Nonlinear excitations in multi-dimensional nonlocal lattices

Abstract

We study the formation of breathers in multi-dimensional lattices with long-range interactions. By variational methods, the exact relationship between various parameters (dimension, nonlinearity, nonlocal parameter $\alpha$) that defines positive excitation thresholds is characterized. We establish a sharp mass-threshold dichotomy: no positive threshold in the mass-subcritical regime, and a strictly positive threshold at and above the critical regime. In the anti-continuum regime, a family of unique ground states characterizes the excitation thresholds, enabling explicit computations. Analytic formulas of the excitation thresholds, determined by the ground states, are derived and corroborated with numerical simulations. We not only characterize the sharp spatial decay of ground states, which varies continuously in $\alpha$, but also identify the time decay of dispersive waves, which undergoes a discontinuous transition in $\alpha$.