Localization and global dynamics in the long-range discrete nonlinear Schr\"odinger equation
We study localization, pinning, and mobility in the fractional discrete nonlinear Schr\"odinger equation (fDNLS) with generalized power-law coupling. A finite-dimensional spatial-dynamics reduction of the nonlocal recurrence yields onsite and offsite stationary profiles; their asymptotic validity, orbital stability of onsite solutions, and $\ell^2$ proximity to the exact lattice solutions are established. Using the explicit construction of localized states, it is shown that the spatial tail behavior is algebraic for all $\alpha$ > 0. The Peierls-Nabarro barrier (PNB) is computed, and the parameter regimes are identified where it nearly vanishes; complementary numerical simulations explore mobility/pinning across parameters and exhibit scenarios consistent with near-vanishing PNB. We also analyze modulational instability of plane waves, locate instability thresholds, and discuss the role of nonlocality in initiating localization. Finally, we establish small-data scattering, and quantify how fDNLS dynamics approximates the nearest-neighbor DNLS on bounded times while exhibiting distinct global behavior for any large $\alpha$.