# Spatial dynamics methods for solitary waves on a ferrofluid jet

## TL;DR

This paper develops a mathematical framework to prove the existence of axisymmetric solitary waves on a ferrofluid jet influenced by magnetic fields, using Hamiltonian systems and dynamical systems techniques.

## Contribution

It introduces a novel application of center-manifold reduction and dynamical systems methods to analyze solitary waves in ferrofluid jets with nonlinear magnetization laws.

## Key findings

- Existence of solitary wave solutions is established.
- A reduction to finite-dimensional Hamiltonian systems is achieved.
- Homoclinic solutions corresponding to solitary waves are identified.

## Abstract

This paper presents existence theories for several families of 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. The ferrohydrodynamic problem for axisymmetric travelling waves is formulated as an infinite-dimensional Hamiltonian system in which the axial direction is the time-like variable. A centre-manifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite number of degrees of freedom, and homoclinic solutions to the reduced system, which correspond to solitary waves, are detected by dynamical-systems methods.