Modulational instability and localized breather modes in the discrete nonlinear Schr\"odinger equation with helicoidal hopping

TL;DR

This paper investigates a discrete nonlinear Schrödinger model with helicoidal hopping, analyzing modulational instability, localized breather modes, and stability thresholds, revealing a dimensional crossover to two-dimensional behavior in certain limits.

Contribution

It introduces a detailed analysis of modulational instability and breather modes in a helicoidal hopping model, including analytical estimates and the discovery of a dimensional crossover.

Findings

Identification of multi-stage instability cascades due to helicoidal hopping

Bistability of localized breather modes and stability insights from plane-wave analysis

Prediction of a dimensional crossover to two-dimensional lattice behavior for large N

Abstract

We study a one-dimensional discrete nonlinear Schr\"odinger model with hopping to the first and a selected N-th neighbor, motivated by a helicoidal arrangement of lattice sites. We provide a detailed analysis of the modulational instability properties of this equation, identifying distinctive multi-stage instability cascades due to the helicoidal hopping term. Bistability is a characteristic feature of the intrinsically localized breather modes, and it is shown that information on the stability properties of weakly localized solutions can be inferred from the plane-wave modulational instability results. Based on this argument, we derive analytical estimates of the critical parameters at which the fundamental on-site breather branch of solutions turns unstable. In the limit of large N, these estimates predict the emergence of an effective threshold behavior, which can be viewed as the result of a dimensional crossover to a two-dimensional square lattice.