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

TL;DR

This paper employs algebraic methods to analyze and classify singular steady solitary waves in shallow water models, revealing a richer variety of solutions including those with angular points.

Contribution

It introduces an algebraic geometric approach to classify singular solitary waves in nonlinear water wave equations, expanding understanding of wave solution diversity.

Findings

Classification of singular solitary waves with angular points

Identification of new solution types in capillary-gravity waves

Application of algebraic geometry to nonlinear wave analysis

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.