In this project, I derive integrable systems from inelastic curve flows in 2 and 3 dimensional Minkowskian space by using Hasimoto variables. I introduce a Lorentzian version of a moving parallel frame and show that its structure equations encode the Hasimoto variables in natural way. For timelike/spacelike curves in the Minkowskian plane, I obtain the defocusing mKdV equation and its bi-Hamiltonian structure. For null curves, I find Burgers equation. For timelike curves in 3 dimensional Minkowskian space, I derive the complex defocusing mKdV equation and the NLS equation, whereas for spacelike curves, I find similar equations with complex numbers replaced by hyperbolic numbers.