# Algebraic method for constructing singular steady solitary waves: A case study: Singular solitary waves

3 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , NKUA - National and Kapodistrian University of Athens
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.
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Journal articles
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Cited literature [24 references]

https://hal.archives-ouvertes.fr/hal-01290471
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Submitted on : Friday, June 17, 2016 - 9:42:07 AM
Last modification on : Saturday, June 25, 2022 - 11:20:40 PM
Long-term archiving on: : Sunday, September 18, 2016 - 10:57:06 AM

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### Identifiers

• HAL Id : hal-01290471, version 2
• ARXIV : 1603.07472

### Citation

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. ⟨hal-01290471v2⟩

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