Steady dark solitary flexural gravity waves

Paul A. Milewski, Jean-Marc Vanden-Broeck, Zhan Wang

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The nonlinear Schrödinger (NLS) equation describes the modulational limit of many surface water wave problems. Dark solitary waves of the NLS equation asymptote to a constant in the far field and have a localized decrease to zero amplitude at the origin, corresponding to water wave solutions that asymptote to a uniform periodic Stokes wave in the far field and decreasing oscillations near the origin. It is natural to ask whether these dark solitary waves can be found in the irrotational Euler equations. In this paper, we find such solutions in the context of flexural-gravity waves, which are often used as a model for waves in ice-covered water. This is a situation in which the NLS equation predicts steadily travelling dark solitons. The solution branches of dark solitons are continued, and one branch leads to fully localized solutions at large amplitudes.
Original languageEnglish
Article number20120485
JournalProceedings of the Royal Society A
Issue number2150
Early online date28 Nov 2012
Publication statusPublished - Feb 2013


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