Smirnov Valeri V, Manevitch Leonid I
Institute of Chemical Physics, RAS, 4 Kosygin Street, Moscow 119991, Russia.
Phys Rev E. 2017 Feb;95(2-1):022212. doi: 10.1103/PhysRevE.95.022212. Epub 2017 Feb 17.
We present an analytical description of the large-amplitude stationary oscillations of the finite discrete system of harmonically coupled pendulums without any restrictions on their amplitudes (excluding a vicinity of π). Although this model has numerous applications in different fields of physics, it was studied earlier in the infinite limit only. The discrete chain with a finite length can be considered as a well analytical analog of the coarse-grain models of flexible polymers in the molecular dynamics simulations. The developed approach allows to find the dispersion relations for arbitrary amplitudes of the nonlinear normal modes. We emphasize that the long-wavelength approximation, which is described by well-known sine-Gordon equation, leads to an inadequate zone structure for the amplitudes of about π/2 even if the chain is long enough. An extremely complex zone structure at the large amplitudes corresponds to multiple resonances between nonlinear normal modes even with strongly different wave numbers. Due to the complexity of the dispersion relations the modes with shorter wavelengths may have smaller frequencies. The stability of the nonlinear normal modes under condition of the resonant interaction are discussed. It is shown that this interaction of the modes in the vicinity of the long wavelength edge of the spectrum leads to the localization of the oscillations. The thresholds of instability and localization are determined explicitly. The numerical simulation of the dynamics of a finite-length chain is in a good agreement with obtained analytical predictions.
我们给出了一个有限离散系统的分析描述，该系统由谐波耦合摆组成，对其振幅没有任何限制（不包括接近π的区域）。尽管该模型在物理学的不同领域有许多应用，但之前仅在无限极限情况下进行过研究。在分子动力学模拟中，具有有限长度的离散链可被视为柔性聚合物粗粒化模型的良好解析类似物。所开发的方法能够找到非线性正常模式任意振幅下的色散关系。我们强调，由著名的正弦 - 戈登方程描述的长波近似，即使链足够长，对于约π/2的振幅也会导致不充分的能带结构。在大振幅下极其复杂的能带结构对应于即使波数差异很大的非线性正常模式之间的多重共振。由于色散关系的复杂性，波长较短的模式可能具有较小的频率。讨论了非线性正常模式在共振相互作用条件下的稳定性。结果表明，频谱长波边缘附近模式的这种相互作用会导致振荡的局域化。明确确定了不稳定性和局域化的阈值。有限长度链动力学的数值模拟与所获得的解析预测结果吻合良好。
