The Hasimoto Transformation for a Finite Length Vortex Filament and its Application

TL;DR

This paper establishes an equivalence between certain boundary value problems for the Localized Induction Equation and the nonlinear Schrödinger Equation via the Hasimoto transformation, and applies this to prove stability of solutions.

Contribution

It demonstrates the first use of the Hasimoto transformation to connect the Localized Induction Equation with the nonlinear Schrödinger Equation for boundary value problems.

Findings

Proves equivalence of solvability for boundary value problems

Establishes orbital stability of plane wave solutions

Provides new insights into the analysis of the nonlinear Schrödinger equation

Abstract

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.