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Updated: Apr 10, 2026
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Published on: February 1, 2017
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Quantum Hasimoto transformation and nonlinear waves on a superfluid vortex filament under the quantum local induction
1Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom.
Summary
The quantum Hasimoto transformation connects quantum vortex filaments to a nonlinear dispersive PDE, enabling new solutions. This method reveals unique behaviors under friction, including solitons and vortex rings, not seen in classical models.
Area of Science:
- Quantum fluid dynamics
- Nonlinear physics
- Mathematical modeling
Background:
- The Hasimoto transformation links classical local induction approximation (LIA) to the nonlinear Schrödinger equation (NLS), facilitating new LIA solutions from known NLS solutions.
- The classical LIA models thin vortex filament motion, but omits quantum effects and mutual friction.
Purpose of the Study:
- To establish a quantum Hasimoto transformation for the quantum LIA, incorporating mutual friction effects.
- To derive and analyze quantum vortex filament solutions using a corresponding complex nonlinear dispersive partial differential equation (PDE).
- To investigate the behavior of specific wave solutions and topological structures under varying friction conditions.
Main Methods:
- Developed a quantum Hasimoto transformation to map the quantum LIA (with friction) to a cubic nonlinear dispersive PDE.
- Analyzed solutions to the derived PDE to determine quantum vortex filament behaviors, including Stokes waves, solitons, and traveling waves.
- Investigated solutions unique to the quantum LIA, arising from normal fluid velocity and mutual friction.
Main Results:
- Derived quantum vortex filament solutions, including Stokes waves, solitons, and similarity solutions, under normal and binormal friction.
- Identified novel solutions existing only with normal fluid velocity and mutual friction, such as normal fluid driven helices and a radius-varying vortex ring.
- Demonstrated that chaos is unlikely to arise from traveling waves in the quantum LIA formulation.
Conclusions:
- The quantum Hasimoto transformation is effective for weak normal fluid velocities, providing insights into quantum vortex dynamics.
- The inclusion of normal fluid velocity and mutual friction terms leads to distinct physical phenomena not present in the classical LIA.
- The study offers a framework for understanding complex behaviors of quantum vortex filaments in superfluids.
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