We investigate a higher-order topological insulator (HOTI) under strong nonlinearity, focusing on the existence and stability of high-amplitude corner states, which can find applications in optics, acoustics, elastodynamics, and other wave-based systems. Our study centers on a breathing kagome lattice composed of point masses and springs, known to exhibit edge and corner states in its linear regime. By introducing on-site cubic nonlinearity, we analyze its impact on both edge and corner states. The nonlinear continuation of the corner state unveils stable high-amplitude corner states within the lattice, featuring nonzero displacements at even sites from the corner—a characteristic absent in the linear limit. Interestingly, the nonlinear continuation of the edge state reveals its transformation into distinct families of high-amplitude corner states via two pitchfork bifurcations. While some states maintain stability, others become unstable through real instability and Neimark-Sacker bifurcation. These unstable corner states dissipate their energy into the edges and the bulk over an extended period, as corroborated by long-time dynamical simulations. Consequently, our study provides insights into achieving significant energy localization at the corners of HOTIs through various classes of nonlinear states.

中文翻译：

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拓扑非平凡 kagome 晶格中的非线性角态


我们研究强非线性下的高阶拓扑绝缘体（HOTI），重点关注高振幅角态的存在和稳定性，这可以在光学、声学、弹性动力学和其他基于波的系统中找到应用。我们的研究集中在由点质量和弹簧组成的呼吸戈薇晶格上，已知其在线性状态下表现出边缘和角状态。通过引入现场三次非线性，我们分析了它对边缘和角状态的影响。角态的非线性延续揭示了晶格内稳定的高振幅角态，其特征是在角点的偶数位置处具有非零位移——这是线性极限中不存在的特征。有趣的是，边缘态的非线性延续揭示了它通过两个干草叉分叉转变为不同的高振幅角态族。虽然一些国家保持稳定，但其他国家却因真正的不稳定和内马克-萨克尔分歧而变得不稳定。这些不稳定的角态会在很长一段时间内将能量耗散到边缘和主体中，这一点已被长时间的动态模拟所证实。因此，我们的研究提供了通过各种非线性状态在 HOTI 角落实现显着能量局部化的见解。