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The Hasimoto transformation between the classical LIA (local inductionapproximation, a model approximating the motion of a thin vortex filament) andthe 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 tothe NLS is known. In the present paper, the quantum form of the LIA (whichincludes mutual friction effects) is put into correspondence with a type ofcomplex nonlinear dispersive partial differential equation (PDE) with cubicnonlinearity (similar in form to a Ginsburg-Landau equation, with additionalnonlinear terms). Transforming the quantum LIA in such a way enables one toobtain quantum vortex filament solutions once solutions to this dispersive PDEare known. From our quantum Hasimoto transformation, we determine the form andbehavior of Stokes waves and a standing 1-soliton solution under normal andbinormal friction effects. The soliton solution on a quantum vortex filament isa natural generalization of the classical 1-soliton solution constructedmathematically by Hasimoto (which motivated subsequent real-world experiments).The quantum Hasimoto transformation is useful when normal fluid velocity isrelatively weak, so for the case where the normal fluid velocity is dominant weresort to other approaches. We consider the dynamics of the tangent vector tothe vortex filament directly from the quantum LIA, and this approach, whileless elegant than the quantum Hasimoto transformation, enables us to studywaves primarily driven by the normal fluid velocity.

Quantum Hasimoto transformation and nonlinear waves on a superfluid vortex filament under the quantum local induction approximation