Quasi-Continuum Approximation for Discrete Breathers in Fermi–Pasta–Ulam Atomic Chains

Abstract

A quasi-continuum model is proposed and studied for the discrete breathers (DBs) or the intrinsic localized modes (ILMs) in one-dimensional anharmonic lattices on the basis of the Padé approximation. Two conservation laws are obtained in the model equation, and the corresponding quantities in discrete system are numerically found to be conserved as well. The exact stationary breather solution is found to the model equation for the Fermi–Pasta–Ulam-β (FPU-β) atomic chains with purely hard quartic anharmonicity, and also found is the approximate stationary breather solution in case the quadratic interaction is included by means of the averaged Lagrangian method. The application of the multiple scales method to the model equation indicates the movability of the breather solutions in the small-amplitude limit. The results of numerical simulations fully support the analytical results mentioned above. A detailed comparison between the FPU-β lattice and its continuum model in properties of the breather collision achieves a good agreement except for a resonant collision case when the discreteness predominates.