Nonlinear discrete Schr\"odinger equations with a point defect

TL;DR

This paper investigates the effects of point defects and nonlinearity on the localization and scattering of solutions in discrete nonlinear Schr"odinger equations across multiple dimensions.

Contribution

It provides new analytical results on the existence, thresholds, and asymptotic behavior of localized states influenced by delta potentials and nonlinearities.

Findings

Existence of exponentially localized, time-periodic ground states for focusing nonlinearity.

Explicit lower excitation thresholds for creating localized states.

Solutions below the excitation threshold scatter to linear solutions.

Abstract

We study the $d$-dimensional discrete nonlinear Schr\"odinger 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 $l^{p>2}$.