任意两端固定单原子链中原子大振幅振动非线性动力学方程的精确解
Exact solutions of nonlinear dynamical equations for large-amplitude atomic vibrations in arbitrary monoatomic chains with fixed ends
摘要：
一维非线性周期结构中的模间相互作用已经被许多作者研究，从上个世纪中叶Fermi，Pasta和Ulam（FPU）的经典工作开始。然而，不同对称性的非线性振动模式之间的能量转移的对称性选择规则，这导致了一些这样的模式的灌木丛的激发的可能性，没有被揭示。每个衬套确定所考虑的系统的非线性动力学方程的精确解。给定衬套的模态集合不随时间变化，而这些模态之间存在连续的能量交换。丛的非线性简正模（NNM）的构造与群论方法的帮助下，因此，它们可以存在的情况下，大振幅的原子振动和任何类型的原子间的相互作用。在大多数出版物中，在周期边界条件下研究一维系统中的NNM或类似动力学对象的丛。在本文中，我们提出了一个详细的研究在单原子链的固定边界条件的情况下，这揭示了一系列新的性质的模间相互作用在这样的系统中的NNMS的灌木丛。我们证明了一些定理，证明了一种方法来构建灌木丛的NNMs的延续传统的正常模式的情况下，大的原子振荡。在密度泛函理论的框架下，我们对FPU链、具有Lennard-Jones原子间相互作用势的链以及碳链（卡宾）进行了研究。对于一维套筒（Rosenberg非线性简正模），给出了幅频图，并简要讨论了其调制不稳定性的可能性。我们还主张赞成的事实，我们的方法和主要结果是有效的任何单原子链。
英文摘要：
Intermode interactions in one-dimensional nonlinear periodic structures have been studied by many authors, starting with the classical work by Fermi, Pasta, and Ulam (FPU) in the middle of the last century. However, symmetry selection rules for the energy transfer between nonlinear vibrational modes of different symmetry, which lead to the possibility of excitation of some bushes of such modes, were not revealed. Each bush determines an exact solution of nonlinear dynamical equations of the considered system. The collection of modes of a given bush does not change in time, while there is a continuous energy exchange between these modes. Bushes of nonlinear normal modes (NNMs) are constructed with the aid of group-theoretical methods and therefore they can exist for the case of large amplitude atomic vibrations and for any type of interatomic interactions. In most publications, bushes of NNMs or similar dynamical objects in one-dimensional systems are investigated under periodic boundary conditions. In this paper, we present a detailed study of the bushes of NNMs in monoatomic chains for the case of fixed boundary conditions, which sheds light on a series of new properties of the intermode interactions in such systems. We prove some theorems that justify a method for constructing bushes of NNMs by continuation of conventional normal modes to the case of large atomic oscillations. Our study was carried out for FPU chains, for the chains with the Lennard-Jones interatomic potential, as well as for the carbon chains (carbynes) in the framework of the density functional theory. For one-dimensional bushes (Rosenberg nonlinear normal modes), the amplitude-frequency diagrams are presented and the possibility of their modulational instability is briefly discussed. We also argue in favor of the fact that our methods and main results are valid for any monoatomic chain.
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