Nonlinear discrete Schrödinger equations with a point defect
Dirk Hennig
TL;DR
We address the DNLS on $\mathbb{Z}^d$ with a point defect and general power nonlinearity, studying how linear defect modes and nonlinear self-trapping interact. The authors develop a variational framework on a weighted Nehari manifold to prove the existence of spatially localized ground states (breathers) for focusing nonlinearity and derive explicit excitation thresholds, while also characterizing persistence and upper bounds of linear defect modes under defocusing nonlinearity. They show that, below a mass threshold, solutions scatter to solutions of the linear problem in $l^p$ with decay rates, and they identify a Weinstein-type threshold that governs the scattering vs non-scattering dichotomy. The results highlight how discreteness and defect strength shape localization, threshold phenomena, and long-time dynamics in lattice nonlinear waves with impurities, with potential implications for optical lattices and discrete photonic systems.
Abstract
We study the $d$-dimensional discrete nonlinear Schrödinger equation with general power nonlinearity and a delta potential. Our interest lies in the interplay between two localization mechanisms. On the one hand, the attractive (repulsive) delta potential acting as a point defect breaks the translational invariance of the lattice so that a linear staggering (non-staggering) bound state is formed with negative (positive) energy. On the other hand, focusing nonlinearity may lead to self-trapping of excitation energy. For focusing nonlinearity we prove the existence of a spatially exponentially localized and time-periodic ground state and investigate the impact of an attractive respectively repulsive delta potential on the existence of an excitation threshold, i.e. supercritical $l^2$ norm, for the creation of such a ground states. Explicit expressions for the lower excitation thresholds are given. Reciprocally, we discuss the influence of defocusing nonlinearity on the durability of the linear bound states and provide upper thresholds of the $l^2-$norm for their preservation. Regarding the asymptotic behavior of the solutions we establish that for a $l^2-$norm below the excitation threshold the solutions scatter to a solution of the linear problem in $l^{p>2}$.