Algebraic method for constructing singular steady solitary waves: A case study
Résumé
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.
Domaines
Mécanique des fluides [physics.class-ph] Systèmes Solubles et Intégrables [nlin.SI] Equations aux dérivées partielles [math.AP] Analyse classique [math.CA] Géométrie algébrique [math.AG] Formation de Structures et Solitons [nlin.PS] Physique Atmosphérique et Océanique [physics.ao-ph] Dynamique des Fluides [physics.flu-dyn]
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