I will try to explain, without going into too much detail, how one can consider a non-linear wave equation as a dynamical system and what it brings to the study of its solutions. We begin by considering our model case, the non-linear Klein-Gordon equation and state its basic properties. We will then see what happens for solutions with energies below that of the ground state. After that, we place ourselves energetically around the ground state and we show the apparition of the so-called invariant manifolds. Finally, we consider the critical "pure" (without the mass term) wave equation and describe some of its interesting solutions. The last part will be concerned with an attempt to rely what we have learn so far with the critical case.
