Algebraic method for constructing singular steady solitary waves: a case study

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 equation with surface tension, because it provides a tractable model that, at the same time, is not too simple, so interest in the method can be emphasized. 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 localized infinitely smooth travelling wave solutions—solitary waves of elevation and of depression. However, if we allow the solitary waves to have an angular point, then the ‘zoology’ of solutions becomes much richer, and the main goal of this study is to provide a complete classification of such singular localized solutions using the methods of the effective algebraic geometry.