非线性动力格中的非近邻相互作用
Non-Nearest-Neighbor Interactions in Nonlinear Dynamical Lattices
摘要：
我们重新讨论了非线性动力学晶格中的非最近邻相互作用的主题，在离散非线性薛定谔方程的原型设置。我们的方法提供了一个系统的方法来分析在所谓的反连续极限的零耦合附近的系统的解决方案的存在性和稳定性。这为我们提供了许多分析见解，例如，次近邻相互作用允许无限一维晶格中具有非平凡相结构的解决方案;在纯最近邻相互作用的情况下，这种相结构是不允许的。另一方面，这种非最近邻相互作用会严重影响不稳定结构的稳定性，例如拓扑电荷S=2的离散涡。这些分析预测证实了数值分岔和稳定性计算。
英文摘要：
We revisit the theme of non-nearest-neighbor interactions in nonlinear dynamical lattices, in the prototypical setting of the discrete nonlinear Schrodinger equation. Our approach offers a systematic way of analyzing the existence and stability of solutions of the system near the so-called anti-continuum limit of zero coupling. This affords us a number of analytical insights such as the fact that, for instance, next-nearest-neighbor interactions allow for solutions with nontrivial phase structure in infinite one-dimensional lattices; in the case of purely nearest-neighbor interactions, such phase structure is disallowed. On the other hand, such non-nearest-neighbor interactions can critically affect the stability of unstable structures, such as topological charge S=2 discrete vortices. These analytical predictions are corroborated by numerical bifurcation and stability computations.
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