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In this Letter we describe a novel class of dynamical excitations --accelerating oscillatory fronts in a new genre of nonlinear sonic vacua withstrongly non-local effects. Indeed, it is surprising that such models naturallyarise in dynamics of common and popular lattices. In this study, we address achain of particles oscillating in the plane and coupled by linear springs, withfixed ends. When one end of this system is harmonically excited in thetransverse direction, one observes accelerated propagation of the excitationfront, accompanied by an almost monochromatic oscillatory tail. The frontpropagation obeys the scaling law $l \sim t^{4/3}$. The frequency of theoscillatory tail remains constant, and the wavelength scales as $\lambda \simt^{1/3}$. These scaling laws result from the nonlocal effects; we derive themanalytically (including the scaling coefficients) from a continuumapproximation. Moreover, a certain threshold excitation amplitude is requiredin order to initiate the front propagation. The initiation threshold isrationalized on the basis of a simplified discrete model. This model is furtherreduced to a new completely integrable nonlinear system. The Letter introducesa new and yet unexplored class of nonlinear sonic vacua and explores theeffects of strong non-locality on the initiation and propagation of oscillatingfronts in these media. Given their simplicity, nonlinear sonic vacua of thetype considered herein should be common in periodic lattices.

Accelerating oscillatory fronts in a nonlinear sonic vacuum with strong non-local effects