Differential geometric aspects of nonlinear Schrödinger equation

Melek Erdoğdu; Ayşe Yavuz

doi:10.31801/cfsuasmas.724634

Research Article

Year 2021, Volume: 70 Issue: 1, 510 - 521, 30.06.2021

Melek Erdoğdu Ayşe Yavuz

https://doi.org/10.31801/cfsuasmas.724634

Cited By: 7

Abstract

References

Ding, Q., Inoguchi, J., Schrödinger flows, binormal motion for curves and second AKNS-hierarchies, Chaos Solitons and Fractals, 21 (3) (2004), 669-677. https://doi.org/10.1016/j.chaos.2003.12.092
Erdogdu, M., Özdemir, M., Geometry of Hasimoto surfaces in Minkowski 3-space, Math. Phys. Anal. Geom., 17 (1) (2014), 169-181. DOI:10.1007/s11040-014-9148-3
Fujika, A., Inoguchi, J., Spacelike surfaces with harmonic inverse mean curvature, J. Math. Sci. Univ. Tokyo, 7 (4) (2000), 657-698.
Grbovic, M. and Nesovic, E., On Bäcklund transformation and vortex filament equation for pseudo null curves in Minkowski 3-space, Int. J. Geom. Methods Mod. Phys. 13 (6) (2016), 1-14. https://doi.org/10.1142/S0219887816500778
Gürbüz, N., Intrinstic geometry of NLS equation and heat system in 3-dimensional Minkowski space, Adv. Stud. Theor., 4 (1) (2010), 557-564.
Gürbüz, N., The motion of timelike surfaces in timelike geodesic coordinates, Int. J. Math. Anal., 4 (2010), 349-356.
Hasimoto, H., A soliton on a vortex filament, J. Fluid. Mech., 51 (3) (1972), 477-485. https://doi.org/10.1017/S0022112072002307
Inoguchi, J., Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo J. Math., 21 (1) (1998), 141-152.DOI:10.3836/tjm/1270041992
Inoguchi, J., Biharmonic curves in Minkowski 3-space, Int. J. Math. Math. Sci., (2003), 1365-1368. https://doi.org/10.1155/S016117120320805X
Kelleci, A., Bekta¸s, M., Ergüt, M., The Hasimoto surface according to bishop frame, Adıyaman University Journal of Science, 9 (2019 ), 13-22.
Özdemir, M., Ergin, A.A., Rotations with unit timelike quaternions in Minkowski 3-space, J. Geom. Phys., 56 (2) (2006), 322-336. https://doi.org/10.1016/j.geomphys.2005.02.004
Özdemir, M., Ergin, A.A., Parallel frames of non-lightlike curves, Missouri Journal of Mathematical Sciences, 20 (2) (2008), 127-137. DOI:10.35834/mjms/1316032813
Rogers, C., Schief, W.K., Intrinsic geometry of the NLS equation and its backlund transformation, Stud. Appl. Math., 101 (3) (1998), 267-288. https://doi.org/10.1111/14679590.00093
Rogers, C., Schief, W.K., Backlund and Darboux Transformations: Geometry of Modern Applications in Soliton Theory, Cambridge University Press, 2002. https://doi.org/10.1017/CBO9780511606359
Schief, W.K., Rogers, C., Binormal motion of curves of constant curvature and torsion, generation of soliton surfaces, Proc. R. Soc. Lond. A., 455 (1988) (1999), 3163-3188. https://doi.org/10.1098/rspa.1999.0445
Marris, A. W., Passman, S. L., Vector fields and flows on developable surfaces, Arch. Ration. Mech. Anal., 32 (1) (1969), 29-86. https://doi.org/10.1007/BF00253256
Rogers, C., Kingston, J. G. Nondissipative magneto-hydrodynamic flows with magnetic and velocity field lines orthogonal geodesics, SIAM J. Appl. Math., 26 (1) (1974), 183-195. https://doi.org/10.1137/0126015

Differential geometric aspects of nonlinear Schrödinger equation

Year 2021, Volume: 70 Issue: 1, 510 - 521, 30.06.2021

Melek Erdoğdu Ayşe Yavuz

https://doi.org/10.31801/cfsuasmas.724634

Cited By: 7

Abstract

The main scope of this paper is to examine the smoke ring (or vortex filament) equation which can be viewed as a dynamical system on the space curve in E³. The differential geometric properties the soliton surface accociated with Nonlinear Schrödinger (NLS) equation, which is called NLS surface or Hasimoto surface, are investigated by using Darboux frame. Moreover, Gaussian and mean curvature of Hasimoto surface are found in terms of Darboux curvatures k_{n}, k_{g} and τ_{g.}. Then, we give a different proof of that the s- parameter curves of NLS surface are the geodesics of the soliton surface. As applications we examine two NLS surfaces with Darboux Frame.

