An existence theory for solitary waves on a ferrofluid jet
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909899141677056 |
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| author | Groves, Mark D. Nilsson, Dag Schütz, Leon |
| author_facet | Groves, Mark D. Nilsson, Dag Schütz, Leon |
| contents | 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 $η$. 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 $η$ to a perturbation of a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrödinger 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08799 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An existence theory for solitary waves on a ferrofluid jet Groves, Mark D. Nilsson, Dag Schütz, Leon Analysis of PDEs 35Q35, 35Q53, 35Q55, 76B15 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 $η$. 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 $η$ to a perturbation of a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrödinger 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. |
| title | An existence theory for solitary waves on a ferrofluid jet |
| topic | Analysis of PDEs 35Q35, 35Q53, 35Q55, 76B15 |
| url | https://arxiv.org/abs/2511.08799 |