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

TL;DR

This paper introduces a new class of nonlinear sonic vacua exhibiting accelerating oscillatory fronts with unique scaling laws, derived analytically, and demonstrates their emergence in common lattice models with strong non-local effects.

Contribution

It presents the discovery and analytical description of accelerating oscillatory fronts in nonlinear sonic vacua with non-local effects, including a new integrable nonlinear system.

Findings

Front propagation follows a $l \\sim t^{4/3}$ scaling law.

Oscillatory tail frequency remains constant over time.

A threshold excitation amplitude is necessary to initiate the front.

Abstract

In this Letter we describe a novel class of dynamical excitations -- accelerating oscillatory fronts in a new genre of nonlinear sonic vacua with strongly non-local effects. Indeed, it is surprising that such models naturally arise in dynamics of common and popular lattices. In this study, we address a chain of particles oscillating in the plane and coupled by linear springs, with fixed ends. 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. The front propagation obeys the scaling law $l \sim t^{4/3}$. The frequency of the oscillatory tail remains constant, and the wavelength scales as $\lambda \sim t^{1/3}$. 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 rationalized on the basis of a simplified discrete model. This model is further reduced to a new completely integrable nonlinear system. The Letter introduces a new and yet unexplored class of nonlinear sonic vacua and explores the effects of strong non-locality on the initiation and propagation of oscillating fronts in these media. Given their simplicity, nonlinear sonic vacua of the type considered herein should be common in periodic lattices.