Special Session 11: Dynamics of fluids and nonlinear waves
Contents
The water-wave problem has small-amplitude line solitary-wave solutions which to leading order are described by the Korteweg-deVries equation (for strong surface tension) or nonlinear
Schr\"{o}dinger equation (for weak surface tension).
We present an existence theory for three-dimensional \emph{periodically modulated solitary-wave solutions} to the water-wave problem 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 and are described to leading order by the Kadomtsev-Petviashvili equation (for strong surface tension) or Davey-Stewartson equation (for weak surface tension). The term \emph{dimension-breaking phenomenon} describes the spontaneous emergence of a spatially inhomogeneous solution of a partial differential equation from a solution which is homogeneous in one or more spatial dimensions.