2018
DOI: 10.22436/jnsa.012.02.02
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A motion of complex curves in $C^3$ and the nonlocal nonlinear Schrödinger equation

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., 110 (2013), 5 pages] describe a motion of three distinct complex curves in C 3 with initial data being suitably restricted. This gives a geometric interpretation of NNLS.

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Cited by 5 publications
(3 citation statements)
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“…The complex-valued spin vector S induced that the unit vectors e j become also complex-valued. It means that the curvature κ(t, x), the torsion τ (t, x) and ω j are complex-valued functions [68]. Since in the nonlocal case, the spin vector S is the complex-valued vector function, so that we may decompose it as S = M + iL, where M and L are already real valued vector functions [69].…”
Section: Nonlocal Landau-lifshitz Equationmentioning
confidence: 99%
“…The complex-valued spin vector S induced that the unit vectors e j become also complex-valued. It means that the curvature κ(t, x), the torsion τ (t, x) and ω j are complex-valued functions [68]. Since in the nonlocal case, the spin vector S is the complex-valued vector function, so that we may decompose it as S = M + iL, where M and L are already real valued vector functions [69].…”
Section: Nonlocal Landau-lifshitz Equationmentioning
confidence: 99%
“…One of them is to use the Lax equations of certain integrable equations to obtain the associated parametrization of two surfaces in three dimensional Euclidean or Minkowski spaces [1]- [7]. The other one is to use the Serret-Frenet equations for curves in three dimensional spaces and defining two surfaces as the traces of the motion of the curves [8]- [21]. In this work we shall follow the second approach and study the motion of null curves in Minkowski 3-space M 3 and determine surfaces swept by such curves.…”
Section: Introductionmentioning
confidence: 99%
“…2 Serret-Frenet equations in M 3 Two dimensional surfaces in Euclidean 3-space R 3 and in Minkowski 3-space M 3 have been studied for many purposes. One of the main interest in these surfaces is the relation between the integrable evolution equations and these surfaces [8]- [21].…”
Section: Introductionmentioning
confidence: 99%