Variational proof for hard Discrete breathers in some classes of Hamiltonian dynamical systems

Abstract

We present a new method for proving the existence of Discrete Breathers in translationally invariant Hamiltonian systems describing massive particles interacting by a short range covex potential provided their frequency is above the linear phonon spectrum. The method holds for systems either with optical phonons (with a phonon gap) or with acoustic phonons (without phonon gap but with nonvanishing sound velocities), and does not use the concept of anticontinuous limit as most early methods. Discrete Breathers are obtained as loops in the phase space which maximize a certain average energy function for a fixed pseudoaction appropriately defined. It suffices to exhibit a trial loop with energy larger than the linear phonon energy at the same pseudoaction to prove the existence of a Discrete Breather with a frequency above the linear phonon spectrum. As a straightforward application of the method, Discrete Breathers are proven to exist at any energy (even small) in the quartic (or $\beta$) one-dimensional FPU model, which up to now was lacking a rigorous existence proof. The method can also work for piezoactive DBs in one or more dimensions and in many more complex models.