Analysis of the response of damped and parametrically driven, strongly anharmonic Klein-Gordon chain - Part 2: Beat-waves

Abstract

This paper continues a series of publications devoted to the analysis of strongly nonlinear wave phenomena, emerging in genuinely anharmonic, Klein-Gordon cyclic chains, subjected to a traveling-wave parametric excitation. In this study, we analyze a special type of nonlinear waves, which will be further referred to as beat-waves. These are spatially extended, and strongly modulated waves manifested by the extreme amplitude modulation, which can be of stationary or nonstationary type. Stationary beat-waves of Klein-Gordon chains propagate with constant group velocity, that depends on the beat-wave amplitude. In the vicinity of 2:1 parametric resonance, we derive the parametrically driven, discrete p-Schrödinger (PD-DpS) model, which depicts the slow modulation of the response of a parametric oscillator chain. Special coordinates' transformation applied on the PD-DpS model allows the exact reduction of extremely modulated beat-waves to an invariant manifold of this slow-flow system. The system dynamics reduced to the beat-wave invariant manifold is depicted by three collective coordinates. In the undamped case, the analysis of the slow-flow system dynamics on the invariant manifold reveals a special family of nonlinear beat-waves characterized by an extreme amplitude modulation. Among these beat-wave solutions lying on the manifold, our analysis identifies some special solutions manifested by the solitary-pulse like, temporal evolution of the beat-wave amplitude which can either decay to zero or to a constant nonzero value. We derive the exact analytical solutions for these special types of waves as well as obtain an explicit analytical approximation for the beat-waves of other types. In the damped case, analysis of the slow-flow system dynamics on the invariant manifold fully recovers the mechanism of system synchronization on the special, stationary beat-wave solutions characterized by the harmonically modulated envelope. Importantly, the derived reduced order model accurately captures the transient evolution stage as well as the steady-state stage of the collective beat-wave response of PD-DpS model, initialized on the beat-wave invariant manifold. A detailed bifurcation analysis of the stationary undamped as well as the stationary damped beat-waves recovers their entire bifurcation structure and enables to fully characterize the zones of system parameters guaranteeing the existence of this special type of non-linear wave solutions. The results of analytical study of various types of beat wave solutions are in excellent agreement with the results of numerical simulation of both PD-DpS as well as the original, parametrically driven, genuinely an-harmonic, discrete Klein-Gordon model.