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 space curve in E³. differential geometric properties soliton surface accociated with Nonlinear Schrödinger (NLS) equation, called NLS or Hasimoto surface, are investigated by using Darboux frame. Moreover, Gaussian and mean curvature found terms curvatures k_{n}, k_{g} ?_{g.}. Then, we give different proof that s- parameter curves geodesics surface. As applications two surfaces Frame.