Statistical Approach to Modulational Instability in Nonlinear Discrete Systems

We use a statistical approach to investigate the modulational instability (Benjamin–Feir instability) in several nonlinear discrete systems: the discrete nonlinear Schrodinger (NLS) equation, the Ablowitz–Ladik equation, and the discrete deformable NLS equation. We derive a kinetic equation for the two-point correlation function and use a Wigner–Moyal transformation to write it in a mixed space-wave-number representation. We perform a linear stability analysis of the resulting equation and discuss the obtained integral stability condition using several forms of the initial unperturbed spectrum (Lorentzian and $\delta$-spectrum). We compare the results with the continuum limit (the NLS equation) and with previous results.