Geometric Methodology for Analyzing Timelike Curve Flows in Minkowski Space

The present study introduces an innovative link between integrable equations and the motion of timelike curves within a three-dimensional Minkowski space. This study aims to establish an anology between the modified generalizations of the Heisenberg spin chain model equation, a complex Korteweg–de Vries equation, and the Ablowitz–Kaup–Newell–Segur hierarchy systems of real type, respectively. This is accomplished through the application of specific functions, which are derived based on the curvatures and torsions of three distinct curves and their corresponding Frenet frames in a 3-dimensional Minkowski space. Making use of this method, the geometric derivation of the integrable equation has been demonstrated with success.