Inextensible Flows of Null Cartan Curves in Minkowski Space

• Published in: Universe (Basel) Vol. 9; no. 3; p. 125
• Main Authors: Samah Gaber, Abeer Al Elaiw
• Format: Journal Article
• Language: English
• Published: MDPI AG 01.02.2023
• Subjects: evolution of curves; inextensible flows; motion of curves; null Cartan curves; time evolution equations
• ISSN: 2218-1997
• DOI: 10.3390/universe9030125

This research focused on studying the flows of a null Cartan curve specified by the velocity and acceleration fields. We have proven that the tangential and normal velocities are influenced by the binormal velocity along the motion. The velocity fields are used to drive the time evolution equations for the Cartan frame and the torsion of the null curve. The objective of this work is to construct a family of inextensible null Cartan curves from the flows of the initial null Cartan curve. The surface formed by this family of inextensible flows of the null Cartan curve is obtained numerically and visualized. In this paper, we refer to the surface traced out by the family of the null Cartan curve as the generated or constructed surface. We present a novel model for the inextensible null Cartan curve, which moves with a constant binormal velocity to describe the process of constructing a family of null Cartan curves. Through this model, the time evolution equation for the torsion of the inextensible null Cartan curve arises as the Korteweg-de Vries (K-dV) equation, and we obtain and visualize the soliton solutions. The soliton solutions represent the torsion of the family of null Cartan curves at various time values. We construct the family of inextensible null Cartan curves and visualize the generated surface. In addition, we investigate the flows of inextensible null Cartan curves specified by acceleration fields, and we obtain the explicit relationships between the acceleration and velocity functions. Finally, we provide an application for the inextensible flows of the null Cartan curve with constant normal acceleration.