Compactification tuning for nonlinear localized modes in sawtooth lattices

TL;DR

This paper explores how nonlinear effects can induce and tune compact localized modes in sawtooth lattices, revealing conditions for their existence, stability, and controllability through nonlinearity and lattice parameters.

Contribution

It provides analytic conditions for compact mode existence in nonlinear sawtooth lattices and demonstrates how nonlinearity enables mode compactification across various parameters.

Findings

Nonlinear compact modes can be continued from linear flat band modes.

Nonlinearity allows compactification for a range of coupling ratios.

Stable localized modes are achievable over large parameter regimes.

Abstract

We discuss the properties of nonlinear localized modes in sawtooth lattices, in the framework of a discrete nonlinear Schr\"odinger model with general on-site nonlinearity. Analytic conditions for existence of exact compact three-site solutions are obtained, and explicitly illustrated for the cases of power-law (cubic) and saturable nonlinearities. These nonlinear compact modes appear as continuations of linear compact modes belonging to a flat dispersion band. While for the linear system a compact mode exists only for one specific ratio of the two different coupling constants, nonlinearity may lead to compactification of otherwise non-compact localized modes for a range of coupling ratios, at some specific power. For saturable lattices, the compactification power can be tuned by also varying the nonlinear parameter. Introducing different on-site energies and anisotropic couplings yield further possibilities for compactness tuning. The properties of strongly localized modes are investigated numerically for cubic and saturable nonlinearities, and in particular their stability over large parameter regimes is shown. Since the linear flat band is isolated, its compact modes may be continued into compact nonlinear modes both for focusing and defocusing nonlinearities. Results are discussed in relation to recent realizations of sawtooth photonic lattices.