Modulational instabilities in discrete lattices

We study analytically and numerically modulational instabilities in discrete nonlinear chains, taking the discrete Klein-Gordon model as an example. We show that discreteness can drastically change the conditions for modulational instability; e.g., at small wave numbers a nonlinear carrier wave is unstable to all possible modulations of its amplitude if the wave amplitude exceeds a certain threshold value. Numerical simulations show the validity of the analytical approach for the initial stage of the time evolution, provided that the harmonics generated by the nonlinear terms are considered. The long-term evolution exhibits chaoticlike states.