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Existence and stability of solitary water waves with surface tension

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Existence and stability of solitary water waves with surface tension

Dr. Erik Wahl´en

Solitary water waves with surface tension on water of infinite depth are constructed by minimising the energy subject to the constraint of fixed mo- mentum. The stability of the set of minimisers follows by a standard principle since the energy and momentum are conserved quantities. The stability must however be understood in a qualified sense due to the lack of a global well- posedness theory for the initial value problem. The variational method relies on the concentration-compactness principle and a penalisation argument, which is needed because of the quasilinear structure. To leading order the solitary waves take the form of a periodic wave train modulated by a small-amplitude decaying envelope described by the focusing nonlinear Schr¨odinger equation.

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