非线性局域模的紧化调谐
Compactification tuning for nonlinear localized modes in sawtooth lattices
摘要：
在具有一般在位非线性的离散非线性薛定谔模型的框架下，讨论了非线性晶格中的非线性局域模的性质。获得了精确紧致三点解存在的解析条件，并针对幂律（三次）和饱和非线性的情况进行了明确说明。这些非线性紧模表现为属于平坦色散带的线性紧模的延续。虽然对于线性系统，仅对于两个不同的耦合常数的一个特定比率存在紧凑模式，但是非线性可能导致在某一特定功率下对于耦合比率范围的非紧凑局部模式的紧凑化。对于饱和格，紧化幂也可以通过改变非线性参数来调节。引入不同的现场能量和各向异性耦合为致密性调整提供了进一步的可能性。强本地化模式的性质进行了研究数值立方和饱和非线性，特别是它们的稳定性在大参数制度。由于线性平带是孤立的，它的紧凑模式可以继续到紧凑的非线性模式的聚焦和散焦的非线性。结果进行了讨论，在最近实现的光子晶格。
英文摘要：
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 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.
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