Two-dimensional capillary liquid drop: Craig-Sulem formulation on $\mathbb{T}^1$ and bifurcations from multiple eigenvalues of rotating waves

TL;DR

This paper formulates a 2D capillary liquid drop problem on the torus, reveals its Hamiltonian structure, and demonstrates the existence of symmetric rotating wave solutions bifurcating from multiple eigenvalues.

Contribution

It derives a Craig-Sulem formulation for the 2D capillary drop on the torus, analyzes its Hamiltonian structure, and proves bifurcation of rotating wave solutions from multiple eigenvalues.

Findings

Formulation of the problem over the circle and torus.

Identification of Hamiltonian structure and conserved quantities.

Existence of symmetric rotating wave solutions bifurcating from multiple eigenvalues.

Abstract

We consider the free boundary problem for a two-dimensional, incompressible, perfect, irrotational liquid drop of nearly circular shape with capillarity: that is, we consider the 2D version of the 3D capillary drop problem treated in Baldi-Julin-La Manna [11] and Baldi-La Manna-La Scala [12]. In particular, we derive its Craig-Sulem formulation firstly over the circle, then over the one-dimensional flat torus; the arising equations are similar to the pure capillary Water Waves for the ocean problem, apart from conformal factors and additional terms due to curvature terms. Then, we show its Hamiltonian structure and we derive constants of motions from symmetries, one of which is the invariance by the torus action. Thanks to this invariance, we show the existence of orbits of rotating wave solutions (which are the analogous of travelling waves of the ocean problem) by bifurcation from multiple eigenvalues in the spirit of Moser-Weinstein [44, 56] and Craig-Nicholls [22] variational approaches; in particular, we can parametrize such orbits by the angular momentum, and for each value of it they are unique. This will imply that each orbit is generated by symmetric rotating waves.