Discrete Breathers: Localization and transfer of energy in discrete Hamiltonian nonlinear systems

We present a partial review of results concerning the theory of Discrete Breathers in nonlinear hamiltonian and discrete systems. These special time-periodic solutions gained much interest during the last decade because they may be involved in numerous and various phenomena in physics and biophysics where they could produce nontrivial effects of energy focusing and transfer. We first review the principles which govern their existence and which are used in the existence proofs available up to now. Next we discuss their linear stability and the interaction of Discrete Breathers with small amplitude waves, showing also how they could grow or decay. We also briefly discuss the existence of intraband DBs in systems with a linear discrete spectrum which are not spatially periodic (random or otherwise) and where linear waves cannot propagate. We also show that nonlinearity may restore the existence of solutions that propagate energy. We review some results concerning energy transportation by DBs and especially the main features associated with their mobility. We briefly discuss new perspectives opened up by the theory for applications that, in particular, look especially interesting for biophysics.