Helical Kelvin Waves for the Three-Dimensional Euler Equations

Abstract. Helical Kelvin waves were predicted to exist for the three-dimensional (3D) Euler equations in Lucas and Dritschel [ J. Fluid Mech., 634 (2009), pp. 245–268] (as well as in [T. Chu and S. G. Llewellyn Smith, Regul. Chaotic Dyn., 26 (2021), pp. 600–617]) by studying a dispersion relation for infinitesimal linear perturbations of a circular helically symmetric vortex patch. This paper aims to rigorously establish the existence of these [Formula: see text]-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 the aforementioned papers in the no-swirl case and extend [Formula: see text]-waves of Kelvin from the 2D Euler equations to the 3D helically symmetric Euler equations.