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

Abstract

<abstract><p>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.</p></abstract>

Geometry and evolution of Hasimoto surface in Minkowski 3-space

“…In the case of r q is spacelike curve, it has quasi-frame in the following form The variation frame of r q with respect to time can be written as where σ , ϕ and θ are the velocities. For further information, we refer to [ 12 , 14 , 15 , 20 – 24 ].…”

“…where σ, ϕ and θ are the velocities. For further information, we refer to [12,14,15,[20][21][22][23][24]. Specifically, we define the quasi-curvatures as…”

“…If r = r ( s , t ) is a timelike curve for every t , then the motion that creates a timelike Hasimoto surface is the motion that satisfies condition ( 1 ), see [ 14 , 15 ].…”

On Some Quasi-Curves in Galilean Three-Space

“…The vector that is perpendicular to both the tangent vector and the quasi-normal vector is referred to as the quasi-binormal vector. Much research on the quasi-frame has been conducted in a variety of Euclidean and Minkowski spaces and may be found in [7][8][9][10].…”