Breathers on diatomic Fermi–Pasta–Ulam lattices

We prove the existence of breathers (spatially localized and time-periodic oscillations) in diatomic Fermi–Pasta–Ulam (FPU) chains with arbitrary mass ratio. This completes an existence result by Livi, Spicci and MacKay valid for large mass ratio. The problem is formulated as a mapping in a loop space and analyzed via a discrete spatial centre manifold reduction. Plane wave solutions of the linearized system have frequencies in a higher “optic” band or a lower “acoustic” band. For frequencies close to band edges, all small amplitude solutions of the nonlinear system lie on a finite-dimensional centre manifold, which reduces the problem locally to the study of a finite-dimensional mapping. For good parameter values, the map admits homoclinic orbits to 0 corresponding to discrete breathers. When the FPU interaction potential satisfies a hardening condition, we find breathers with frequencies slightly above the optic band, or in the gap slightly above the acoustic band. For a potential satisfying the opposite softening condition, we obtain breathers with frequencies in the gap slightly below the optic band.