Differential geometric approach of Betchov-Da Rios soliton equation

Year 2023, Volume: 52 Issue: 1, 114 - 125, 15.02.2023

Yanlin Li Melek Erdoğdu , Ayşe Yavuz

https://doi.org/10.15672/hujms.1052831

Cited By: 24

Abstract

In the present paper, we investigate differential geometric properties the soliton surface $M$ associated with Betchov-Da Rios equation. Then, we give derivative formulas of Frenet frame of unit speed curve $\Phi=\Phi(s,t)$ for all $t$. Also, we discuss the linear map of Weingarten type in the tangent space of the surface that generates two invariants: $k$ and $h$. Moreover, we obtain the necessary and sufficient conditions for the soliton surface associated with Betchov-Da Rios equation to be a minimal surface. Finally, we examine a soliton surface associated with Betchov-Da Rios equation as an application.

Keywords

Betchov-Da Rios equation, localized induction equation (LIE), smoke ring equation, vortex filament equation, nonlinear Schrodinger (NLS) equation

Supporting Institution

National Natural Science

Project Number

12101168

Thanks

This work was funded by the National Natural Science Foundation of China (Grant No. 12101168).

References

• [1] M. Barros, A. Ferrández and P. Lucas, M.A. Merono, Hopf cylinders, B-scrolls and solitons of the Betchov-Da Rios equation in the 3-dimensional anti-De Sitter space, CR Acad. Sci. Paris, Série I 321, 505-509, 1995.
• [2] M. Barros, A. Ferrández, P. Lucas and M.A. Merono, Solutions of the Betchov-Da Rios soliton equation: a Lorentzian approach, Journal of Geometry and Physics 31 (2-3), 217-228, 1999.
• [3] M. Barros, A. Ferrández, P. Lucas and M.A. Merono, Solutions of the Betchov-Da Rios soliton equation in the anti-De Sitter 3-space, New Approaches in Nonlinear Analysis, Hadronic Press, Florida, USA, 1999.
• [4] Q. Ding and J. Inoguchi, Schrödinger flows, binormal motion for curves and second AKNS-hierarchies, Chaos Solitons and Fractals 21 (3), 669-677, 2004.
• [5] M. Erdoğdu and M. Özdemir, Geometry of Hasimoto surfaces in Minkowski 3-space, Math. Phys. Anal. Geom. 17 (1), 169-181, 2014.
• [6] M. Erdoğdu and A. Yavuz, Differential geometric aspects of nonlinear Schrödinger equation, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70 (1), 510-521, 2021.
• [7] G. Ganchev and M. Velichka, On the theory of surfaces in the four-dimensional Euclidean space, Kodai Mathematical Journal 31, 183-198, 2008.
• [8] H. Hasimoto, A soliton on a vortex filament, J. Fluid. Mech. 51 (3), 477-485, 1972.
• [9] A. W. Marris and S.L. Passman, Vector fields and flows on developable surfaces, Arch. Ration. Mech. Anal. 32 (1), 29-86, 1969.
• [10] C. Rogers and W.K. Schief, Intrinsic geometry of the NLS equation and its backlund transformation, Stud. Appl. Math. 101 (3), 267-288, 1998.
• [11] C. Rogers and W.K. Schief, Backlund and Darboux transformations: Geometry of modern applications in soliton theory, Cambridge University Press, 2002.
• [12] W.K. Schief and C. Rogers, Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces, Proc. R. Soc. Lond. A. 455, 3163-3188, 1999.
• [13] A. Yavuz, Construction of binormal motion and characterization of curves on surface by system of differential equations for position vector, Journal of Science and Arts 4 (57), 1043-1056, 2021.