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Modulational instabilities in lattices with power-law hoppings and interactions

We study the occurrence of modulational instabilities in lattices with non-local, power-law hoppings and interactions. Choosing as a case study the discrete nonlinear Schrödinger equation, we consider one-dimensional chains with power-law decaying interactions (with exponent α) and hoppings (with exponent β): An extensive energy is obtained for α, β>1. We show that the effect of power-law interactions is that of shifting the onset of the modulational instabilities region for α>1. At a critical value of the interaction strength, the modulational stable region shrinks to zero. Similar results are found for effectively short-range nonlocal hoppings (β> 2): At variance, for longer-ranged hoppings (1 < β< 2) there is no longer any modulational stability. The hopping instability arises for q = 0 perturbations, thus the system is most sensitive to the perturbations of the order of the system&#39;s size. We also discuss the stability regions in the presence of the interplay between competing interactions - (e.g., attractive local and repulsive nonlocal interactions). We find that noncompeting nonlocal interactions give rise to a modulational instability emerging for a perturbing wave vector q = πwhile competing nonlocal interactions may induce a modulational instability for a perturbing wave vector 0 < q < π. Since for α> 1 and β> 2 the effects are similar to the effect produced on the stability phase diagram by finite range interactions and/or hoppings, we conclude that the modulational instability is &#34;genuinely&#34; long-ranged for 1 < β< 2 nonlocal hoppings.