Solitons in one-dimensional nonlinear Schr\"{o}dinger lattices with a local inhomogeneity

TL;DR

This paper investigates how localized solutions in a one-dimensional discrete nonlinear Schrödinger system behave in the presence of a linear defect, revealing effects on stability, existence, and dynamics of solitons.

Contribution

It provides a comprehensive analysis of soliton existence, stability, and mobility in a lattice with a defect, highlighting new bifurcation phenomena and interaction regimes.

Findings

Destabilization of on-site mode at the defect in repulsive case

Localized modes vanish near the defect due to saddle-node bifurcations

Threshold for amplitude formation varies with defect type and strength

Abstract

In this paper we analyze the existence, stability, dynamical formation and mobility properties of localized solutions in a one-dimensional system described by the discrete nonlinear Schr\"{o}dinger equation with a linear point defect. We consider both attractive and repulsive defects in a focusing lattice. Among our main findings are: a) the destabilization of the on--site mode centered at the defect in the repulsive case; b) the disappearance of localized modes in the vicinity of the defect due to saddle-node bifurcations for sufficiently strong defects of either type; c) the decrease of the amplitude formation threshold for attractive and its increase for repulsive defects; and d) the detailed elucidation as a function of initial speed and defect strength of the different regimes (trapping, trapping and reflection, pure reflection and pure transmission) of interaction of a moving localized mode with the defect.