The non-linear Schrödinger equation associated with the soliton surfaces in Minkowski 3-space

Ayman Elsharkawy; Clemente Cesarano; Abdelrhman Tawfiq; Abdul Aziz Ismail; Ayman Elsharkawy; Clemente Cesarano; Abdelrhman Tawfiq; Abdul Aziz Ismail

doi:10.3934/math.2022985

AIMS Mathematics

2022, Volume 7, Issue 10: 17879-17893. doi: 10.3934/math.2022985

Previous Article Next Article

Research article

The non-linear Schrödinger equation associated with the soliton surfaces in Minkowski 3-space

1.
Mathematics Department, Faculty of Science, Tanta University, 31511 Tanta, Egypt
2.
Section of Mathematics, International Telematic University Uninettuno, 001186 Roma, Italy
3.
Mathematics Department, Faculty of Education, Ain-Shams University, 11566 Cairo, Egypt
4.
Mechanical Engineering Department, College of Engineering and Islamic Architecture, Umm Al-Qura University, 5555 Makkah, Saudi Arabia. Email: aiismail@uqu.edu.sa

Received: 25 May 2022 Revised: 29 June 2022 Accepted: 07 July 2022 Published: 04 August 2022
MSC : 35Q51, 51B20, 53A35, 76B47

The quasi frame is more efficient than the Frenet frame in investigating surfaces, and it is regarded a generalization frame of both the Frenet and Bishop frames. The geometry of quasi-Hasimoto surfaces in Minkowski 3-space $ \mathbb{E}_1^3 $ is investigated in this paper. For the three situations of non-lightlike curves, the geometric features of the quasi-Hasimoto surfaces in $ \mathbb{E}_1^3 $ are examined and the Gaussian and mean curvatures for each case are determined. The quasi-Hasimoto surfaces in $ \mathbb{E}_1^3 $ must satisfy a necessary and sufficient condition to be developable surfaces. As a result, the parameter curves of quasi-Hasimoto surfaces in $ \mathbb{E}_1^3 $ is described. Thus, the $ s $-parameter and $ t $-parameter curves of quasi-Hasimoto surfaces in $ \mathbb{E}_1^3 $ are said to be geodesics, asymptotic, and curvature lines under necessary and sufficient circumstances are proved. Finally, quasi curves and associated quasi-Hasimoto surface correspondences are discussed.
- Minkowski space,
- Hasimoto surfaces,
- smoke ring equation,
- Gaussian and mean curvatures
Citation: Ayman Elsharkawy, Clemente Cesarano, Abdelrhman Tawfiq, Abdul Aziz Ismail. The non-linear Schrödinger equation associated with the soliton surfaces in Minkowski 3-space[J]. AIMS Mathematics, 2022, 7(10): 17879-17893. doi: 10.3934/math.2022985

Related Papers:

Abstract

The quasi frame is more efficient than the Frenet frame in investigating surfaces, and it is regarded a generalization frame of both the Frenet and Bishop frames. The geometry of quasi-Hasimoto surfaces in Minkowski 3-space $ \mathbb{E}_1^3 $ is investigated in this paper. For the three situations of non-lightlike curves, the geometric features of the quasi-Hasimoto surfaces in $ \mathbb{E}_1^3 $ are examined and the Gaussian and mean curvatures for each case are determined. The quasi-Hasimoto surfaces in $ \mathbb{E}_1^3 $ must satisfy a necessary and sufficient condition to be developable surfaces. As a result, the parameter curves of quasi-Hasimoto surfaces in $ \mathbb{E}_1^3 $ is described. Thus, the $ s $-parameter and $ t $-parameter curves of quasi-Hasimoto surfaces in $ \mathbb{E}_1^3 $ are said to be geodesics, asymptotic, and curvature lines under necessary and sufficient circumstances are proved. Finally, quasi curves and associated quasi-Hasimoto surface correspondences are discussed.

