  The Hasimoto transformation between the classical LIA (local induction
approximation, a model approximating the motion of a thin vortex filament) and
the nonlinear Schr\"odinger equation (NLS) has proven very useful in the past,
since it allows one to construct new solutions to the LIA once a solution to
the NLS is known. In the present paper, the quantum form of the LIA (which
includes mutual friction effects) is put into correspondence with a type of
complex nonlinear dispersive partial differential equation (PDE) with cubic
nonlinearity (similar in form to a Ginsburg-Landau equation, with additional
nonlinear terms). Transforming the quantum LIA in such a way enables one to
obtain quantum vortex filament solutions once solutions to this dispersive PDE
are known. From our quantum Hasimoto transformation, we determine the form and
behavior of Stokes waves and a standing 1-soliton solution under normal and
binormal friction effects. The soliton solution on a quantum vortex filament is
a natural generalization of the classical 1-soliton solution constructed
mathematically by Hasimoto (which motivated subsequent real-world experiments).
The quantum Hasimoto transformation is useful when normal fluid velocity is
relatively weak, so for the case where the normal fluid velocity is dominant we
resort to other approaches. We consider the dynamics of the tangent vector to
the vortex filament directly from the quantum LIA, and this approach, while
less elegant than the quantum Hasimoto transformation, enables us to study
waves primarily driven by the normal fluid velocity.

Fluid Dynamics;Chaotic Dynamics;Pattern Formation and Solitons;Exactly Solvable and Integrable Systems