Weak Coupling Limit and Localized Oscillations in Euclidean Invariant Hamiltonian Systems

We prove the existence of time-periodic and spatially localized oscillations (discrete breathers) in a class of planar Euclidean-invariant Hamiltonian systems consisting of a finite number of interacting particles. This result is obtained in an “anticontinuous” limit, where atomic masses split into two groups that have different orders of magnitude (the mass ratio tending to infinity) and several degrees of freedom become weakly coupled. This kind of approach was introduced by MacKay and Aubry (Nonlinearity 7:1623–1643, 1994) (and further developed by Livi et al. in Nonlinearity 10:1421–1434, 1997) for one-dimensional Hamiltonian lattices. We extend their method to planar Euclidean-invariant systems and prove the existence of reversible discrete breathers in a general setting. In addition, we show the existence of nonlinear normal modes near the anticontinuous limit.