Karve Nachiket, Rose Nathan, Campbell David
Department of Physics, Boston University, Boston, Massachusetts 02215, USA.
Chaos. 2024 Sep 1;34(9). doi: 10.1063/5.0223767.
The Fermi-Pasta-Ulam-Tsingou (FPUT) paradox is the phenomenon whereby a one-dimensional chain of oscillators with nonlinear couplings shows long-lived nonergodic behavior prior to thermalization. The trajectory of the system in phase space, with a long-wavelength initial condition, closely follows that of the Toda model over short times, as both systems seem to relax quickly to a non-thermal, metastable state. Over longer times, resonances in the FPUT spectrum drive the system toward equilibrium, away from the Toda trajectory. Similar resonances are observed in q-breather spectra, suggesting that q-breathers are involved in the route toward thermalization. In this article, we first review previous important results related to the metastable state, solitons, and q-breathers. We then investigate orbit bifurcations of q-breathers and show that they occur due to resonances, where the q-breather frequencies become commensurate as mΩ1=Ωk. The resonances appear as peaks in the breather energy spectrum. Furthermore, they give rise to new "composite periodic orbits," which are nonlinear combinations of multiple q-breathers that exist following orbit bifurcations. We find that such resonances are absent in integrable systems, as a consequence of the (extensive number of) conservation laws associated with integrability.
费米-帕斯塔-乌拉姆-津戈（FPUT）悖论是一种现象，即具有非线性耦合的一维振子链在热化之前呈现出长寿命的非遍历行为。在相空间中，具有长波长初始条件的系统轨迹在短时间内与托达模型的轨迹紧密跟随，因为这两个系统似乎都能迅速弛豫到一个非热的亚稳态。在更长的时间里，FPUT谱中的共振会驱使系统趋向平衡，偏离托达轨迹。在q呼吸子谱中也观察到了类似的共振，这表明q呼吸子参与了热化过程。在本文中，我们首先回顾与亚稳态、孤子和q呼吸子相关的先前重要结果。然后，我们研究q呼吸子的轨道分岔，并表明它们是由共振引起的，其中当mΩ1 = Ωk时，q呼吸子频率变得可公度。这些共振在呼吸子能谱中表现为峰值。此外，它们会产生新的“复合周期轨道”，这些轨道是轨道分岔后存在的多个q呼吸子的非线性组合。我们发现，由于与可积性相关的（大量）守恒定律，可积系统中不存在此类共振。
