The differential formula of Hasimoto transformation in Minkowski 3-space

Hasimoto [10] introduced the map from vortex filament solutions of Euler’s equations for incompressible fluids in the local induction approximation to solutions of the nonlinear Schrödinger equation and he showed vortex filament equation is equivalent nonlinear Schrödinger equation. After this discovering of Hasimoto, several authors [1, 5, 9, 12, 13, 14, 15, 17, 20, 21, 22, 23, 24] studied the connection between the integrable nonlinear Schrödinger equation and the nonstretching vortex filament equation. Ding and Inoguchi also presented this connection in Minkowski 3-space [6, 7, 8]. Langer and Perline derived the formula for the differential of the Hasimoto transformation in 3D spaces [16]. We also present a formula for the differential formula of Hasimoto transformation in Minkowski 3-space in this paper. Since this construction has potential applications to further investigation using the inverse scattering scheme and finite-gap solutions, much works have been revived by several authors. In recent years, Langer and Perline found a recursion relation which generates the hierarchy of space curve equations which maps by Hasimoto transformation and nonlinear Schrödinger equation [18]. Calini and Ivey [2, 3, 4] studied finite-gap solutions of the vortex filament equation. Holm and Stechmann also investigated vortex solution motion driven by fluid helicity [11].