Nonlinear excitations in multi-dimensional nonlocal lattices

TL;DR

This paper investigates the formation and thresholds of breathers in multi-dimensional nonlocal lattices, deriving explicit formulas and analyzing how parameters like dimension and nonlocality influence excitations.

Contribution

It provides a comprehensive analysis of excitation thresholds in nonlocal lattices, including explicit formulas and the impact of parameters like dimension and nonlocality.

Findings

Sharp mass-threshold dichotomy established

Explicit formulas for excitation thresholds derived

Ground state decay properties characterized

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 $\alpha$.