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.

Keywords

Bibliographic record

BibTeX Citation

@article{Clamond2016algebraicmethod,
  author = {Clamond, D. and Dutykh, D. and Galligo, A.},
  title = {Algebraic method for constructing singular steady solitary waves: A case study},
  journal = {Proc. R. Soc. A.},
  year = {2016},
  volume = {472},
  number = {2191},
  pages = {20160194},
  doi = {10.1098/rspa.2016.0194},
  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.},
  keywords = {solitary waves, effective algebraic geometry, surface tension, Serre-Green-Naghdi equations, shallow water, algebraic method, phase plane analysis, ordinary differential equations, singular solutions, algebraic geometry, capillary-gravity waves}
}