Energy localization on $q$-tori, long-term stability, and the interpretation of Fermi-Pasta-Ulam recurrences

We focus on two approaches that have been proposed in recent years for the explanation of the so-called Fermi-Pasta-Ulam (FPU) paradox, i.e., the persistence of energy localization in the “low-$q$” Fourier modes of Fermi-Pasta-Ulam nonlinear lattices, preventing equipartition among all modes at low energies. In the first approach, a low-frequency fraction of the spectrum is initially excited leading to the formation of “natural packets” exhibiting exponential stability, while in the second, emphasis is placed on the existence of “$q$ breathers,” i.e., periodic continuations of the linear modes of the lattice, which are exponentially localized in Fourier space. Following ideas of the latter, we introduce in this paper the concept of “$q$-tori” representing exponentially localized solutions on low-dimensional tori and use their stability properties to reconcile these two approaches and provide a more complete explanation of the FPU paradox.

Figure 1

Comparison of numerical (points) versus analytical (solid line) solutions, using the Poincaré-Lindstedt series up to order $O\left({\sigma }^2\right)$, for the temporal evolution of the modes (a) $q=1$, (b) $q=3$, and (c) $q=7$, when $A_1=1$, $A_2=0.5$ and $N=8$, $\beta =0.1$ and (d) the time evolution of the GALI indices $G_2–G_6$ up to a time $t={10}^6$ show that the motion lies on two-dimensional torus.

Figure 2

Same as in Fig. 1, showing the existence of a four-torus in the system with $N=16$, $\beta =0.1$, when $A_1=1$, $A_2=0.5$, $A_3=0.333\dots$, and $A_4=0.25$. The temporal evolution $Q_q\left(t\right)$ is shown for the modes (a) $q=1$, (b) $q=5$, and (c) $q=13$. (d) The GALI indices $G_k$ (with $k=2,3,4$) are seen to stabilize after $t\sim {10}^4$, while for $k\ge 5$ the indices continue to decrease by power laws.

Figure 3

(Color online) The average harmonic energy $E_q$ of the $q\text{th}$ mode as a function of $q$, after a time $T={10}^6$ for the (a) two-torus and (b) four-torus solutions (open circles) corresponding to the initial conditions used in Figs. 1, 2, respectively. The stars are $E_q$ values calculated via the analytical representation of the solutions $Q_q\left(t\right)$ by the Poincaré-Lindstedt series. The filled circles show a theoretical estimate based on the average energy of suitably defined groups of modes [see Eq. (35) and relevant discussion in the text].

Figure 4

(Color online) The average harmonic energy $E_q$ of the $q\text{th}$ mode over a time span $T={10}^6$ as a function of $q$ in various examples of FPU trajectories, for $\beta =0.3$, in which the $s\left(=N/16\right)$ first modes are only excited initially via $Q_q\left(0\right)=A_q$, ${\dot{Q}}_q\left(0\right)=0$, and $q=1,\dots ,s$, with $A_q$ selected so that the total energy is equal to the value $E=H$ indicated in each panel. We thus have (a) $N=64$, $E={10}^{-4}$; (b) $N=128$, $E=2\times {10}^{-4}$; (c) $N=256$, $E=4\times {10}^{-4}$; (d) $N=64$, $E={10}^{-3}$; (e) $N=128$, $E=2\times {10}^{-3}$; and (f) $N=256$, $E=4\times {10}^{-3}$. The specific energy is constant in each of the two rows, i.e., $\epsilon =1.5625\times {10}^{-6}$ in the top row and $\epsilon =1.5625\times {10}^{-5}$ in the bottom row. The dashed lines represent the average exponential profile $E_q$ obtained theoretically by the hypothesis that the depicted FPU trajectories lie close to $q$-tori governed by profile (35).

Figure 5

(Color online) Same as in Fig. 4a but for larger energies, namely, (a) $E=0.05$, (b) $E=0.1$, (c) $E=0.2$, (d) $E=0.3$, (e) $E=0.4$, and (f) $E=0.5$. Beyond the threshold $E\simeq 0.05$, theoretical profiles of form (33) yield the correct exponential slope if $s$ is gradually increased from $s=4$ in (a) and (b) to $s=6$ in (c), $s=7$ in (d), $s=10$ in (e), and $s=12$ in (f).

Figure 6

(Color online) Assuming that the critical energy $E_c$, at which the GALI index $G_2$ shows that a $q$-torus destabilizes is fitted by the law $E_c=A{\beta }^{-1}$, we show in the upper curve (triangles) the dependence of $A$ on $N$ for an FPU trajectory started by exciting initially only the $q=1,2$ modes. The middle curve (filled circles) corresponds to a similar calculation for FPU trajectories started near a $q$-breather solution, where only the $q=1$ mode is excited. The dashed line corresponds to $A\sim N^{-1}$, according to the law of Eq. (36).

Figure 7

(Color online) (a) Time evolution of the GALI index $G_2$ up to $t={10}^7$ for an FPU trajectory started by exciting the $q=1$ and 2 modes in the system $N=32$, $\beta =0.1$, with a total energy $E=2$, i.e., higher than $E_c=1.6$. (b) Instantaneous localization profile of the FPU trajectory of (a) at $t={10}^7$. (c) Same as in (a) but for $N=128$, $E=1.323$ (in this case the critical energy is $E_c=1.24$). (d) Same as in (b) but for the trajectory of (c).