Abstract
We discuss the properties of nonlinear localized modes in sawtooth lattices, in the framework of a discrete nonlinear Schrödinger 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 noncompact 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 yields 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.
Highlights
The creation and manipulation of nonlinear localized lattice excitations (“discrete solitons” or “discrete breathers”) is of interest in many areas of physics [1], and in particular within nonlinear optics [2]
In this work we showed the existence of compact modes for generic realizations of the sawtooth lattice, considering power law and saturable nonlinearities
These modes are compact nonlinear continuations of the linear flat-band modes found in the sawtooth model
Summary
The creation and manipulation of nonlinear localized lattice excitations (“discrete solitons” or “discrete breathers”) is of interest in many areas of physics [1], and in particular within nonlinear optics [2]. It is well known that in certain lattices geometries, frustration effects make possible the existence of strictly compact modes even in the absence of any nonlinearity Such modes can be seen to result from destructive interference of couplings between different sites, and are typically associ-. Of coupling constants of these lattices on polarization [22] provides ample and tunable opportunities beyond geometric considerations This is important since, in contrast to the kagome lattice, compactification in a linear sawtooth lattice appears only at one specific (irrational) ratio between the two different coupling constants (horizontal and diagonal), which again poses an experimental challenge for fine-tuning. III, illustrating explicitly how certain, linearly stable, localized modes compactify when the appropriate relation between coupling ratios and power is fulfilled
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