An existence theory for solitary waves on a ferrofluid jet

TL;DR

The paper develops a rigorous existence theory for axisymmetric solitary waves on a ferrofluid jet surrounding a current-carrying wire by recasting the ferrohydrodynamic problem into a single nonlocal equation for the surface elevation using a Dirichlet-Neumann operator in a Zakharov-Craig-Sulem framework. A detailed analytic setup in radial Sobolev-type spaces yields an analytic dependence of the nonlocal operator on the surface, permitting a Lyapunov-Schmidt reduction that, under strong surface tension, reduces to a full-dispersion KdV equation, and under weak surface tension, to a full-dispersion nonlinear Schrödinger equation. In both regimes, the authors construct small-amplitude solitary-wave solutions via an implicit-function theorem, proving the existence of symmetric KdV-type solitons for 1<γ<9 and symmetric NLS-type solitons for γ>9, with explicit leading-order profiles and parameter relations. This provides a rigorous foundation for axisymmetric solitary waves in ferrofluid jets in the presence of an azimuthal magnetic field, with potential implications for ferrofluidic waveguides and related magnetofluidic systems.

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 $η$. 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.

Figures (6)

• Figure 1: Waves on the surface of a ferrofluid jet surrounding a current-carrying wire
• Figure 2: Korteweg-de Vries solitary waves of elevation (left) and of depression (right) depending on the sign of $d_0$
• Figure 3: Nonlinear Schrödinger solitary waves of elevation (left) and of depression (right) depending on the sign in equation \ref{['eta from zeta']}
• Figure 4: Dispersion relation in the cases $1<\gamma\leq 9$ (left) and $\gamma>9$ (right); the minimum value of $c^2$ is denoted by $c_0^2$
• Figure 5: (a) The support of $\hat{\eta}_1$ is contained in the set $S$, where $S=(-\delta,\delta)$ for $1<\gamma<9$ (left) and $S=(-\omega-\delta,-\omega+\delta) \cup (\omega-\delta,\omega+\delta)$ for $\gamma>9$ (right).

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.3
• Theorem 2.1
• Lemma 2.2
• Theorem 2.3
• Corollary 2.4
• Corollary 2.5
• Proposition 2.6
• Proposition 2.7