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The problem of showing the existence of localised modes in nonlinear latticeshas attracted considerable efforts from the physical but also from themathematical viewpoint where a rich variety of methods has been employed. Inthis paper we prove that a fixed point theory approach based on the celebratedSchauder's Fixed Point Theorem may provide a general method to establishconcisely not only the existence of localised structures but also a requiredrate of spatial localisation. As a case study we consider lattices of coupledparticles with nonlinear nearest neighbour interaction and prove the existenceof exponentially spatially localised breathers exhibiting either even-parity orodd-parity symmetry under necessary non-resonant conditions accompanied withthe proof of energy bounds of the solutions.

journal of mathematical physics

Existence of exponentially spatially localised breather solutions for  lattices of nonlinearly coupled particles: Schauder's fixed point theorem  approach