Giacomo Gori, Tommaso Macri, Andrea Trombettoni
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 Schr\"odinger equation, we consider one-dimensional chains with power-law decaying interactions (with exponent \alpha) and hoppings (with exponent \beta): an extensive energy is obtained for \alpha, \beta>1. We show that the effect of power-law interactions is that of shifting the onset of the modulational instabilities region for \alpha>1: at a critical value of the interaction strength, the modulational stable region shrinks to zero. Similar results are found for effectively short-range non-local hoppings (\beta>2): at variance, for longer ranged hoppings (1<\beta<2) there is no longer any modulational stability. We also discuss the stability regions in presence of the interplay between competing interactions - e.g., attractive local and repulsive non-local interactions. We find that non-competing (competing) non-local interactions give rise to a modulational instability emerging for a perturbing wavevector q=\pi (0 < q < \pi). The hopping instability instead arises for q=0 perturbations, thus the system is most sensitive to perturbations of the order of the system's size. Since for \alpha>1 and \beta>2 these effects are similar to the effect produced on the stability phase diagram by a finite-range interactions and/or hoppings, we conclude that the modulational instability is "genuinely" long-ranged for 1<\beta<2 non-local hoppings.
View original:
http://arxiv.org/abs/1212.0495
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