From modulational instability to self-trapping in nonlinear chains with power-law hopping amplitudes

Abstract

The modulational instability problem in tight-binding nonlinear chains with long-ranged hopping amplitudes decaying as 1∕rα is investigated. Exploring that the spectral dimension of the corresponding linear chain deff=2∕(α−1) for α≥3, we demonstrate that the nonlinear strength χMI above which a uniform solution becomes unstable scales with the chain size N as χMI∝N2−α for α≤3 while χMI∝1∕N for α≥3. We also present numerical data unveiling that the intermediate regimes of breathing and chaotic-like solutions appearing during the crossover from the stable uniform solution to self-trapping are fully suppressed when the hopping amplitudes decay slower than 1∕r2.