References

[1]	N. H. Abdel-All, R. A. Hussien, T. Youssef, Hasimoto surfaces, Life Sci. J., 9 (2012), 556–560.
[2]	Q. Ding, J. Inoguchi, Schrödinger flows, binormal motion for curves and the second AKNS-hierarchies, Chaos Soliton. Fract., 21 (2004), 669–677. https://doi.org/10.1016/j.chaos.2003.12.092 doi: 10.1016/j.chaos.2003.12.092
[3]	M. Elzawy, Hasimoto surfaces in Galilean space $G_{3} $, J. Egypt. Math. Soc., 29 (2021), 5. https://doi.org/10.1186/s42787-021-00113-y doi: 10.1186/s42787-021-00113-y
[4]	M. Erdogdu, M. Özdemir, Geometry of Hasimoto surfaces in Minkowski 3-space, Math. Phys. Anal. Geom., 17 (2014), 169–181. https://doi.org/10.1007/s11040-014-9148-3 doi: 10.1007/s11040-014-9148-3
[5]	N. Gürbüz, Intrinstic geometry of NLS equation and heat system in 3-dimensional Minkowski space, Adv. Studies Theor. Phys., 4 (2010), 557–564.
[6]	N. Gürbüz, The motion of timelike surfaces in timelike geodesic coordinates, Int. J. Math. Anal., 4 (2010), 349–356.
[7]	E. Hamouda, O. Moaaz, C. Cesarano, S. Askar, A. Elsharkawy, Geometry of solutions of the quasi-vortex filament equation in Euclidean 3-space $\mathbb{E}^3$, Mathematics, 10 (2022), 891. https://doi.org/10.3390/math10060891 doi: 10.3390/math10060891
[8]	H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech., 51 (1972), 477–485. https://doi.org/10.1017/S0022112072002307 doi: 10.1017/S0022112072002307
[9]	K. Hosseini, M. Mirzazadeh, D. Baleanu, S. Salahshour, L. Akinyemi, Optical solitons of a high-order nonlinear Schrödinger equation involving nonlinear dispersions and Kerr effect, Opt. Quant. Electron., 54 (2022), 177. https://doi.org/10.1007/s11082-022-03522-0 doi: 10.1007/s11082-022-03522-0
[10]	K. Hosseini, S. Salahshour, M. Mirzazadeh, Bright and dark solitons of a weakly nonlocal Schrödinger equation involving the parabolic law nonlinearity, Optik, 227 (2021), 166042. https://doi.org/10.1016/j.ijleo.2020.166042 doi: 10.1016/j.ijleo.2020.166042
[11]	J. Inoguchi, Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo J. Math., 21 (1998), 141–152. https://doi.org/10.3836/tjm/1270041992 doi: 10.3836/tjm/1270041992
[12]	A. Kelleci, M. Bektas, M. Ergüt, The Hasimoto surface according to bishop frame, Adıyaman Univ. J. Sci., 9 (2019), 13–22.
[13]	E. Kemal, A. K. Akbay, On the harmonic evolute surfaces of hasimoto surfaces, Adi. Uni. J. Sci. 11 (2019), 87–100. https://doi.org/10.37094/adyujsci.820698 doi: 10.37094/adyujsci.820698
[14]	T. Körpınar, R. C. Demirkol, Z. Körpınar, Approximate solutions for the inextensible Heisenberg antiferromagnetic flow and solitonic magnetic flux surfaces in the normal direction in Minkowski space, Optik, 238 (2021), 166403. https://doi.org/10.1016/j.ijleo.2021.166403 doi: 10.1016/j.ijleo.2021.166403
[15]	M. Özdemir, A. A. Ergin, Rotations with unit timelike quaternions in Minkowski 3-space, J. Geom. Phys., 56 (2006), 322–336. https://doi.org/10.1016/j.geomphys.2005.02.004 doi: 10.1016/j.geomphys.2005.02.004
[16]	C. Rogers, W. K. Schief, Bäcklund and Darboux transformations: Geometry and modern applications in soliton theory, Cambridge University Press, 2002. https://doi.org/10.1017/CBO9780511606359
[17]	C. Rogers, W. K. Schief, Intrinsic geometry of the NLS equation and its auto-Bäcklund transformation, Stud. Appl. Math., 101 (1998), 267–287. https://doi.org/10.1111/1467-9590.00093 doi: 10.1111/1467-9590.00093
[18]	W. K. Schief, C. Rogers, Binormal motion of curves of constant curvature and torsion, Math. Phys. Eng. Sci., 455 (1988), 3163–3188. https://doi.org/10.1098/rspa.1999.0445 doi: 10.1098/rspa.1999.0445

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)