In this article we investigate the nonlinear stability of Hasimoto solitons, in energy space, for a fourth order Schrödinger equation (4NLS) which arises in the context of the vortex filament. The proof relies on a suitable Lyapunov functional, at the $H^2$ level, which allows us to describe the dynamics of small perturbations. This stability result is also extended to Sobolev spaces $H^m$ for all $m∈\mathbb{Z}_+$ by employing the infinite conservation laws of 4NLS.
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