Pace Salvatore D, Campbell David K
Department of Physics, Boston University, Boston, Massachusetts 02215, USA.
Chaos. 2019 Feb;29(2):023132. doi: 10.1063/1.5079659.
We numerically investigate the existence and stability of higher-order recurrences (HoRs), including super-recurrences, super-super-recurrences, etc., in the α and β Fermi-Pasta-Ulam-Tsingou (FPUT) lattices for initial conditions in the fundamental normal mode. Our results represent a considerable extension of the pioneering work of Tuck and Menzel on super-recurrences. For fixed lattice sizes, we observe and study apparent singularities in the periods of these HoRs, speculated to be caused by nonlinear resonances. Interestingly, these singularities depend very sensitively on the initial energy and the respective nonlinear parameters. Furthermore, we compare the mechanisms by which the super-recurrences in the two models breakdown as the initial energy and respective nonlinear parameters are increased. The breakdown of super-recurrences in the β-FPUT lattice is associated with the destruction of the so-called metastable state and thus with relaxation towards equilibrium. For the α-FPUT lattice, we find this is not the case and show that the super-recurrences break down while the lattice is still metastable and far from equilibrium. We close with comments on the generality of our results for different lattice sizes.
我们对α和β费米-帕斯塔-乌拉姆-津古（FPUT）晶格中高阶递归（HoRs）的存在性和稳定性进行了数值研究，其中包括超递归、超超递归等，初始条件为基本简正模式。我们的结果极大地扩展了塔克和门泽尔关于超递归的开创性工作。对于固定的晶格尺寸，我们观察并研究了这些HoRs周期中明显的奇点，推测是由非线性共振引起的。有趣的是，这些奇点对初始能量和各自的非线性参数非常敏感。此外，我们比较了随着初始能量和各自的非线性参数增加，两个模型中超递归的崩溃机制。β-FPUT晶格中超递归的崩溃与所谓亚稳态的破坏相关，从而与向平衡态的弛豫相关。对于α-FPUT晶格，我们发现情况并非如此，并表明在晶格仍处于亚稳态且远离平衡态时，超递归就会崩溃。最后，我们对不同晶格尺寸下结果的普遍性进行了评论。
