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

2020 ◽  
Vol 80 (2) ◽  
pp. 1034-1056
Author(s):  
Francisco de la Hoz ◽  
Sandeep Kumar ◽  
Luis Vega
Keyword(s):  
Vortex Filament ◽  
Vortex Filament Equation ◽  
Nonzero Torsion

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 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.

scholarly journals Vortex filament equation for a regular polygon

Nonlinearity ◽  
2014 ◽  
Vol 27 (12) ◽  
pp. 3031-3057 ◽  
Author(s):  
Francisco de la Hoz ◽  
Luis Vega
Keyword(s):  
Regular Polygon ◽  
Vortex Filament ◽  
Vortex Filament Equation

scholarly journals The Vortex Filament Equation as a Pseudorandom Generator

2014 ◽  
Vol 138 (1) ◽  
pp. 135-151 ◽  
Author(s):  
Francisco de la Hoz ◽  
Luis Vega
Keyword(s):  
Vortex Filament ◽  
Pseudorandom Generator ◽  
Vortex Filament Equation

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.

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