Abstract : This article describes the use of algebraic methods in a phase plane analysis of ordinary differential equations. The method is illustrated by the study of capillary-gravity steady surface waves propagating in shallow water. We consider the (fully nonlinear, weakly dispersive) Serre-Green-Naghdi equations with surface tension, because it provides a tractable model that, in the same time, is not too simple so the interest of the method can be emphasised. In particular, we analyse a special class of solutions, the solitary waves, which play an important role in many fields of Physics. In capillary-gravity regime, there are two kinds of localised infinitely smooth travelling wave solutions -- solitary waves of elevation and of depression. However, if we allow the solitary waves to have an angular point, the ``zoology'' of solutions becomes much richer and the main goal of this study is to provide a complete classification of such singular localised solutions using the methods of the effective Algebraic Geometry.
https://hal.archives-ouvertes.fr/hal-01290471
Contributor : Denys Dutykh
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Submitted on : Friday, June 17, 2016 - 9:42:07 AM
Last modification on : Thursday, May 3, 2018 - 1:32:58 PM
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Didier Clamond, Denys Dutykh, André Galligo. Algebraic method for constructing singular steady solitary waves: A case study: Singular solitary waves. Proceedings of the Royal Society of London, Royal Society, The, 2016, 472 (2191), pp.20160194. ⟨http://rspa.royalsocietypublishing.org/content/472/2191/20160194⟩. ⟨hal-01290471v2⟩
