The paper proves convergence of concentrated helical vortices to an explicit ODE and claims the first rigorous derivation of leapfrogging of Kelvin waves, but a coefficient in the ODE contradicts the paper's own derivation.
Helical kelvin waves for the 3D Euler equation
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abstract
Helical Kelvin waves were conjectured to exist for the 3D Euler equations in Lucas and Dritschel \cite{LucDri} (as well as in \cite{Chu}) by studying dispersion relation for infinitesimal linear perturbations of a circular helically symmetric vortex patch. This paper aims to rigorously establish the existence of these $m$-fold symmetric helical Kelvin waves, in both simply and doubly connected cases, for the 3D Euler equations. The construction is based on linearization of contour dynamics equations and bifurcation theory. Our results rigorously verify the prediction in aforementioned papers and extend $m$-waves of Kelvin from the 2D Euler equations to the 3D helically symmetric Euler equations.
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Dynamics and leapfrogging phenomena of multiple helical vortices for 3D incompressible Euler equations
The paper proves convergence of concentrated helical vortices to an explicit ODE and claims the first rigorous derivation of leapfrogging of Kelvin waves, but a coefficient in the ODE contradicts the paper's own derivation.