This book chapter is published open access.
In this survey we shall consider Hamiltonian dispersive partial differential equations on compact manifolds and discuss the existence, close to an elliptic fixed point, of special recursive solutions, which are superpositions of oscillating motions, together with their stability/instability properties. One can envision such equations as chains of harmonic oscillators coupled with a small nonlinearity, thus one expects a complicated interplay between chaotic and recursive phenomena due to resonances and small divisors, which are studied with methods from KAM theory.
We shall concentrate mainly on the stability properties of the fixed point, as well as the existence and stability of quasiperiodic and almost periodic solutions. After giving an overview on the literature, we shall present some promising recent results and discuss possible extensions and open problems.