Keywords

Smoke ring equation, Vortex Filament equation, NLS surface, Darboux Frame

References

Ding, Q., Inoguchi, J., Schrödinger flows, binormal motion for curves and second AKNS-hierarchies, Chaos Solitons and Fractals, 21 (3) (2004), 669-677. https://doi.org/10.1016/j.chaos.2003.12.092
Erdogdu, M., Özdemir, M., Geometry of Hasimoto surfaces in Minkowski 3-space, Math. Phys. Anal. Geom., 17 (1) (2014), 169-181. DOI:10.1007/s11040-014-9148-3
Fujika, A., Inoguchi, J., Spacelike surfaces with harmonic inverse mean curvature, J. Math. Sci. Univ. Tokyo, 7 (4) (2000), 657-698.
Grbovic, M. and Nesovic, E., On Bäcklund transformation and vortex filament equation for pseudo null curves in Minkowski 3-space, Int. J. Geom. Methods Mod. Phys. 13 (6) (2016), 1-14. https://doi.org/10.1142/S0219887816500778
Gürbüz, N., Intrinstic geometry of NLS equation and heat system in 3-dimensional Minkowski space, Adv. Stud. Theor., 4 (1) (2010), 557-564.
Gürbüz, N., The motion of timelike surfaces in timelike geodesic coordinates, Int. J. Math. Anal., 4 (2010), 349-356.
Hasimoto, H., A soliton on a vortex filament, J. Fluid. Mech., 51 (3) (1972), 477-485. https://doi.org/10.1017/S0022112072002307
Inoguchi, J., Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo J. Math., 21 (1) (1998), 141-152.DOI:10.3836/tjm/1270041992
Inoguchi, J., Biharmonic curves in Minkowski 3-space, Int. J. Math. Math. Sci., (2003), 1365-1368. https://doi.org/10.1155/S016117120320805X
Kelleci, A., Bekta¸s, M., Ergüt, M., The Hasimoto surface according to bishop frame, Adıyaman University Journal of Science, 9 (2019 ), 13-22.
Özdemir, M., Ergin, A.A., Rotations with unit timelike quaternions in Minkowski 3-space, J. Geom. Phys., 56 (2) (2006), 322-336. https://doi.org/10.1016/j.geomphys.2005.02.004
Özdemir, M., Ergin, A.A., Parallel frames of non-lightlike curves, Missouri Journal of Mathematical Sciences, 20 (2) (2008), 127-137. DOI:10.35834/mjms/1316032813
Rogers, C., Schief, W.K., Intrinsic geometry of the NLS equation and its backlund transformation, Stud. Appl. Math., 101 (3) (1998), 267-288. https://doi.org/10.1111/14679590.00093
Rogers, C., Schief, W.K., Backlund and Darboux Transformations: Geometry of Modern Applications in Soliton Theory, Cambridge University Press, 2002. https://doi.org/10.1017/CBO9780511606359
Schief, W.K., Rogers, C., Binormal motion of curves of constant curvature and torsion, generation of soliton surfaces, Proc. R. Soc. Lond. A., 455 (1988) (1999), 3163-3188. https://doi.org/10.1098/rspa.1999.0445
Marris, A. W., Passman, S. L., Vector fields and flows on developable surfaces, Arch. Ration. Mech. Anal., 32 (1) (1969), 29-86. https://doi.org/10.1007/BF00253256
Rogers, C., Kingston, J. G. Nondissipative magneto-hydrodynamic flows with magnetic and velocity field lines orthogonal geodesics, SIAM J. Appl. Math., 26 (1) (1974), 183-195. https://doi.org/10.1137/0126015

There are 17 citations in total.

Details

Primary Language	English
Subjects	Applied Mathematics
Journal Section	Research Articles
Authors	Melek Erdoğdu 0000-0001-9610-6229 Ayşe Yavuz 0000-0002-0469-3786
Publication Date	June 30, 2021
Submission Date	April 21, 2020
Acceptance Date	February 1, 2021
Published in Issue	Year 2021 Volume: 70 Issue: 1

Cite

APA	Erdoğdu, M., & Yavuz, A. (2021). Differential geometric aspects of nonlinear Schrödinger equation. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(1), 510-521. https://doi.org/10.31801/cfsuasmas.724634
AMA	Erdoğdu M, Yavuz A. Differential geometric aspects of nonlinear Schrödinger equation. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2021;70(1):510-521. doi:10.31801/cfsuasmas.724634
Chicago	Erdoğdu, Melek, and Ayşe Yavuz. “Differential Geometric Aspects of Nonlinear Schrödinger Equation”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70, no. 1 (June 2021): 510-21. https://doi.org/10.31801/cfsuasmas.724634.
EndNote	Erdoğdu M, Yavuz A (June 1, 2021) Differential geometric aspects of nonlinear Schrödinger equation. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70 1 510–521.
IEEE	M. Erdoğdu and A. Yavuz, “Differential geometric aspects of nonlinear Schrödinger equation”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 70, no. 1, pp. 510–521, 2021, doi: 10.31801/cfsuasmas.724634.
ISNAD	Erdoğdu, Melek - Yavuz, Ayşe. “Differential Geometric Aspects of Nonlinear Schrödinger Equation”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70/1 (June 2021), 510-521. https://doi.org/10.31801/cfsuasmas.724634.
JAMA	Erdoğdu M, Yavuz A. Differential geometric aspects of nonlinear Schrödinger equation. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70:510–521.
MLA	Erdoğdu, Melek and Ayşe Yavuz. “Differential Geometric Aspects of Nonlinear Schrödinger Equation”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 70, no. 1, 2021, pp. 510-21, doi:10.31801/cfsuasmas.724634.
Vancouver	Erdoğdu M, Yavuz A. Differential geometric aspects of nonlinear Schrödinger equation. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70(1):510-21.