Bright and gap solitons in membrane-type acoustic metamaterials

We study analytically and numerically envelope solitons (bright and gap solitons) in a one-dimensional, nonlinear acoustic metamaterial, composed of an air-filled waveguide periodically loaded by clamped elastic plates. Based on the transmission line approach, we derive a nonlinear dynamical lattice model which, in the continuum approximation, leads to a nonlinear, dispersive, and dissipative wave equation. Applying the multiple scales perturbation method, we derive an effective lossy nonlinear Schrödinger equation and obtain analytical expressions for bright and gap solitons. We also perform direct numerical simulations to study the dissipation-induced dynamics of the bright and gap solitons. Numerical and analytical results, relying on the analytical approximations and perturbation theory for solions, are found to be in good agreement.

Figure 1

(a) Waveguide loaded with an array of elastic plates; (b) the unit cell of the system; (c) corresponding unit-cell circuit.

Figure 2

Panels (a) and (b), respectively, show the real and imaginary parts of the complex dispersion relation. The black circles (red crosses) show the results from the TL (TMM) approach from Eq. (13) [Eq. (14)]. The blue dashed (light pink continuous) line shows the lossless dispersion relation obtained from the TL (TMM) approach from the lossless limit of Eq. (13) [Eq. (14)]. (c) The frequency dependence of $q{k}^{\prime \prime }$, the product of the dispersion and nonlinearity coefficients of the NLS equation. The light (dark) green region corresponds to the focusing (defocusing) case, with $q{k}^{\prime \prime }>0$ ($q{k}^{\prime \prime }<0$).

Figure 3

(a) Three-dimensional plot depicting the evolution of a bright soliton of the form of Eq. (27), obtained by numerically integrating the lossless version of Eq. (6) ($R_{\omega }=0$) with $\epsilon =0.018$ ($2\epsilon \eta P_0=5471$ Pa), $f=369$ Hz. (b) Numerical spatial profiles of the bright soliton measured at $t_1=2$ s (light blue line), $t_2=2.5$ s (light pink line), $t_3=3$ s (light green line), $t_4=4$ s (light yellow line), and $t_5=5.7$ s (red line). The blue dashed line, pink dashed line, green dashed line, yellow dashed line, and dark red dashed line denote the analytical envelope results of Eq. (27) at $t_1,t_2,t_3,t_4$, and $t_5$ respectively. The black dash-dotted line denotes the nonlinear length $L_{\mathrm{NL}}$ and the dispersion length $L_D$, where $L_{\mathrm{NL}}=L_D=16$ m. (c) The nonlinearity effect turned off, 3D plot depicting the dispersive effect numerically obtained. (d) Numerical spatial profile of dispersive effect measured at $t_1$ (light blue line), $t_2$ (light pink line), $t_3$ (light green line), $t_4$ (light yellow line), and $t_5$ (red line).

Figure 4

(a) Spectra of the different drivers, of the form of Eq. (27), introduced at $x=0$: $\epsilon =0.008$ ($2\epsilon \eta P_0=2431$ Pa) at 369 Hz (red dashed line); $\epsilon =0.018$ ($2\epsilon \eta P_0=5471$ Pa) at 369 Hz (thick light blue continuous line) and $\epsilon =0.018$ at $307.3$ Hz (thin black continuous line). (b) Space-time diagrams of the different wave packet generated from the different drivers in (a). Symbols stand for the analytical space-time diagrams at 369 Hz and $307.3$ Hz. The slopes of the lines depict the corresponding group velocities. (c) Numerical time evolutions of the maximum pressure value of the solitons for the different drivers.

Figure 5

Effect of viscothermal losses on traveling bright solitons. Evolution of the maximum pressure in time for the lossy bright soliton (continuous red line for numerical results and blue crosses for the analytical ones) and for linear lossy dispersive wave (dashed yellow line for numerical results). The driver corresponds to $\epsilon =0.018$ ($2\epsilon \eta P_0=5471$ Pa) and $f=369$ Hz. (a) Propagation in a weakly lossy medium where $R_{\omega }=6.8$ ohm; (b) Propagation in a real lossy medium where $R_{\omega }=68.04$ ohm.

Figure 6

(a) Three-dimensional plot depicting the evolution of a gap soliton of the form of Eq. (35), obtained by numerically integrating the lossless version of Eq. (6) ($R_{\omega }=0$) with $\epsilon =0.04$ ($2\epsilon \eta P_0=12158$ Pa), in a lattice with a length of 150 m. (b) Numerical spatial profiles of gap soliton measured from $t_0$ (at which gap soliton has a maximal amplitude) to $t_0+T/2$ (at which gap soliton has a minimal amplitude).

Figure 7

(a) Amplitude dependence of the frequency of the gap soliton. The blue line denotes the analytical results, Eq. (34). the red crosses denote the numerical results, where the numerical values of the amplitudes and the frequencies are getting from the main peak of the spectrum of the different gap solitons obtained by numerically integrating the lossless version of Eq. (6) ($R_{\omega }=0$) with different initial amplitudes. The black dashed line denotes the cut-off frequency of the system. (b) Time evolution of the middle point of the gap soliton in Fig. 6.

Figure 8

Numerical study of the effect of viscothermal losses on gap solitons. Equation (35) with $\epsilon =0.04$ ($2\epsilon \eta P_0=12158$ Pa) is the initial condition for the gap soliton. Panels (a) and (c) represent the time evolution of the middle point of the gap soliton propagating in a weakly lossy medium where $R_{\omega }=6.18$ ohm and $R_{\omega }=61.8$ ohm, respectively (the blue lines denote the numerical results and red crosses denote the analytical time evolution of the maximum pressure for the lossy gap solitons). [(b) and (d)] The spectra of the gap soliton propagating in a lossy medium. The continuous line is the fast Fourier transform (FFT) of the first part of the signal in (a) and (c) and dashed line is the FFT of the last part of the signal in (a) and (c), respectively.