Nonlinear excitations in multi-dimensional nonlocal lattices

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 $α$) 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 $α$, but also identify the time decay of dispersive waves, which undergoes a discontinuous transition in $α$.

💡 Research Summary

The paper investigates the formation of discrete breathers—spatially localized, temporally periodic excitations—in multi‑dimensional lattices with long‑range (nonlocal) interactions. The authors consider a generalized discrete nonlinear Schrödinger (DNLS) model

 i ∂ₜ uₙ = κ L uₙ − |uₙ|^{p‑1}uₙ, n ∈ ℤᵈ,

where the linear operator L is defined by a symmetric, summable coupling kernel Jₙ that decays algebraically as |n|^{-(d+α)} for a nonlocal exponent α ∈ (0,∞]. The case α = ∞ corresponds to the usual nearest‑neighbour (NNI) DNLS. The main goal is to determine for which combinations of spatial dimension d, nonlinearity exponent p, and nonlocality α a positive excitation threshold ν₀ exists, i.e. a minimal ℓ²‑mass that must be supplied to generate a non‑trivial breather that persists for all time.

The analysis is variational. The authors introduce the constrained minimisation problem

 I_ν = inf{ H