Intrinsic localized modes in parametrically driven arrays of nonlinear resonators

We study intrinsic localized modes (ILMs), or solitons, in arrays of parametrically driven nonlinear resonators with application to microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). The analysis is performed using an amplitude equation in the form of a nonlinear Schrödinger equation with a term corresponding to nonlinear damping (also known as a forced complex Ginzburg-Landau equation), which is derived directly from the underlying equations of motion of the coupled resonators, using the method of multiple scales. We investigate the creation, stability, and interaction of ILMs, show that they can form bound states, and that under certain conditions one ILM can split into two. Our findings are confirmed by simulations of the underlying equations of motion of the resonators, suggesting possible experimental tests of the theory.

Figure 1

(Color online) The integral measure $I\left\{f\left(X\right)\right\}$ as a function of $h$. Red outer solid and dashed lines represent the exact analytical solutions of the PDNLS equation without nonlinear damping $A_{+}{\text{cos}}^2{\Theta }_{+}$ and $A_{-}{\text{cos}}^2{\Theta }_{-}$, respectively. Black inner solid and dashed lines represent the approximate analytical solutions with nonlinear damping $a_{+}{\text{cos}}^2{\theta }_{+}$ and $a_{-}{\text{cos}}^2{\theta }_{-}$, respectively. These analytical lines end at the upper stability boundary $h=\sqrt{1+{\gamma }^2}$. The points designated by crosses and circles ($+$ and $\circ$) are taken, respectively, from the stationary numerical solution of the amplitude equation (8), and from the numerical solution of the equations of motion (2), as elaborated in the text. The inset shows the absolute value of the profile of the solution for $h=0.87$ where $\circ \text{s}$ are results of the numerical solution of the equations of motion (2). The solid line shows the real part of the numerical solution $\psi$ of the amplitude equation (8), scaled by a factor of $2\sqrt{2ϵ\omega \Omega /3}$. The scaled analytical approximation (22) is indistinguishable from the solid line in this plot. The dot-dashed line shows the scaled analytical solution (9) in the absence of nonlinear damping. The parameters are $\gamma =0.5$, $D=0.25$, $ϵ=0.01$, ${\omega }_p=1.002\omega$, $\eta =0.1$, and $N=399$.

Figure 2

(Color online) The phase of the numerical solution of the amplitude equation (8), and the analytical exact and approximate phases, as indicated in the legend. Near the peak of the soliton, the phase of the numerical solution is close to ${\theta }_{+}$ and it asymptotes to ${\Theta }_{-}$ as the soliton’s amplitude decays to zero. Parameters are the same as in the inset of Fig. 1.

Figure 3

(Color online) Stability diagram for localized solutions of the amplitude equation (8) in the $h$ vs $\gamma$ plane. The dotted line is the lower existence boundary for $\eta =0$, namely $h=\gamma$. The dashed-dotted line is the approximate low boundary for $\eta =0.1$, given by Eq. (21). Above the solid line the ${\Psi }_{+}$ solution of the PDNLS equation with $\eta =0$ is unstable with respect to a Hopf bifurcation [56]. The dashed line is the line $h=\sqrt{1+{\gamma }^2}$ above which the zero solution is unstable. Red asterisks (*) are points for which the matrix $J^{-1}H$ has a pair of complex conjugate eigenvalues with a positive real part for $\eta =0.1$, hence the soliton solution $\psi \left(X\right)$ is unstable. Blue dots represent points for which the solution $\psi \left(X\right)$ is stable according to the linear analysis. The four black circles labeled (a)–(d) indicate parameter values corresponding to the numerical simulations of the equations of motion (2) shown in Figs. 4a, 4b, 4c, 4d.

Figure 4

(Color online) Results of the numerical solution of the equations of motion (2) for $\eta =0.1$, $\gamma =0.1225$, and values of $h$ labeled as (a)–(d) in Fig. 3. The solutions are plotted at times $t_m=2\pi m/{\omega }_p$, with integer values $m$ shown on the vertical axis. (a) $h=0.17$ is below the approximate low boundary (21) but above the low boundary for $\eta =0$. One sees that the localized structure decays to zero. (b) $h=0.1988$ is above the approximate low boundary, where linear stability analysis predicts that the soliton is stable. (c) For $h=0.35$ the stationary soliton is unstable, and an oscillating localized solution is formed instead. In (d) $h=0.85$ and the soliton is stable again. The stability is due to nonlinear damping, without which the soliton is unstable as demonstrated in (e) where $h=0.85$ and $\eta =0$. All unspecified parameters are the same as in Fig. 1.

