Modulation instability and wavenumber bandgap breathers in a time layered phononic lattice

Abstract

We provide the first experimental realization of wavenumber bandgap $\left(q\text{−}\text{gap}\right)$ breathers. Experiments are obtained in the setting of a time-periodic phononic lattice where the model and experiment exhibit good qualitative agreement. $q\text{−}\text{gap}$ breathers are localized in time and periodic in space, and are the counterparts to the classical breathers found in space-periodic systems. We derive an exact condition for modulation instability that leads to the opening of wavenumber bandgaps in which the $q\text{−}\text{gap}$ breathers can arise. The $q\text{−}\text{gap}$ breathers become more narrow and larger in amplitude as the wavenumber goes further into the bandgap. In the presence of damping, these structures acquire a nonzero, oscillating tail. The controllable temporal localization that $q\text{−}\text{gap}$ breathers make possible has potential applications in the creation of phononic frequency combs, energy harvesting or acoustic signal processing.

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