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

Abstract

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.

Keywords

• minkowski space
• hasimoto surfaces
• smoke ring equation
• gaussian and mean curvatures