Hasimoto Maps for Nonlinear Schrödinger Equations in Minkowski Space

Abstract

AbstractIn this paper, we study the vortex filament flow for timelike and spacelike curves in Minkowski 3-space. The vortex filament flow equations of the timelike and the spacelike curves are equivalent to the nonlinear Schrödinger equation and the heat equation, respectively. As a consequentce, we prove that a soliton of the nonlinear Schrödinger equations of the timelike curve gives a solution of a traveling wave on a line at infinity. Also, we study a solution of a traveling wave of the nonlinear Schrödinger equations of the spacelike curve in terms of a new complex frame. Finally, we discuss the method to find the exact shape of the timelike and the spacelike curves from the vortex filament by solving the Frenet vectors of these curves and provide applications to illustrate the method.