Fully localised three-dimensional solitary water waves on Beltrami flows with strong surface tension

TL;DR

The paper proves the existence of fully localised three-dimensional solitary water waves on Beltrami flows with strong surface tension by reformulating the problem via a Groves–Horn Dirichlet–Neumann framework and reducing it to a perturbation of the KP-I equation. It develops a careful KP-scale reduction, decomposing the surface profile into low- and high-frequency parts, solving the high-frequency part through a contraction argument, and then applying an implicit-function theorem to obtain fully localised waves from symmetric KP lumps. The construction yields waves that, in the formal limit, converge to KP lump profiles, with nondegeneracy of the lumps ensuring persistence. The results extend irrotational solitary-wave theories to rotational Beltrami flows and provide a rigorous mechanism for generating a family of fully localised 3D solitary waves under large surface tension (β) and suitable α.

Abstract

Fully localised three-dimensional solitary waves are steady water waves which are evanescent in every horizontal direction. This paper presents an existence theory for such waves under the assumptions that the relative vorticity and velocity fields are parallel (`Beltrami flows'), that the free surface of the water takes the form $\{z=η(x,y)\}$ for some function $η: {\mathbb R}^2\rightarrow{\mathbb R}$, and that the influence of surface tension is sufficiently strong. The governing equations are formulated as a single equation for $η$, which is then reduced to a perturbation of the KP-I equation. This equation has recently been shown to have a family of nondegenerate localised solutions, and an application of a suitable variant of the implicit-function theorem shows that they persist under perturbations.

Figures (3)

• Figure 1: The trivial flow \ref{['ABC flow']}
• Figure 2: The KP lumps $\zeta_1^\star$ (left) and $\zeta_2^\star$ (right).
• Figure 3: The set $S=\{(k_1,k_2): |k_1| \leq \delta, \left|\frac{k_2}{k_1}\right|\leq \delta\}$.

Theorems & Definitions (39)

• Lemma 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Proposition 1.7
• Proposition 1.8
• Corollary 1.9
• Proposition 2.1