Driven Localized Excitations in the Acoustic Spectrum of Small Nonlinear Macroscopic and Microscopic Lattices

Both bright and dark traveling, locked, intrinsic localized modes (ILMs) have been generated with a spatially uniform driver at a frequency in the acoustic spectrum of a nonlinear micromechanical cantilever array. Complementary numerical simulations show that a minimum density of modes, hence array size, is required for the formation of such locked smoothly running excitations. Additional simulations on a small 1D antiferromagnetic spin system are used to illustrate that such uniformly driven running ILMs should be a generic feature of a nanoscale atomic lattice.

Figure 1

Several kinds of ILMs produced experimentally with the driver at different frequencies in a cantilever versus time plot. (a) Stationary bright ILM is recorded as a thick black horizontal band, (b) bright traveling ILM appears as a dark zigzag pattern, (c) dark traveling ILM zigzag pattern is identified by the arrows. Where the dark ILM excitation travels the large amplitude standing wave pattern vanishes. The bright regions at its boundaries are due to the overlap of the incident and reflected hole amplitudes at those locations. Driving frequencies: (a) 137.8, (b) 110.086, and (c) 81.26 kHz.

Figure 2

(a) Linear and nonlinear dispersion map of the dielement cantilever array showing various locked ILM positions. Open circles identify the linear modes. The three horizontal dashed lines identify characteristic driving frequencies. (1) Solid diamonds, illustration of a Fourier transform (FT) of a locked, stationary ILM generated above the top of the upper branch, (2) open squares, FT of a locked traveling bright ILM, and (3) open squares, FT of a locked traveling dark ILM. Solid circles indicate carrier modes of traveling ILMs. These locked ILMs occur where the dispersion balances the nonlinearity and map as groups of points on straight lines. Because of the hard nonlinearity the bright ILM, (2), appears above the linear dispersion curve while the dark ILM, (3), appears below the nonlinear dispersion curve (dotted curve that is shifted up by the background excitation of the dark ILM). The hole feels a soft nonlinearity in the strongly excited background. The two arrows indicate the frequency region examined numerically. (b) Numerical simulation results for the dependence of traveling locked bright ILMs on system size. Solid and dotted horizontal lines show strongly and weakly active vibrational modes, respectively. Open circles identify frequencies where locked smoothly traveling ILMs are found. Crosses correspond to frequency regions of instability, involving intermittent modes and complex traveling modes. The smoothly traveling locked ILMs appear at frequencies above these unstable regions.

Figure 3

Energy of cantilever per site averaged over the large amplitude cantilevers as a function of driver frequency for $N=100$. The complex traveling mode pattern (inset a) occurs in the frequency region indicated by the two heaeded arrow. The smooth traveling ILM (inset b) is observed for frequencies between 112.54–113.6 kHz. The frequencies of insets (a) and (b) are 112.45 and 112.6 kHz, respectively. The time window of each inset is 4 ms. The spikes occur at instabilities within the averaging 4000 period time duration.

Figure 4

(a) Linear dispersion curve of the quasi 1D antiferromagnet $({\mathrm{C}}_2{\mathrm{H}}_5{\mathrm{NH}}_3{)}_2{\mathrm{CuCl}}_4$. The two magnetically active $k=0$ modes are linearly polarized in orthogonal directions. Because of the soft nonlinearity of magnetic systems the lower frequency region between the two arrows will be probed numerically for locked traveling bright magnetic ILMs. Inset: Spin site versus time showing a single locked traveling magnetic ILM at 1.74 GHz for the 100 spin case. The inset time window is 57 ns. (b) Numerical simulation results for the dependence of traveling locked ILMs on system size. Horizontal lines identify spin wave modes of the lower branch that can be excited by the ac field polarized along the upper branch $k=0$ direction due to the finite lattice size with free boundary conditions. Crosses indicate frequencies for the complex traveling modes and open circles identify smoothly traveling mode locations. The frequency of the smoothly running mode is always lower than the complex mode, when they stem from the same carrier mode.