Algebraic method for constructing singular steady solitary waves: A case study
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.
Domains
Fluid mechanics [physics.class-ph] Exactly Solvable and Integrable Systems [nlin.SI] Analysis of PDEs [math.AP] Classical Analysis and ODEs [math.CA] Algebraic Geometry [math.AG] Pattern Formation and Solitons [nlin.PS] Atmospheric and Oceanic Physics [physics.ao-ph] Fluid Dynamics [physics.flu-dyn]Origin | Files produced by the author(s) |
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