larger than 4/3 and that it has nowhere a tangent.

On self-similar solutions of the vortex filament equation

We study self-similar solutions of the binormal curvature flow which governs the evolution of vortex filaments and is equivalent to the Landau-Lifshitz equation. The corresponding dynamics is described by the real solutions of the σ-Painlevé IV equation with two real parameters. Connection formulae for Painlevé IV transcendents allow for a complete characterization of the asymptotic properties of the curvature and torsion of the filament. We also provide compact hypergeometric expressions for self-similar solutions corresponding to corner initial conditions.

Hasimoto variables, generalized vortex filament equations, Heisenberg models and Schrödinger maps arising from group-invariant NLS systems

The deep geometrical relationships holding among the NLS equation, the vortex filament equation, the Heisenberg spin model, and the Schrödinger map equation are extended to the general setting of Hermitian symmetric spaces. New results are obtained by utilizing a generalized Hasimoto variable which arises from applying the general theory of parallel moving frames. The example of complex projective space ℂPN=SU(N+1)∕U(N) is used to illustrate the method and results.

On involutes of order k of a null Cartan curve in Minkowski spaces

In this paper, we define an involute and an evolving involute of order k of a null Cartan curve in Minkowski space En1 for n ? 3 and 1 ? k ? n-1. In relation to that, we prove that if a null Cartan helix has a null Cartan involute of order 1 or 2, then it is Bertrand null Cartan curve and its involute is its Bertrand mate curve. In particular, we show that Bertrand mate curve of Bertrand null Cartan curve can also be a non-null curve and find the relationship between the Cartan frame of a null Cartan curve and the Frenet or the Cartan frame of its non-null or null Cartan involute of order 1 ? k ? 2. We show that among all null Cartan curves in E31 , only the null Cartan cubic has two families of involutes of order 1, one of which lies on B-scroll. We also give some relations between involutes of orders 1 and 2 of a null Cartan curve in Minkowski 3-space. As an application, we show that involutes of order 1 of a null Cartan curve in E31 , evolving according to null Betchov-Da Rios vortex filament equation, generate timelike Hasimoto surfaces.

On Bishop frame of a null Cartan curve in Minkowski space-time

In this paper, we define the Bishop frame of a null Cartan curve in Minkowski space-time [Formula: see text]. We obtain the Bishop’s frame equations of a null Cartan curve which lies in the timelike hyperplane of [Formula: see text]. We show that a null Cartan cubic lying in the timelike hyperplane of [Formula: see text] has two Bishop frames, one of which coincides with its Cartan frame. We also derive the Bishop’s frame equation of the null Cartan curve which has the third Cartan curvature [Formula: see text]. As an application, we find a solution of the null Betchov-Da Rios vortex filament equation in terms of a null Cartan curve and its Bishop frame, which generates a timelike Hasimoto surface.