A dimension-breaking phenomenon for water waves with weak surface tension
arXiv:1411.2475 · doi:10.1007/s00205-015-0941-3
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 Schrödinger 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.
The final publication is available at Springer via http://dx.doi.org/10.1007/s00205-015-0941-3
Cited by in corpus (6)
- Solitary waves on rotational flows with an interior stagnation point
- An existence theory for small-amplitude doubly periodic water waves with vorticity
- Transverse instability of periodic and generalized solitary waves for a fifth-order KP model
- Periodic solitons for the elliptic-elliptic focussing Davey-Stewartson equations
- On the mathematical description of time-dependent surface water waves
- Two dimensional solitary water waves with constant vorticity, Part I: the deep gravity case