  We consider two nonlinear equations, the Localized Induction Equation and the
cubic nonlinear Schr\"odinger Equation, and prove that the solvability of
certain initial-boundary value problems for each equation is equivalent through
the generalized Hasimoto transformation. As an application, we prove the
orbital stability of plane wave solutions of the nonlinear Schr\"odinger
equation based on stability estimates obtained for the Localized Induction
Equation by the author in a paper in preparation. As far as the author knows,
this is the first time that the analysis of the Localized Induction Equation,
along with the Hasimoto transformation, provided new insight for the nonlinear
Schr\"odinger equation.
