Dynamical Systems in MathematicalPhysics

The purpose of this article is to describe some basic problems related to the interplay between Dynamical Systems and Mathematical Physics. Since it is impossible to be exhaustive in these topics, we choose to focus on water waves models. These mathematical models are described by partial differential equations that can be understood as dynamical systems in a suitable infinite-dimensional phase space. Wewill not address the original equations for 2D surface water waves, even if we know that dynamical systems methods can help to exhibit some solitary waves for the equations; we refer here to the works of Iooss, Kirchgarsner, Mielke. Another approach is to seek these 2D surface water waves as saddle points for some Hamiltonian energies; see Buffoni, Séré, Tolland. We rather present these arguments on some asymptotical models for the propagation of surface water waves. 2. Asymptotical models in hydrodynamics