Geometric Analysis of the NLS Surface with the Pseodu Null Darboux Frame of Spacelike Curve
Öz
In this study, examination of the vortex filament equation, known as the smoke ring equation, is given by considering it as a dynamic system associated with a unit speed spacelike curve in Minkowski space. Using the Darboux framework, the differential geometric features of the soliton surface that relate to the nonlinear Schrödinger (NLS) equation were investigated. This surface is often called the NLS surface or Hasimoto surface. Additionally, Gaussian and the mean curvatures of the NLS surface were determined using the Darboux curvatures. By providing an in-depth look at the geometric properties of the NLS surface, this analysis will make a significant contribution to increasing knowledge in this field and shedding light on potential applications.
Anahtar Kelimeler
Darboux frame, Hasimoto surface, Nonlinear Schrödinger equation
Kaynakça
- H. Hasimoto, A soliton on a vortex filament, Journal of Fluid Mechanics. 51(3) (1972), 477-485.
- S. Izumiya, S. Otani, Flat approximations of surfaces along curves, Demonstratio Mathematica. 48(2) (2015), 217-241.
- M. Erdoğdu, A. Yavuz, Differential geometric aspects of nonlinear Schrödinger equation. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 70(1) (2021), 510-521.
- J. Djordjević, E. Nešović, On generalized Darboux frame of a pseudo null curve lying on a lightlike surface in Minkowski 3-Space. International Electronic Journal of Geometry. 16(1) (2023), 81-94.
- N. Gürbüz, W. Y. Dae, Hasimoto surfaces for two classes of curve evolution in Minkowski 3-space, Demonstratio Mathematica. 53(1) (2020), 277-284.
- E. S. Y. Topbaş, İ. Gök, N. Ekmekçi, Y. Yaylı, Darboux frame of a curve lying on a lightlike surface, Mathematical Sciences and Applications E-Notes. 4(2) (2016), 121–130.
- E. Hamouda, O. Moaaz, C. Cesarano, S. Askar, A. Elsharkawy, Geometry of solutions of the quasi vortex filament equation in Euclidean 3-Space E^3, Mathematics. 10(6) (2022), 891.
- W. K. Schief, C. Rogers, Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences. 455 (1988), 3163-3188.
- C. Rogers, W. K. Schief, Intrinsic geometry of the NLS equation and its auto‐Bäcklund transformation. Studies in Applied Mathematics. 101(3) (1998), 267-287.
- A. W. Marris, S.L. Passman, Vector fields and flows on developable surfaces, Archive for Rational Mechanics and Analysis. 32 (1969), 29-86.