Water-waves as a spatial dynamical system

Frédéric Dias (1) , Gérard Iooss (2)

1 CMLA - Centre de Mathématiques et de Leurs Applications
2 INLN - Institut Non Linéaire de Nice Sophia-Antipolis

The mathematical study of travelling waves, in the context of two dimensional potential flows in one or several layers of perfect fluid(s) and in the presence of free surface and interfaces, can be formulated as an ill-posed evolution problem, where the horizontal space variable plays the role of ``time''. In the finite depth case, the study of near equilibria waves reduces to a low dimensional "reversible ordinary differential equation." In most cases, it appears that the problem is a "perturbation of an integrable system", where all types of solutions are known. We describe the method of study and review typical results. In addition, we study the infinite depth limit, which is indeed a case of physical interest. In such a case, the above reduction technique fails because the linearized operator possesses an "essential spectrum" filling the whole real axis, and new adapted tools are necessary. We also discuss the latest results on the existence of travelling waves in stratified fluids and on three dimensional travelling waves, in the same spirit of reversible dynamical systems. Finally, we review the recent results on the classical two-dimensional standing wave problem.

Mots clés: Infinite dimensional Dynamical systems; Bifurcations in reversible systems; Water waves; Spatial dynamics; Normal forms of reversible systems

Domaines: Equations aux dérivées partielles [math.AP]

Frédéric Dias, Gérard Iooss. Water-waves as a spatial dynamical system. S. Friedlander, D. Serre. Handbook of Mathematical Fluid Dynamics, Elsevier, pp.443-499, 2003, ⟨10.1016/S1874-5792(03)80012-5⟩. ⟨hal-00017484⟩