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

We prove the existence of time-periodic solutions representing breathing patterns in general nonlinear Hamiltonian finite-size lattices with global coupling. As a first step the existence of localised solutions of a two site segment, where one oscillator performs larger-amplitude motion compared to the other one, is established. To this end the existence problem is converted into a fixed point problem for an operator on some appropriate function space which is solved by means of Schauder's Fixed Point Theorem. For isoenergetic states the degree of localisation can be tuned in a wide range by changing the initial data and the coupling strength. Subsequently, it is proven that a related localised state can be excited 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 also spatial patterns are possible that are built up from any combination of the small-amplitude state and the large-amplitude state.