Bulletin of the
Korean Mathematical Society BKMS

The aim of this paper is to investigate Betchov-Da Rios equation by using null Cartan, pseudo null and partially null curve in Minkowski spacetime. Time derivative formulas of frame of $s$ parameter null Cartan, pseudo null and partially null curve are examined, respectively. By using the obtained derivative formulas, new results are given about the solution of Betchov-Da Rios equation. The differential geometric properties of these solutions are obtained with respect to Lorentzian causal character of $s$ parameter curve. For a solution of Betchov-Da Rios equation, it is seen that null Cartan $s$ parameter curves are space curves in three-dimensional Minkowski space. Then all points of the soliton surface are flat points of the surface for null Cartan and partially null curve. Thus, it is seen from the results obtained that there is no surface corresponding to the solution of Betchov-Da Rios equation by using the pseudo null $s$ parameter curve.

Keywords: Betchov-Da Rios equation, localized induction equation (LIE), null Cartan curve, pseudo null curve, partially null curve, Minkowski spacetime

MSC numbers: Primary 35Q55, 53A05, 53Z05

Supported by: This work was funded by the National Natural Science Foundation of China (Grant No. 12101168) and Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ22A010014).