Discrete breathers in a triangular $\beta$-Fermi-Pasta-Ulam-Tsingou lattice

A practical approach to the search for (quasi-) discrete breathers (DBs) in a triangular $\beta$-FPUT lattice (after Fermi, Pasta, Ulam, and Tsingou) is proposed. DBs are obtained by superimposing localizing functions on delocalized nonlinear vibrational modes (DNVMs) having frequencies above the phonon spectrum of the lattice. Zero-dimensional and one-dimensional DBs are obtained. The former ones are localized in both spatial dimensions, and the latter ones are only in one dimension. Among the one-dimensional DBs, two families are considered: the first is based on the DNVMs of a triangular lattice, and the second is based on the DNVMs of a chain. We speculate that our systematic approach on the triangular $\beta$-FPUT lattice reveals all possible types of spatially localized oscillations with frequencies bifurcating from the upper edge of the phonon band as all DNVMs with frequencies above the phonon band are analyzed.

Figure 1

Computational cell for the triangular lattice with the lattice spacing $h$ including $N_x\times N_y$ particles numbered by indices $i=1,2,...,N_x$ and $j=1,2,...,N_y$. Particles in their equilibrium positions are shown by yellow circles, and red lines show the primitive translational cells. The particles moving in the $x\text{−}y$ plane have two degrees of freedom, the components of the displacement vector $(u_{i,j},v_{i,j})$. Each particle interacts with its six nearest neighbors via the $\beta$-FPUT potential (3). The bonds are shown only for one translational cell of the lattice; they are designated as $l_1$, $l_2$, and $l_3$.

Figure 2

Three DNVMs of the triangular lattice considered in this study: (a) one-component DNVM 2, (b) one-component DNVM 4, and (c) two-component time-periodic DNVM $\gamma 2$ [94]. The distance between nearest particles is $h$. Particles in the equilibrium positions are shown by the yellow circles. Red lines show the primitive translational cells of the lattice. Translational cells of DNVMs include four particles and are shown by the blue lines. The bonds are shown only for one translational cell of the DNVM; they are designated as $l_k$. The black dots show the initial particle displacements used to excite the DNVMs. In (a) and (b), all displacement vectors have a length $A$, while in (c), horizontal and vertical displacements have magnitudes $A$ and $B$, respectively. In (a) and (c), all particles are displaced at $t=0$; in (b), one of the four particles in the translational cell is not displaced initially and remains at rest at any $t$. Each translational cell includes 12 bonds shown by the solid black lines. The bonds have lengths $l_i$.

Figure 3

Dependence of $p$ on $A$ in Eq. (31) when the two-component DNVM $\gamma 2$ becomes time periodic. The inset shows the coefficients of the polynomial approximation by the method of least squares. The horizontal dashed line shows the synchronized conditions with $p=\sqrt{3}$ for small $A$.

Figure 4

(a) Frequency $\omega$ and (b) energy per particle $E$ as functions of amplitudes of the DNVM 2 (black solid dots), DNVM 4 (red open dots), and DNVM $\gamma 2$ (blue squares). The horizontal dashed line in (a) indicates the upper edge of the phonon band with ${\omega }_{max}=\sqrt{6}\approx 2.449$ [94].

Figure 5

(a) DB (shown by the red cross) based on the DNVM 2 centered between two particles in a horizontal close-packed row. Particle trajectories are shown in black. (b) Vertical displacement as a function of time for the particle indicated in (a) by the arrow. The DB was excited with the use of the localizing function (32) with $A=0.1$, ${\beta }_x=0.8$, and ${\beta }_y=0.7$. Displacements of the particles are scaled by a factor of 3.

Figure 6

DBs obtained by applying the localizing function (33) upon DNVM 4 with the amplitude $A=0.1$. Depending on the location of the center of the localizing function ${\mathbf{r}}_0$ shown by the red crosses, (a) six-, (b) three-, or (c) twofold symmetry DBs are obtained. Stable long-lived oscillations are observed for (a) $\beta =0.830$, (b) $\beta =0.730$, and (c) $\beta =0.838$ (Fig. 7). Particle trajectories are shown in black. Displacements of the particles are scaled by a factor of 3.

Figure 7

(a–c) Displacements along the $y$-axis of particles marked by the arrows in Figs. 6–6, respectively.

Figure 8

DBs based on the DNVM $\gamma 2$. DBs were obtained by applying the localizing function (32) upon DNVM $\gamma 2$ with the amplitude $A=0.2$. The center of the localizing function is shown by the red crosses. In (a) it is between particles vibrating horizontally and in (b) on a particle vibrating horizontally. Stable long-lived oscillations in both cases are observed for ${\beta }_x=1.35$ and ${\beta }_y=1.10$. Particle trajectories are shown in black. Particle displacements are multiplied by a factor of 1.5.

Figure 9

(a, b) Displacements along the $x$-axis of particles marked by the arrows in Figs. 8 and 8, respectively.

Figure 10

One-dimensional horizontal DBs based on (a) DNVM 2 and (b) DNVM $\gamma 2$, obtained by applying the localizing function (34) upon DNVMs with the amplitude $A=0.1$. Long-lived oscillations are observed for (a) ${\beta }_x=0.65$ and (b) ${\beta }_x=0.79$. Particle trajectories are shown in black. Displacements of the particles are magnified by a factor of 3.

Figure 11

(a, b) Displacements of particles indicated by the arrows in Figs. 10 and 10, respectively. (a) Along the $y$-axis and (b) along the $x$-axis as functions of time

Figure 12

One-dimensional vertical DBs based on (a) DNVM 2, (b) DNVM 4, and (c) DNVM $\gamma 2$, obtained by applying the localizing function (35) upon DNVMs with the amplitude $A=0.1$. Long-lived oscillations are observed for (a) ${\beta }_x=1.15$, (b) ${\beta }_x=1.07$, and (c) ${\beta }_x=0.687$. Particle trajectories are shown in black. Displacements of the particles are magnified by a factor of 3.

Figure 13

(a)-(c) Displacements as functions of time for particles indicated by the arrows in Figs. 12–12, respectively. The displacements in (a) and (b) are along the $y$-axis and the displacements in (c) are along the $x$-axis.

Figure 14

One-dimensional DBs based on DNVMs of a chain, which have periodic patterns (a) $[A,-A]$ and (b) $[A,0,-A,0]$. The DBs are excited by introducing initial displacements to the particles in a close-packed row according to these patterns. Spatially localized oscillations are observed with (a) $A=0.2$ and (b) $A=0.8$. Particle trajectories are shown in black. The particle displacements in (b) are scaled by a factor of 0.5 to prevent trajectory overlapping of particles vibrating with large amplitudes.

Figure 15

(a), (b) Displacements $\mathrm{\Delta }x$ as functions of time for particles indicated by the arrows in Figs. 14 and 14, respectively.