Large-Amplitude Steady Electrohydrodynamic Solitary Waves with Constant Vorticity

TL;DR

This work analyzes large-amplitude solitary electrohydrodynamic waves on a two-dimensional dielectric fluid surface under gravity and a vertical electric field, in the presence of constant vorticity $\omega$. By combining conformal mapping, non-dimensionalization, and a center-manifold reduction near the critical mode, the authors construct a global continuum $\mathscr{C}$ of solitary-wave solutions and prove that, along $\mathscr{C}$, either stagnation, conformal degeneracy, or unbounded wave speed occurs as the curve is extended. They further exclude monotone bore solutions and establish lower bounds on the Froude number, providing a detailed global bifurcation picture for the fully nonlinear, non-Fredholm problem and achieving explicit small-amplitude profiles via center-manifold reduction. The results significantly advance the mathematical understanding of electrohydrodynamic interfacial waves with vorticity, enabling rigorous prediction of end-state behaviors and qualitative wave-structure properties for practical fluid-physics applications. The combination of conformal-geometry methods with global analytic bifurcation theory yields a robust framework for fully nonlinear EHD solitary waves with overhanging profiles and interior stagnation points.

Abstract

This paper investigates solitary water waves propagating on the surface of a two-dimensional dielectric fluid subject to an electric field. The system is formulated as a nonlinear free boundary problem, with interfacial dynamics governed by the strong coupling between the Euler equations with constant vorticity and the electric potential equations. We aim to explore the effects of the electric field and constant vorticity on the nonlinear wave interactions in such a system, specifically examining whether large-amplitude solitary waves analogous to those in reference \cite{SVHMHW2023} exist. Although the inclusion of the electric field considerably complicates the analysis, we establish the existence of a continuous branch of large-amplitude solitary wave solutions. Moreover, along the global bifurcation curve, one of the following must occur: (i) the formation of an equilibrium stagnation point, (ii) the degeneration of the conformal mapping, (iii) the onset of flow stagnation, or (iv) an unbounded increase in the dimensionless wave speed.

Figures (4)

• Figure 1: Schematic of the problem.
• Figure 2: The conformal parametrization of the fluid domain $\Omega$
• Figure 3: An electrohydrodynamic bore, with distinct asymptotic velocities, electric fields and depths at $x \to \pm\infty$.
• Figure 4: Phase portrait of ODE \ref{['0067.4']} with $\varepsilon=0, \gamma=1, \epsilon_1 \in \{0, 0.1, 0.5, 1, 5, 10\}$

Theorems & Definitions (57)

• Remark 2.1
• Theorem 2.2
• Remark 2.3
• Remark 2.4
• Theorem 2.5: Nonexistence of bore solutions
• Theorem 2.6
• Remark 2.7
• Remark 3.1
• Lemma 3.2
• proof