A motion of complex curves in \(\mathbb C^3\) and the nonlocal nonlinear Schrödinger equation
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2016
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Authors
Shiping Zhong
- School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, P. R. China.
Abstract
This paper shows that soliton solutions to the nonlocal nonlinear Schrödinger equation (NNLS) proposed recently by Ablowitz and Musslimani [M. J. Ablowitz, Z. H. Musslimani, Phys. Rev. Lett., \(\bf 110\) (2013), 5 pages] describe a motion of three distinct complex curves in \(\mathbb C^3\) with initial data being suitably restricted. This gives a geometric interpretation of NNLS.
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ISRP Style
Shiping Zhong, A motion of complex curves in \(\mathbb C^3\) and the nonlocal nonlinear Schrödinger equation, Journal of Nonlinear Sciences and Applications, 12 (2019), no. 2, 75--85
AMA Style
Zhong Shiping, A motion of complex curves in \(\mathbb C^3\) and the nonlocal nonlinear Schrödinger equation. J. Nonlinear Sci. Appl. (2019); 12(2):75--85
Chicago/Turabian Style
Zhong, Shiping. "A motion of complex curves in \(\mathbb C^3\) and the nonlocal nonlinear Schrödinger equation." Journal of Nonlinear Sciences and Applications, 12, no. 2 (2019): 75--85
Keywords
- Complex moving curve
- geometric interpretation
- uniqueness
MSC
References
-
[1]
M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, Method for solving the Sine-Gordon equation, Phys. Rev. Lett., 30 (1973), 1262–1264.
-
[2]
M. J. Ablowitz, Z. H. Musslimani, Integrable nonlocal nonlinear Schrödinger equation , Phys. Rev. Lett., 110 (2013), 5 pages.
-
[3]
M. J. Ablowitz, H. Segur , Solitons and the inverse scattering transform, SIAM , Philadelphia (1981)
-
[4]
N.-H. Chang, J. Shatah, K. Unlenbeck, Schrödinger maps, Comm. Pure Appl. Math., 53 (2000), 590–602.
-
[5]
L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo., 22 (1906), 117–135.
-
[6]
Q. Ding , A note on the NLS and the Schrödinger flow of maps, Phys. Lett. A, 248 (1998), 49–56.
-
[7]
Q. Ding, J. Inoguchi, Schrödinger flows, binormal motion of curves and the second AKNS hierarchies, Chaos Solitons Fractals, 21 (2004), 669–677.
-
[8]
W. Y. Ding, Y. D. Wang , Schrödinger flows of maps into symplectic manifolds, Sci. China A, 41 (1998), 746–755.
-
[9]
Q. Ding, W. Wang, Y. D. Wang, A motion of spacelike curves in the Minkowski 3-space and the KdV equation, Phys. Lett. A, 374 (2010), 3201–3205.
-
[10]
Q. Ding, S. P. Zhong, The complex 2-sphere in \(\mathbb{C}^3\) and Schrödinger flows, Sci. China Math., (accepted),
-
[11]
A. Doliwa, P. M. Santini , An elementary geometric characterization of the integrable motions of a curve, Phys. Lett. A, 185 (1994), 373–384.
-
[12]
Y. Fukumoto, T. Miyazaki, Three-dimensional distortions of a vortex filament with axial velocity , J. Fluid. Mech., 22 (1991), 369–416.
-
[13]
T. A. Gadzhimuradov, A. M. Agalarov, Towards a gauge-equivalent magnetic structure of the nonlocal nonlinear Schrödinger equation, Phys. Rev. A, 93 (2016), 062124.
-
[14]
H. Gollek , Deformations of minimal curves in \(\mathbb{C}^3\), in Proc: 1-st NOSONGE Conference, Warsaw, (1996), 269– 286.
-
[15]
H. Gollek, Duals of vector fields and of null curves, Result. Math., 50 (2007), 53–79.
-
[16]
A. Gray, Modern differential geometry of curves and surfaces, CRC Press, Boca Raton (1995)
-
[17]
M. Gürses , Motion of curves on two-dimensional surfaces and soliton equations, Phys. Lett. A, 241 (1998), 329–334.
-
[18]
M. Gürses, A. Pekcan, Nonlocal nonlinear Schrödinger equations and their soliton solutions, J. Math. Phys., 59 (2018), 17 pages.
-
[19]
H. Hasimoto , A soliton on a vortex filament, J. Fluid. Mech., 21 (1972), 477–485.
-
[20]
K. M. Lakshmanan, K. M. Tamizhmani, Motion of strings, embedding problem and soliton equations, Appl. Sci. Res., 37 (1981), 127–143.
-
[21]
G. L. Lamb, Solitons on moving space curves, J. Mathematical Phys., 18 (1977), 1654–1661.
-
[22]
V. G. Makhankov, O. K. Pashaev, On the gauge equivalence of the Landau-Lifshitz and the nonlinear Schrödinger equations on symmetric spaces, Phys. Lett. A, 95 (1983), 95–100.
-
[23]
S. Murugesh, R. Balakrishnana, New geometries associated with the nonlinear Schrödinger equation, Eur. Phys. J. B Condens. Matter Phys., 29 (2002), 193–196.
-
[24]
S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, V. E. Zakharov , Theory of solitons: the inverse scattering method, Plenum, New York (1984)
-
[25]
V. Pessers, J. Van der Veken, On holomorphic Riemannian geometry and submanifolds of Wick-related spaces, J. Geom. Phys., 104 (2016), 163–174.
-
[26]
A. K. Sarma, M.-A. Miri, Z. H. Musslimani, D. N. Christodoulides, Continuous and discrete Schrödinger systems with parity-time-symmetric nonlinearities, Phys. Rev. E, 89 (2014), 7 pages.
-
[27]
C.-L. Terng, K. Uhlenbeck, Schrödinger flows on Grassmannians, AMS/IP Stud. Adv. Math., 36 (2006), 235–256.
-
[28]
V. E. Zakharov, L. A. Takhtajan, Equivalence of the nonlinear Schrödinger equation and the equation of a Heisenberg ferromagnet, Theor. Math. Phys., 38 (1979), 17–23.
