Nonlinear excitations in multi-dimensional nonlocal lattices

TL;DR

The paper addresses how multi-dimensional lattices with long-range, algebraically decaying interactions modify breather localization and energy dispersion. It develops a variational framework to characterize ground states and excitation thresholds, revealing a sharp mass-threshold dichotomy that depends on the dimension $d$, nonlinearity $p$, and tail exponent $\\alpha$, with explicit results in the anti-continuum limit. Ground states exhibit fully algebraic decay $|q_n| \lesssim \langle n\rangle^{-(d+\\alpha)}$ (and polynomially faster for $\\alpha=\\infty$), and the analysis extends to anisotropic, non-radial couplings; in 1D, the work links thresholds to dispersive decay rates governed by polylogarithmic structures, with numerical demonstrations of breather formation and algebraic radiation. Overall, the results show that long-range coupling fundamentally reshapes excitation thresholds and decay dynamics, offering guidance for controlling localization in photonic lattices and Bose–Einstein condensate networks with nonlocal interactions.

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 $α$) 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 $α$.

Figures (3)

• Figure 1: Peak intensity vs. time for varying $\alpha$ (left) and $\nu$ (right) with $\kappa = 0.1$. The left plot shows a transition from localization to decay as $\alpha$ increases. The right plot confirms that for $\nu < \nu_0$, solutions exhibit global dispersive decay, while for $\nu > \nu_0$, localized breather-like states persist, validating the threshold predictions of \ref{['p1']}.
• Figure 2: The evolution of a localized initial condition $u(0) = 5 \delta_0$ under $\kappa = 0.1,\ p=3$, and the Dirichlet boundary condition $u_{n} = 0,\ |n| > 100$. Left: $\log |u_n(t)|$ on $t \in [0,30]$ for $\alpha = 0.5$. Right: for $\alpha = 1.5$, the linear fit is $y = -2.499 x - 1.779$, and for $\alpha = 0.5$, $y = -1.501 x - 1.756$. The left plot is a visual illustration of the decomposition into a localized state and radiation. Note that the intensity is not stationary since $u(0)$ is not an exact solution to the nonlinear eigenvalue problem \ref{['main_model3']}. The decay of radiation, whose order is $d+\alpha$, is in excellent agreement with \ref{['algebraic_decay']}.
• Figure 3: Peak intensity versus time with $(\kappa,p) = (1,3)$. For $\alpha = 1.5$, the linear fit is $y = -0.30555 x - 0.67053$, and for $\alpha = 0.5$, $y = -0.45583 x - 0.96974$ computed in Matlab. The vanishing Dirichlet boundary condition, $u_{n} = 0,\ |n| > 100$ is used. As predicted in \ref{['p5']}, the rates of dispersive decay $\rho_{\pm}$ is illustrated by the slopes of the log-log plot. This numerically verifies \ref{['nonlinear_decay', 'scattering2']} where small data disperse over the lattice without self-collapse.

Theorems & Definitions (47)

• Definition 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.1
• proof
• Remark 2.1
• Remark 2.2
• Lemma 2.3