Accelerating oscillatory fronts in a nonlinear sonic vacuum with strong nonlocal effects

We describe and explore accelerating oscillatory fronts in sonic vacua with nonlocal interactions. As an example, a chain of particles oscillating in the plane and coupled by linear springs, with fixed ends, is considered. When one end of this system is harmonically excited in the transverse direction, one observes accelerated propagation of the excitation front, accompanied by an almost monochromatic oscillatory tail. Position of the front obeys the scaling law l(t) ∼ t(4/3). The frequency of the oscillatory tail remains constant, and the wavelength scales as λ ∼ t(1/3). These scaling laws result from the nonlocal effects; we derive them analytically (including the scaling coefficients) from a continuum approximation. Moreover, a certain threshold excitation amplitude is required in order to initiate the front propagation. The initiation threshold is evaluated on the basis of a simplified discrete model, further reduced to a completely integrable nonlinear system. Given their simplicity, nonlinear sonic vacua of the type considered herein should be common in periodic lattices.