On admissible curves and their evolution equations in pseudo-Galilean space

Abstract

The evolution equations of some forms of admissible curves in the pseudo-Galilean Space G 1 3 are investigated in this paper. In more detail, we use two separate methods to obtain coupled nonlinear partial differential equations of time evolution in terms of their curvatures. The first method studies the evolution equations for admissible curves via the frame field, while the second studies the evolution equations via the velocity vector. Then, the position vectors of the evolving curves are formulated. Also, we conduct comparative research of the evolution equations for curves in different spaces. We furthermore present some models as an application of the evolution equations of the curvature and torsion for admissible curves, confirming our theoretical results.

Geometry and evolution of Hasimoto surface in Minkowski 3-space

“…Numerous spaces, including the Euclidean space [ 16 ], Minkowski space [ 17 ], Galilean space [ 18 ], and pseudo-Galilean space [ 5 ], have been used to study the equations of motion of curves and surfaces. Within the scope of our work, we investigate the evolution equations of Hasimoto surface by employing the quasi-frame of spacelike curve with timelike binormal.…”

“…The flowing curve of the sine Gordon equation was analyzed by Rick Mukherjee and Radha Balakrishnan [ 19 ]. In [ 5 , 16 ], the authors investigated the motion of plane curves, hypersurface motion, and the motion of space curves in various spaces. By using the fundamental existence and uniqueness hypothesis of space curves, the authors in [ 13 ] developed Hasimoto surface via integration for Frenet-Serret equations.…”

“…If the arc length or the intrinsic curvature of a surface is preserved throughout the flow of curve or surface, respectively, we then say that the flow is inextensible [1][2][3]. Several scholars have researched geometric flow issues on curves and surfaces in different spaces in recent years (see for instance [1,[4][5][6]). At any point on a curve, there are many frames associated with it.…”

Evolution Equations of Pseudo Spherical Images for Timelike Curves in Minkowski 3-Space

“…These images are obtained by means of Frenet-Serret frame vector fields associated to the meant curves, so this classical topic is a well-known concept in Lorentzian geometry of the curves, see [11,12]. There are many studies of evolution equations that have been done in different spaces (see for instance, [7,9,10,13]). Also, evolution equations for the elliptic partial differential equations and the magnetic geodesics equations have been obtained in [8].…”