Hasimoto Transformation and Vortex Soliton Motion Driven by Fluid Helicity

Details Hasimoto Transformation and Vortex Soliton Motion Driven by Fluid Helicity Hasimoto Transformation and Vortex Soliton Motion Driven by Fluid Helicity

2004-09-20

arxiv.org/abs/nlin/0409040v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

15 pages

Scientific paper

Vorticity filament motions with respect to the Dirac bracket of Rasetti and Regge [1975] are known to be related to the nonlinear Schr\"odinger equation by the Hasimoto transformation (HT), when the Hamiltonian is the Local Induction Approximation (LIA) of the kinetic energy. We show that when the Hamiltonian is the LIA of Euler-fluid helicity $\int\mathbf{u}\cdot {\rm curl} \mathbf{u}$, the vorticity filament equation of motion under the Rasetti-Regge Dirac bracket is mapped by HT to the integrable complex modified Korteweg-de Vries (cmKdV) equation, the second equation in the nonlinear Schr\"odinger hierarchy.

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