Discrete solitons in electromechanical resonators

We consider a particular type of parametrically driven discrete Klein-Gordon system describing microdevices and nanodevices, with integrated electrical and mechanical functionality. Using a multiscale expansion method we reduce the system to a discrete nonlinear Schrödinger equation. Analytical and numerical calculations are performed to determine the existence and stability of fundamental bright and dark discrete solitons admitted by the Klein-Gordon system through the discrete Schrödinger equation. We show that a parametric driving can not only destabilize onsite bright solitons, but also stabilize intersite bright discrete solitons and onsite and intersite dark solitons. Most importantly, we show that there is a range of values of the driving coefficient for which dark solitons are stable, for any value of the coupling constant, i.e., oscillatory instabilities are totally suppressed. Stability windows of all the fundamental solitons are presented and approximations to the onset of instability are derived using perturbation theory, with accompanying numerical results. Numerical integrations of the Klein-Gordon equation are performed, confirming the relevance of our analysis.

Figure 1

(Color online) A sketch of the dynamics of the eigenvalues and the continuous spectrum of a stable onsite bright soliton in the $\mathbf{\left(}\text{Re}\left(\Omega \right),\text{Im}\left(\Omega \right)\mathbf{\right)}$ plane. The arrows indicate the direction of movement as the coupling constant $C$ increases. Note that a soliton is unstable if there is some $\Omega$ with either $\Omega <0$ or $\text{Im}\left(\Omega \right)\ne 0$.

Figure 2

(Color online) As Fig. 1, but for a stable intersite bright soliton.

Figure 3

(Color online) The (in)stability region of onsite bright solitons in $\left(C,\gamma \right)$ space. For each value of $C$ and $\gamma$, the corresponding color indicates the maximum value of $\left|\text{Im}\left(\omega \right)\right|$ (over all eigenvalues $\omega$) for the steady-state solution at that point. Stability is therefore indicated by the region in which $\text{Im}\left(\omega \right)=0$, namely, the dark region. White dashed and dashed-dotted lines give the analytical approximations (20, 21), respectively.

Figure 4

Comparison between the critical eigenvalues of intersite bright solitons obtained numerically (solid lines) and their analytical approximation (dashed lines). The upper and lower curves correspond, respectively, to $\gamma =0.5$ and 0.1, approximated by Eq. (24), whereas the middle one corresponds to $\gamma =0.18$, approximated by Eq. (23).

Figure 5

The structure of the eigenvalues of intersite bright solitons in the complex plane for three values of $\gamma$, as indicated in the caption of each plot. Left and right panels depict the eigenvalues of stable and unstable solitons, respectively.

Figure 6

(Color online) As Fig. 3, but for intersite bright solitons. Our analytical approximations, given by Eqs. (25, 26, 27), are shown as white dashed-dotted, dotted, and dashed lines, respectively.

Figure 7

(Color online) As Fig. 1, but for a stable onsite dark soliton.

Figure 8

(Color online) As Fig. 1, but for a stable intersite dark soliton.

Figure 9

Comparisons between the critical eigenvalue for onsite dark solitons obtained numerically (solid lines) and analytically using Eq. (41) (dashed lines) for $\gamma =0.1$ (upper curves) and $\gamma =0.6$ (lower curves). An approximation that explicitly includes the next term in expansion Eq. (44) is also plotted (dotted lines).

Figure 10

The eigenvalue structure of onsite dark solitons for several values of $\gamma$ and $C$, as indicated in the caption of each panel.

Figure 11

(Color online) The (in)stability region of onsite dark solitons in the two-parameter $\left(C,\gamma \right)$ space. The white solid and dashed lines, respectively, give the analytical approximations (43, 46). White dashed-dotted and dotted lines show Eqs. (42, 45); note that these curves are indistinguishable in this plot.

Figure 12

Comparisons between the critical eigenvalue for intersite dark solitons obtained numerically (solid lines) and analytically (dashed lines) using Eq. (51), for two values of $\gamma$. The upper curves correspond to $\gamma =0.8$, and the lower ones correspond to $\gamma =0.1$.

Figure 13

The eigenvalue structure of intersite dark solitons with parameter values as indicated in the caption for each panel.

Figure 14

(Color online) As Fig. 11, but for an intersite dark soliton. The white dashed line is our analytical approximation (53).

Figure 15

(Color online) The spatiotemporal evolution of an onsite bright soliton governed by the original time-dependent parametrically driven Klein-Gordon system (1), with $ϵ=0.2$. The parameter values are indicated in the caption for each panel. The left and right panels show stable and unstable solitons, respectively.

Figure 16

(Color online) As Fig. 15, but for an intersite bright soliton, with parameter values as indicated in the caption for each panel. The initial profile in each panel corresponds to the same parameters as in Fig. 5.

Figure 17

(Color online) As Fig. 15, but for onsite dark solitons. The parameter values are as in Fig. 10.

Figure 18

(Color online) As Fig. 15, but for an intersite dark soliton with $\gamma =0.1$. The left panel shows the evolution of a stable dark soliton with $C=0.05$, while the right panel shows the evolution of an unstable dark soliton with $C=0.5$.