Oscillatory Instabilities of Standing Waves in One-Dimensional Nonlinear Lattices

In one-dimensional anharmonic lattices, we construct nonlinear standing waves (SWs) reducing to harmonic SWs at small amplitude. For SWs with spatial periodicity incommensurate with the lattice period, a transition by breaking of analyticity versus wave amplitude is observed. As a consequence of the discreteness, oscillatory linear instabilities, persisting for arbitrarily small amplitude in infinite lattices, appear for all wave numbers $Q\ne 0$, $\pi$. Incommensurate analytic SWs with $|Q|>\pi /2$ may however appear as “quasistable,” as their instability growth rate is of higher order.