Existence of breathing patterns in globally coupled finite-size nonlinear lattices

We prove the existence of time-periodic solutions consisting of patterns built up from two states, one with small amplitude and the other one with large amplitude, in general nonlinear Hamiltonian finite-size lattices with global coupling. Utilising the comparison principle for differential equations it is demonstrated that for a two site segment of the nonlinear lattice one can construct solutions that are sandwiched between two oscillatory localised lattice states. Subsequently, it is proven that such a localised state can be embedded in the extended nonlinear lattice forming a breathing pattern with a single site of large amplitude against a background of uniform small-amplitude states. Furthermore, it is demonstrated that spatial patterns are possible that are built up from any combination of the small-amplitude state and the large-amplitude state. It is shown that for soft (hard) on-site potentials the range of allowed frequencies of the in-phase (out-of-phase) breathing patterns extends to values below (above) the lower (upper) value of the bivalued degenerate linear spectrum of phonon frequencies.