A dimension-breaking phenomenon for water waves with weak surface tension

TL;DR

This paper investigates a dimension-breaking bifurcation in water waves with weak surface tension, revealing the existence of three-dimensional modulated solitary waves and demonstrating the instability of line solitary waves to transverse perturbations.

Contribution

It provides an existence theory for 3D modulated solitary waves emanating from line solitary waves via a dimension-breaking bifurcation, and analyzes their stability properties.

Findings

Existence of 3D periodically modulated solitary waves

Line solitary waves are linearly unstable to long-wavelength transverse perturbations

Dimension-breaking bifurcation leads to new wave solutions

Abstract

It is well known that the water-wave problem with weak surface tension has small-amplitude line solitary-wave solutions which to leading order are described by the nonlinear Schr\"odinger equation. The present paper contains an existence theory for three-dimensional periodically modulated solitary-wave solutions which have a solitary-wave profile in the direction of propagation and are periodic in the transverse direction; they emanate from the line solitary waves in a dimension-breaking bifurcation. In addition, it is shown that the line solitary waves are linearly unstable to long-wavelength transverse perturbations. The key to these results is a formulation of the water wave problem as an evolutionary system in which the transverse horizontal variable plays the role of time, a careful study of the purely imaginary spectrum of the operator obtained by linearising the evolutionary system at a line solitary wave, and an application of an infinite-dimensional version of the classical Lyapunov centre theorem.