Figure 5

(Color online) Numerical simulation of the coupled equations of motion (2) showing the dynamical creation of solitons. Linear damping is set to $\gamma =1$ and nonlinear damping to $\eta =0.3$. All other parameters are the same as in Fig. 1. Plotted are the absolute values of the displacements of the resonators, which alternate between positive and negative values. Left panels show the complete time evolution, with $m$ counting the number of drive periods. Right panels show the initial (black dots) and final (blue circles) states along with the analytical form of the solitons (green solid line), using only their central positions $X_0$ as fitting parameters. Top panels: A simulation of 199 resonators with fixed boundary conditions is initiated with random noise and a drive amplitude of $h=5$, which is above the upper stability limit, $h=\sqrt{1+{\gamma }^2}=\sqrt{2}$, for both the zero-state and the solitons. At time $m=600$ drive periods, after some nonzero transient (black $+\text{s}$ in the right panel) has developed, the drive amplitude is lowered to $h=1.35<\sqrt{2}$, yielding stable solitons. Bottom panels: A simulation of 200 resonators with periodic boundary conditions is initiated with the uniform nonzero solution and a drive amplitude of $h=1.3$, which is above the stability threshold (36), $h_{th}\simeq 1.26$, for this state. After $m=10000$ drive periods during which the uniform state remains stable, the drive amplitude is lowered to $h=1.2<h_{th}$, yielding stable solitons.

Figure 6

Stability boundary of the large-amplitude uniform solution ${\overline{\psi }}_{+}$ for $\gamma =1$. The solution exists above the dashed and dot-dashed curves. The dashed curve is the saddle-node $h_{sn}\left(\overline{\eta }\right)$ [Eq. (33)], which is replaced at the point marked with a small square by the dot-dashed curve, indicating the supercritical bifurcation from the zero solution at $h=\sqrt{1+{\gamma }^2}=\sqrt{2}$. The solid curve shows $h_{th}\left(\overline{\eta }\right)$ [Eq. (36)] above which $\mathcal{D}<0$. It coincides with the dashed curve at ${\overline{\eta }}_c=-\gamma +\sqrt{1+{\gamma }^2}=\sqrt{2}-1$, indicated by a small circle. In the dark-gray region ${\overline{\psi }}_{+}$ is stable because both zeros of the quadratic function in (34) are complex. In the light-gray region ${\overline{\psi }}_{+}$ is stable because both zeros are real and negative. This implies that for $\overline{\eta }\ge {\overline{\eta }}_c$ the large-amplitude solution is always stable, and for $\overline{\eta }<{\overline{\eta }}_c$ the drive $h$ has to exceed the threshold value $h_{th}\left(\overline{\eta }\right)$ [Eq. (36)] before the solution becomes stable. Recall that $\overline{\eta }=\eta /2$.

Figure 7

(Color online) Numerical simulations of soliton interaction. Left and right panels display results obtained, respectively, from numerical simulations of the amplitude equation (8) and of the underlying equations of motion (2), with $h=0.7$ and the remaining parameters as in Fig. 1. (a) and (b): attraction between two in-phase solitons and their merger into a single one, after half the energy has been dissipated. (c) and (d): repulsion between out-of-phase solitons.

Figure 8

(Color online) Annihilation of a pair of strongly overlapping solitons. Left and right panels display results obtained, respectively, from numerical simulations of the amplitude equation (8) and of the underlying equations of motion (2), initiated with a separation given by the indicated values of $r_c$ above which annihilation was not observed. (a) and (b): an initial attraction followed by the annihilation of a pair of in-phase solitons. (c) and (d): an initial repulsion followed by the annihilation of a pair of out-of-phase solitons. Parameters are the same as in Fig. 7.

Figure 9

(Color online) Simulations initiated with a boosted soliton. Left and right panels display results obtained, respectively, from numerical simulations of the amplitude equation (8) and of the underlying equations of motion (2), with $\eta =0.02$, $h=1$, and all other parameters as in Fig. 1. Note that the simulations of the equations of motion, displayed on the right, show only the initial stage of the evolution that is simulated with the amplitude equation and displayed on the left (with $T=1$ on the left equivalent to about $m=80$ drive periods on the right, as discussed in the text). (a) and (b): $k=1.35<k_{th}\simeq 1.36$ and the soliton moves slightly and stops. (c) and (d): $k=1.37>k_{th}$ and the soliton splits into two. The solutions eventually settle into the known states of one or two stationary solitons.

Figure 10

(Color online) A stable bound state of two solitons. The $\circ \text{s}$ are absolute values of the displacements of the resonators, obtained by a numerical integration of the discrete equations of motion (2), after a sufficiently long transient time has elapsed. The solid line is the stable solution obtained by solving the amplitude equation (8) as a boundary value problem. The parameters are $\gamma =0.565$, $h=1$, $\eta =0.3$, and all others as in Fig. 1.