Discrete breathers assist energy transfer to ac-driven nonlinear chains

A one-dimensional chain of pointwise particles harmonically coupled with nearest neighbors and placed in sixth-order polynomial on-site potentials is considered. The power of the energy source in the form of single ac driven particle is calculated numerically for different amplitudes $A$ and frequencies $\omega$ within the linear phonon band. The results for the on-site potentials with hard and soft anharmonicity types are compared. For the hard-type anharmonicity, it is shown that when the driving frequency is close to (far from) the upper edge of the phonon band, the power of the energy source normalized to $A^2$ increases (decreases) with increasing $A$. In contrast, for the soft-type anharmonicity, the normalized power of the energy source increases (decreases) with increasing $A$ when the driving frequency is close to (far from) the lower edge of the phonon band. Our further demonstrations indicate that in the case of hard (soft) anharmonicity, the chain can support movable discrete breathers (DBs) with frequencies above (below) the phonon band. It is the energy source quasiperiodically emitting moving DBs in the regime with driving frequency close to the DB frequency that induces the increase of the power. Therefore, our results here support the mechanism that the moving DBs can assist energy transfer from the ac driven particle to the chain.

Figure 1

(a) Schematic plot of the 1D chain of harmonically coupled pointwise particles in the anharmonic on-site potential. (b) The on-site potential as a function of $\xi$ for hard-type (red line) and soft-type (blue line) potentials. The stars show the inflection points in the soft-type case.

Figure 2

Dispersion relation for small-amplitude waves (phonons) supported by the considered chain of particles. The phonon band ranges from ${\omega }_{min}=1$ to ${\omega }_{max}=\sqrt{5}\approx 2.236$.

Figure 3

Power $P$ of the energy source, normalized by $A^2$, as a function of driving frequency for two driving amplitudes, $A=0.005$ and 0.6. The exact result for the linear chain is shown by the thick dashed line. Vertical dashed lines show the edges of the phonon band with ${\omega }_{min}=1$ and ${\omega }_{max}=\sqrt{5}$.

Figure 4

The same result as that shown in Fig. 3 but for the case of soft anharmonicity.

Figure 5

Hard-type anharmonicity: Power of the energy source $p_j$ defined by Eq. (9) and normalized by $A^2$ as a function of the driving period number, $j=t_j/T$, for different driving amplitudes $A$, as indicated for each curve. Driving frequency is inside the phonon band and it is close to (a) the lower edge, $\omega ={\omega }_{min}+0.01$, and (b) the upper edge, $\omega ={\omega }_{max}-0.01$.

Figure 6

The same as Fig. 5, but for the case of soft-type anharmonicity.

Figure 7

Contour plots showing the space-time evolution of normalized energy of the particles, $e_n/A^2$. More intense colors correspond to higher energies, according to the color bar shown at the right of each panel. Particle located at $n=2000$ is driven. (a), (${\mathrm{a}}^{\prime }$), (b), (${\mathrm{b}}^{\prime }$) Hard-type anharmonicity. (c), (${\mathrm{c}}^{\prime }$), (d), (${\mathrm{d}}^{\prime }$) Soft-type anharmonicity. (a), (${\mathrm{a}}^{\prime }$), (c), (${\mathrm{c}}^{\prime }$) Driving frequency is close to the lower edge of the phonon band. (b), (${\mathrm{b}}^{\prime }$), (d), (${\mathrm{d}}^{\prime }$) Driving frequency is close to the upper edge of the phonon band. Left (right) panels correspond to the driving amplitude $A=0.2$ ($A=0.6$).

Figure 8

Normalized energy of the particles at the end of the numerical run at $t=t_{max}=2000$. Particle $n=2000$ is driven. Half of the picture is shown due to the mirror symmetry with respect to the driven site. (a), (b) Hard-type anharmonicity. (c), (d) Soft-type anharmonicity. (a), (c) Driving frequency is close to the lower edge of the phonon band. (b), (d) Driving frequency is close to the upper edge of the phonon band. Blue solid lines show the results of relatively small driving amplitude $A=0.2$, while black dashed lines correspond to the result of $A=0.6$, when nonlinearity comes into play. Trains of DBs moving away from the energy source can be seen in (b), (c), when the driving frequency is close to the DB frequency and the driving amplitude is sufficiently large.

Figure 9

Vibration amplitude of $n=2050$ particle as a function of time (driven particle is $n=2000$). (a), (b) Hard-type anharmonicity. (c), (d) Soft-type anharmonicity. (a), (c) Driving frequency is close to the lower edge of the phonon band. (b), (d) Driving frequency is close to the upper edge of the phonon band. Blue solid (black dashed) lines show the results for $A=0.2$ ($A=0.6$).

Figure 10

The associated frequencies of the vibrations shown in Fig. 9 as a function of the driving amplitude $A$. Here we use the open circles to represent the estimated frequencies within the phonon band, while adopting the crosses and dots to denote those that can appear outside the phonon band. Therefore, in (a) [(b)] the open circles (the crosses and dots) are for the soft- (hard-) type anharmonicity.

Figure 11

Standing DB profiles for two different amplitudes $A_{\mathrm{DB}}=0.5$ and $A_{\mathrm{DB}}=1.5$ for the case of (a) hard-type and (b) soft-type anharmonicities.

Figure 12

Moving DB profile in the case of hard-type anharmonicity. Parameters of the ansatz Eq. (22) are $A_{\mathrm{DB}}=1.5, \theta =0.398, {\omega }_{\mathrm{DB}}=2.270, x_0=2000, \delta =0.3$. DB velocity $v_{\mathrm{DB}}=0.1303$. Time is indicated in each panel.

Figure 13

Moving DB profile in the case of soft-type anharmonicity. Parameters of the ansatz Eq. (23) are $A_{\mathrm{DB}}=1.5, \theta =0.361, {\omega }_{\mathrm{DB}}=0.932, x_0=2000, \delta =0.3$. DB velocity $v_{\mathrm{DB}}=0.2838$. Time is indicated in each panel.

Figure 14

Velocity $v_{\mathrm{DB}}$ of DBs as a function of $\delta$ for (a) hard-type and (b) soft-type anharmonicities. DB amplitude is indicated in the legends. Other DB parameters are taken from Table 1.

Figure 15

$\theta$ and $A_{\mathrm{DB}}$ as functions of ${\omega }_{\mathrm{DB}}$ for standing DB for hard-type [(a), (b)] and soft-type [(c), (d)] anharmonicities, where the dots give the numerical estimates shown in Table 1, the solid lines correspond to the solutions of the sine-Gordon breather, i.e., Eqs. (28) and (29), and the dashed lines are the solutions from Eqs. (31) and (32).