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

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2295-2305
Author(s):  
Muhammad Hanif ◽  
Hou Hua ◽  
Emilija Nesovic
Keyword(s):  
Minkowski Space ◽  
Vortex Filament ◽  
Null Curve ◽  
Minkowski Spaces ◽  
Vortex Filament Equation ◽  
The Relationship

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.

scholarly journals Integrability aspects of the vortex filament equation for pseudo-null curves

2017 ◽  
Vol 14 (06) ◽  
pp. 1750090 ◽  
Author(s):  
José del Amor ◽  
Ángel Giménez ◽  
Pascual Lucas
Keyword(s):  
Space Form ◽  
Burgers Equation ◽  
Three Dimensional ◽  
Recursion Operator ◽  
Vortex Filament ◽  
Lorentzian Space ◽  
Null Curve ◽  
Vortex Filament Equation ◽  
Null Curves ◽  
The Relationship

An algebraic background in order to study the integrability properties of pseudo-null curve motions in a three-dimensional Lorentzian space form is developed. As an application, we delve into the relationship between the Burgers’ equation and the pseudo-null vortex filament equation. A recursion operator for the pseudo-null vortex filament equation is also provided.

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

2018 ◽  
Vol 15 (08) ◽  
pp. 1850142 ◽  
Author(s):  
Kazim Ilarslan ◽  
Emilija Nešović
Keyword(s):  
Minkowski Space ◽  
Space Time ◽  
Vortex Filament ◽  
The Third ◽  
Bishop Frame ◽  
Minkowski Space Time ◽  
Vortex Filament Equation ◽  
Cartan Curvature

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.

On Bäcklund transformation and vortex filament equation for pseudo null curves in Minkowski 3-space

2016 ◽  
Vol 13 (06) ◽  
pp. 1650077 ◽  
Author(s):  
Milica Grbović ◽  
Emilija Nešović
Keyword(s):  
Evolution Equation ◽  
Bäcklund Transformation ◽  
Sufficient Conditions ◽  
Vortex Filament ◽  
Backlund Transformation ◽  
Burger's Equation ◽  
Null Curve ◽  
Vortex Filament Equation ◽  
Null Curves ◽  
The Given

In this paper, we introduce Bäcklund transformation of a pseudo null curve in Minkowski 3-space as a transformation mapping a pseudo null helix to another pseudo null helix congruent to the given one. We also give the sufficient conditions for a transformation between two pseudo null curves in the Minkowski 3-space such that these curves have equal constant torsions. By using the Da Rios vortex filament equation, based on localized induction approximation (LIA), we derive the vortex filament equation for a pseudo null curve and prove that the evolution equation for the torsion is the viscous Burger’s equation. As an application, we show that pseudo null curves and their Frenet frames generate solutions of the Da Rios vortex filament equation.

scholarly journals Vortex Filament Equation for a Regular Polygon in the Hyperbolic Plane

2021 ◽  
Vol 32 (1) ◽  
Author(s):  
Francisco de la Hoz ◽  
Sandeep Kumar ◽  
Luis Vega
Keyword(s):  
Center Of Mass ◽  
Point Of View ◽  
Vortex Filament ◽  
Complete Agreement ◽  
Euclidean Case ◽  
Runge Kutta Method ◽  
Vortex Filament Equation ◽  
The One ◽  
The Relationship ◽  
Planar Polygon

AbstractThe 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.

scholarly journals Special Curves According to Bishop Frame in Minkowski 3-Space

2020 ◽  
Vol 5 (1) ◽  
pp. 237-248
Author(s):  
Muhammad Abubakar Isah ◽  
Mihriban Alyamaç Külahçı
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bishop Frame ◽  
Null Curve ◽  
Minkowski Spaces ◽  
Slant Helix ◽  
Null Curves ◽  
Necessary And Sufficient

AbstractPseudo null curves were studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curve are not considered. In this paper, we study weak AW (k) – type and AW (k) – type pseudo null curve in Minkowski 3-space [E_1^3 . We define helix and slant helix according to Bishop frame in [E_1^3 . Furthermore, the necessary and sufficient conditions for the slant helix and helix in Minkowski 3-space are obtained.

Harmonic evolutes of B-scrolls with constant mean curvature in Lorentz–Minkowski space

2019 ◽  
Vol 16 (05) ◽  
pp. 1950076 ◽  
Author(s):  
Rafael López ◽  
Željka Milin Šipuš ◽  
Ljiljana Primorac Gajčić ◽  
Ivana Protrka
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Ruled Surfaces ◽  
Null Curve ◽  
Base Curve

In this paper, we study harmonic evolutes of [Formula: see text]-scrolls, that is, of ruled surfaces in Lorentz–Minkowski space having no Euclidean counterparts. Contrary to Euclidean space where harmonic evolutes of surfaces are surfaces again, harmonic evolutes of [Formula: see text]-scrolls turn out to be curves. In particular, we show that the harmonic evolute of a [Formula: see text]-scroll of constant mean curvature together with its base curve forms a null Bertrand pair. This allows us to characterize [Formula: see text]-scrolls of constant mean curvature and reconstruct them from a given null curve which is their harmonic evolute.

scholarly journals A New Angular Measurement in Minkowski 3-Space

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 56 ◽  
Author(s):  
Jinhua Qian ◽  
Xueqian Tian ◽  
Jie Liu ◽  
Young Ho Kim
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Angular Measurement ◽  
Frenet Frame ◽  
Measuring Methods ◽  
Null Curve

In Lorentz–Minkowski space, the angles between any two non-null vectors have been defined in the sense of the angles in Euclidean space. In this work, the angles relating to lightlike vectors are characterized by the Frenet frame of a pseudo null curve and the angles between any two non-null vectors in Minkowski 3-space. Meanwhile, the explicit measuring methods are demonstrated through several examples.

Mappings that preserve helices in the n-dimensional Minkowski spaces

2020 ◽  
Vol 17 (10) ◽  
pp. 2050107
Author(s):  
Bülent Altunkaya
Keyword(s):  
Minkowski Space ◽  
Minkowski Spaces ◽  
Dimensional Minkowski Space

We introduce two types of mappings that preserve nonnull helices in Minkowski spaces. The first type constructs helices in the [Formula: see text]-dimensional Minkowski space from helices in the same Minkowski space. The second type constructs helices in the [Formula: see text]-dimensional Minkowski space from helices in the [Formula: see text]-dimensional Minkowski space. Furthermore, we study invariants of these mappings and present examples.

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