Differential geometric aspects of nonlinear Schrödinger equation
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.
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- 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
- ISSN: 1303-5991
- Yayın Aralığı: Yılda 4 Sayı
- Başlangıç: 1948
- Yayıncı: Ankara Üniversitesi
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