Accurate continuous models for the dynamics of a lattice meta-materials with long-range interactions

This paper examines the construction of continuous models for the dynamics of lattice meta-materials with long-range interactions. Lagrange lattices with nearest-neighbour and second-neighbour coupling, and with nearest-neighbour and third-nearest-neighbour coupling, are studied. Particular attention is paid to characteristic points on the dispersion curve, such as rotons, maxons, and stationary inflation points. Multipoint Padé approximants are employed for approximation purposes. Selecting the parameters of the fractional–rational function ensures that the dispersion curves of the discrete and continuum systems coincide at the characteristic points. The resulting partial differential equations have a low order in spatial variables. Importantly, these systems provide a good approximation of the discrete systems for any stiffness of coupling.