An existence theory for solitary waves on a ferrofluid jet

Abstract

We discuss axisymmetric solitary waves on the surface of an otherwise cylindrical ferrofluid jet surrounding a stationary metal rod. The ferrofluid, which is governed by a general (nonlinear) magnetisation law, is subject to an azimuthal magnetic field generated by an electric current flowing along the rod. We treat the governing equations using a modification of the Zakharov-Craig-Sulem formulation for water waves, reducing the problem to a single nonlocal equation for the free-surface elevation variable $\eta$. The nonlocality in the equation takes the form of a Dirichlet-Neumann operator whose analyticity (in standard function spaces) is demonstrated by studying its defining boundary-value problem in newly introduced Sobolev spaces for radial functions.\ Using rudimentary fixed-point arguments and Fourier analysis we rigorously reduce the equation for $\eta$ to a perturbation of a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schr\"{o}dinger equation (for weak surface tension), both of which have nondegenerate explicit solitary-wave solutions. The existence theory is completed using an appropriate version of the implicit-function theorem.