Phononic rogue waves

We present a theoretical study of extreme events occurring in phononic lattices. In particular, we focus on the formation of rogue or freak waves, which are characterized by their localization in both spatial and temporal domains. We consider two examples. The first one is the prototypical nonlinear mass-spring system in the form of a homogeneous Fermi-Pasta-Ulam-Tsingou (FPUT) lattice with a polynomial potential. By deriving an approximation based on the nonlinear Schrödinger (NLS) equation, we are able to initialize the FPUT model using a suitably transformed Peregrine soliton solution of the NLS equation, obtaining dynamics that resembles a rogue wave on the FPUT lattice. We also show that Gaussian initial data can lead to dynamics featuring a rogue wave for sufficiently wide Gaussians. The second example is a diatomic granular crystal exhibiting rogue-wave-like dynamics, which we also obtain through an NLS reduction and numerical simulations. The granular crystal (a chain of particles that interact elastically) is a widely studied system that lends itself to experimental studies. This study serves to illustrate the potential of such dynamical lattices towards the experimental observation of acoustic rogue waves.

Figure 1

(a) Dispersion relation for the monomer FPUT lattice with $K_2=1$, which consists of an acoustic branch only. (b) The spatial profile of the Peregrine soliton at $T=0$ and the temporal profile at $X=0$. (c) Dispersion relation for the dimer FPUT lattice with $K_2=3/2$ and $\rho =0.8$, which consists of an acoustic branch (lower branch) and optical branch (upper branch).

Figure 2

(a) Simulation of Eq. (1) with $\varepsilon =0.02,X\in [-40,40]$ and $T\in [-5,5]$ that is initialized with Eq. (7) and $P_0=1$. Color intensity corresponds to the strain $|y_n\left(t\right)|$. Notice that the space-time evolution here and in the figures that follow is given in terms of the rescaled variables $\varepsilon n$ and ${\varepsilon }^{2}t$ for space and time, respectively. (b) Corresponding NLS prediction $2\varepsilon \left|A\right(X,T\left)\right|$. Note that the background amplitude of $|y_n\left(t_0\right)|$ is the same as $2\varepsilon \left|A\right(X,T_0\left)\right|$.

Figure 3

Simulation of Eq. (1) that is initialized with Eq. (3) with $A$ given by Eq. (8) with the following width parameters $\sigma =20.1$ for panels (a, e, i), $\sigma =10.5$ for panels (b, f, j), $\sigma =2.5$ for panels (c, g, k) and $\sigma =1.3$ for panels (d, h, l). In panels (a)–(d) the perturbation parameter value is $\varepsilon =0.1$. In panels (e)–(h) the perturbation parameter value is $\varepsilon =0.05$. Panels (i)–(l) correspond to the respective numerical simulations of the NLS equation with Gaussian initial data; see, e.g., Ref. [19].

Figure 4

Simulation of Eq. (9) that is initialized with Eq. (17) and Eq. (18) with $A$ given by Eq. (20) with $P_0=1,\varepsilon =0.05, X\in [-40,40]$, and $T\in [-5,5]$. Color intensity corresponds to the strain $|y_n|$. The values of the mass ratio parameter are (a) $\rho =0.1$, (b) $\rho =0.2$, (c) $\rho =0.3$, (d) $\rho =0.4$, (e) $\rho =0.5$, (f) $\rho =0.6$, (g) $\rho =0.7$, and (h) $\rho =0.9$.

Figure 5

Simulation of Eq. (9) that is initialized with Eq. (17) and Eq. (18) with $A$ given by Eq. (21) with the following parameters: The top row has $\rho =0.6$ and $\sigma =10.5$ (a), $\sigma =2.5$ (b), and $\sigma =1.3$ (c). The middle row has $\rho =0.4$ and $\sigma =10.5$ (d), $\sigma =2.5$ (e), and $\sigma =1.3$ (f). The bottom row has $\rho =0.2$ and $\sigma =10.5$ (g), $\sigma =2.5$ (h), and $\sigma =1.3$ (i).

Figure 6

Spatial distribution of velocity profiles is shown at (a) and (d) $t=0.01\mathrm{s}$, (b) $t=0.104\mathrm{s}$, (c) and (f) $t=0.2\mathrm{s}$, and (e) $t=0.106\mathrm{s}$. (a) Velocity of solution after exciting an at rest dimer chain at the boundaries at a frequency at the bottom of the lower optical band. (b) After the precompression is decreased by a factor of 7, the system behaves nonlinearly, causing the onset of MI. This results in the spatial localization of the solution. (c) The localization is not persistent. (d–f) Same as panels (a)–(c) but in a damped system with $\tau =70\mathrm{ms}$.

Figure 7

(a) Space-time contour plot corresponding to panels (a)–(c) of Fig. 6, which shows the onset of MI. (b) Space-time contour plot corresponding to panels (d)–(f) of Fig. 6.