Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

BETCHOV-DA RIOS EQUATION BY NULL CARTAN, PSEUDO NULL AND PARTIALLY NULL CURVE IN MINKOWSKI SPACETIME

  • Melek Erdogdu (Department of Mathematics and Computer Sciences Necmettin Erbakan University) ;
  • Yanlin Li (School of Mathematics Hangzhou Normal University) ;
  • Ayse Yavuz (Department of Mathematics and Science Education Necmettin Erbakan University)
  • Received : 2022.09.19
  • Accepted : 2023.03.16
  • Published : 2023.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

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

Acknowledgement

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).

References

  1. M. Barros, A. Ferrandez, P. Lucas, and M. Merono, Hopf cylinders, B-scrolls and solitons of the Betchov-Da Rios equation in the three-dimensional anti-de Sitter space, C. R. Acad. Sci. Paris Ser. I Math. 321 (1995), no. 4, 505-509.
  2. M. Barros, A. Ferrandez, P. Lucas, and M. Merono, Solutions of the Betchov-Da Rios soliton equation: a Lorentzian approach, J. Geom. Phys. 31 (1999), no. 2-3, 217-228. https://doi.org/10.1016/S0393-0440(99)00005-4
  3. M. Barros, A. Ferrandez, P. Lucas, and M. Merono, Solutions of the Betchov-Da Rios soliton equation in the anti-de Sitter 3-space, New Approaches in Nonlinear Analysis, Hadronic Press, Palm Harbor, 1999.
  4. R. Betchov, On the curvature and torsion of an isolated vortex filament, J. Fluid Mech. 22 (1965), 471-479. https://doi.org/10.1017/S0022112065000915
  5. L. S. Da Rios, Sul moto d'un liquido indefinito con un filetto vorticoso di forma qualunque, Rendiconti del Circolo Matematico di Palermo (1884-1940) 22 (1906), no. 1, 117-135. https://doi.org/10.1007/BF03018608
  6. M. Erdogdu and M. Ozdemir, Geometry of Hasimoto surfaces in Minkowski 3-space, Math. Phys. Anal. Geom. 17 (2014), no. 1-2, 169-181. https://doi.org/10.1007/s11040-014-9148-3
  7. M. Erdogdu and A. Yavuz, Differential geometric aspects of nonlinear Schrodinger equation, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 70 (2021), no. 1, 510-521. https://doi.org/10.31801/cfsuasmas.724634
  8. M. Erdogdu and A. Yavuz, On Backlund transformation and motion of null Cartan curves, Int. J. Geom. Methods Mod. Phys. 19 (2022), no. 1, Paper No. 2250014, 24 pp. https://doi.org/10.1142/s0219887822500141
  9. H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech. 51 (1972), no. 3, 477-485. https://doi.org/10.1017/S0022112072002307
  10. T. Levi-Civita, Attrazione newtoniana dei tubi sottili e vortici filiformi, Ann. Scuola Norm. Super. Pisa Cl. Sci. (2) 1 (1932), no. 1-2, 1-33.
  11. Y. Li, M. Erdogdu, and A. Yavuz, Nonnull soliton surface associated with the Betchov-Da Rios equation, Rep. Math. Phys. 90 (2022), no. 2, 241-255. https://doi.org/10.1016/S0034-4877(22)00068-4
  12. Y. Li, M. Erdogdu, and A. Yavuz, Differential geometric approach of Betchov-Da Rios soliton equation, Hacet. J. Math. Stat. 52 (2023), no. 1, 114-125.
  13. Y. Li and Z. G. Wang, Lightlike tangent developables in de Sitter 3-space, J. Geom. Phys. 164 (2021), Paper No. 104188, 11 pp. https://doi.org/10.1016/j.geomphys.2021.104188
  14. Y. Li, Y. Zhu, and Q.-Y. Sun, Singularities and dualities of pedal curves in pseudohyperbolic and de Sitter space, Int. J. Geom. Methods Mod. Phys. 18 (2021), no. 1, Paper No. 2150008, 33 pp. https://doi.org/10.1142/S0219887821500080
  15. B. O'Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, 103, Academic Press, Inc., New York, 1983.
  16. C. Rogers and W. K. Schief, Backlund and Darboux Transformations, Cambridge Texts in Applied Mathematics, Cambridge Univ. Press, Cambridge, 2002. https://doi.org/10.1017/CBO9780511606359
  17. J. Walrave, Curves and surfaces in Minkowski space, Thesis (Ph.D.), Katholieke Universiteit Leuven (Belgium), 1995.
  18. Z. Yang, Y. Li, M. Erdogdu, and Y. Zhu, Evolving evolutoids and pedaloids from viewpoints of envelope and singularity theory in Minkowski plane, J. Geom. Phys. 176 (2022), Paper No. 104513, 13 pp. https://doi.org/10.1016/j.geomphys.2022.104513
  19. A. Yavuz and M. Erdogdu, Non-lightlike Bertrand W curves: a new approach by system of differential equations for position vector, AIMS Math. 5 (2020), no. 6, 5422-5438. https://doi.org/10.3934/math.2020348
  20. A. Yavuz and M. Erdogdu, A different approach by system of differential equations for the characterization position vector of spacelike curves, Punjab Univ. J. Math. (Lahore) 53 (2021), no. 4, 231-245.