An existence theory for solitary waves on a ferrofluid jet

TL;DR

This paper develops a rigorous mathematical framework for the existence of solitary wave solutions on a ferrofluid jet influenced by magnetic fields, extending classical wave theories to magnetohydrodynamic contexts.

Contribution

It introduces a novel existence theory for solitary waves on ferrofluid jets, combining nonlocal equations, Sobolev space analysis, and perturbation methods to generalize classical wave models.

Findings

Existence of solitary wave solutions established.

Reduction to perturbed KdV and nonlinear Schrödinger equations.

Rigorous mathematical proof using fixed-point and implicit-function theorems.

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.