2014/02/04 by Robert A. Van Gorder, Van Gorder, Robert A.
Engineering · Physics and Astronomy · #Advanced Fiber Optic Sensors #Advanced Frequency and Time Standards #Chaotic Dynamics (nlin.CD) #Cold Atom Physics and Bose-Einstein Condensates #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Pattern Formation and Solitons (nlin.PS)
paper · pdf · doi:10.48550/arxiv.1402.7023
openalex publication_date 2014/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Hasimoto transformation between the classical LIA (local induction\napproximation, a model approximating the motion of a thin vortex filament) and\nthe nonlinear Schr "odinger equation (NLS) has proven very useful in the past,\nsince it allows one to construct new solutions to the LIA once a solution to\nthe NLS is known. In the present paper, the quantum form of the LIA (which\nincludes mutual friction effects) is put into correspondence with a type of\ncomplex nonlinear dispersive partial differential equation (PDE) with cubic\nnonlinearity (similar in form to a Ginsburg-Landau equation, with additional\nnonlinear terms). Transforming the quantum LIA in such a way enables one to\nobtain quantum vortex filament solutions once solutions to this dispersive PDE\nare known. From our quantum Hasimoto transformation, we determine the form and\nbehavior of Stokes waves and a standing 1-soliton solution under normal and\nbinormal friction effects. The soliton solution on a quantum vortex filament is\na natural generalization of the classical 1-soliton solution constructed\nmathematically by Hasimoto (which motivated subsequent real-world experiments).\nThe quantum Hasimoto transformation is useful when normal fluid velocity is\nrelatively weak, so for the case where the normal fluid velocity is dominant we\nresort to other approaches. We consider the dynamics of the tangent vector to\nthe vortex filament directly from the quantum LIA, and this approach, while\nless elegant than the quantum Hasimoto transformation, enables us to study\nwaves primarily driven by the normal fluid velocity.\n