Spatiotemporal Structures of Parametrically Driven Nonlinear Lattices

We theoretically investigate the effects of parametric driving on the one-dimensional Frenkel-Kontorova model, a nonlinear many-body lattice system. It is numerically found that a parametric vibration induces spatiotemporal ordering characterized by the subharmonic frequency ${\omega }_{\mathrm{e}\mathrm{x}}\mathrm{/}2$ and wavenumber $k$ which takes nontrivial values depending on the vibration strength $P_{\mathrm{a}\mathrm{c}}$ and frequency ${\omega }_{\mathrm{e}\mathrm{x}}$. With increasing $P_{\mathrm{a}\mathrm{c}}$, $k$ gradually increases and discontinuously jumps up to $k=\pi$. Based on an extended linear stability analysis, we show that the former $k$ resonance (the latter $\pi$ resonance) corresponds to the unstable (stable) mode and that the $k$ resonance is made possible by the interplay between the parametric amplification and the collective fluctuation developing through the nonlinear effect.