Internal modes of discrete solitons near the anti-continuum limit of the dNLS equation

TL;DR

This paper investigates the spectral stability of discrete solitons in the dNLS equation near the anti-continuum limit, showing that internal modes do not exist under certain conditions, thus clarifying their stability properties.

Contribution

The study proves the absence of internal modes for simply connected discrete solitons near the anti-continuum limit by analyzing the resolvent operator's boundedness.

Findings

Resolvent operator is uniformly bounded near the continuous spectrum.

Internal modes are ruled out for simply connected discrete solitons.

Provides spectral stability insights near the anti-continuum limit.

Abstract

Discrete solitons of the discrete nonlinear Schr\"odinger (dNLS) equation become compactly supported in the anti-continuum limit of the zero coupling between lattice sites. Eigenvalues of the linearization of the dNLS equation at the discrete soliton determine its spectral and linearized stability. All unstable eigenvalues of the discrete solitons near the anti-continuum limit were characterized earlier for this model. Here we analyze the resolvent operator and prove that it is uniformly bounded in the neighborhood of the continuous spectrum if the discrete soliton is simply connected in the anti-continuum limit. This result rules out existence of internal modes (neutrally stable eigenvalues of the discrete spectrum) of such discrete solitons near the anti-continuum limit.