Three-dimensional water waves
Seminar Room 1, Newton Institute
AbstractThe existence of solitary-wave solutions to the three-dimensional water-wave problem with is predicted by the Kadomtsev-Petviashvili (KP) equation for strong surface tension and Davey-Stewartson (DS) equation for weak surface tension.The term solitary wave describes any solution which has a pulse-like profile in its direction of propagation, and these model equations admit three types of solitary waves. A line solitary wave is spatially homogeneous in the direction transverse to its direction of propagation, while a periodically modulated solitary wave is periodic in the transverse direction. A fully localised solitary wave on the other hand decays to zero in all spatial directions. In this talk I outline mathematical results which confirm the existence of all three types of solitary wave for the full gravity-capillary water-wave problem in its usual formulation as a free-boundary problem for the Euler equations. Both strong and weak surface tension are treated. The line solitary waves are found by establishing the existence of a low-dimensional invariant manifold containing homoclinic orbits. The periodically modulated solitary waves are created when a line solitary wave undergoes a dimension-breaking bifurcation in which it spontaneously loses its spatial homogeneity in the transverse direction; an infinite-dimensional version of the Lyapunov centre theorem is the main ingredient in the existence theorem. The fully localised solitary waves are obtained by finding critical points of a variational functional.
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