Emergence of relaxation beat-waves in genuinely nonlinear Klein-Gordon chain with bi-harmonic parametric excitation

This paper provides analytical investigation of nonstationary regimes in a strongly anharmonic Klein-Gordon chain subjected to the two-component parametric excitation. We explore the mechanisms of formation and provide a comprehensive analytical characterization of the dynamics of two distinct highly nonstationary beat-wave regimes, namely the weakly- and strongly- modulated beat-waves. To this end, we derive the double parametrically driven discrete p-Schrodinger model in the neighborhood of 2:2:1 parametric resonance. The obtained non-autonomous slow-flow model depicts the low-energy complex amplitude modulations of coupled oscillators in the vicinity of 2:2:1 resonance. Through a special coordinate transformation, we exactly reduce the slow-flow system dynamics to a beat-wave slow invariant manifold governed by three collective coordinates. To study the complex nonstationary dynamics of beat-waves, we further reduce the overall system dynamics onto the super-slow invariant manifold (SSIM) by applying an additional multi-scale procedure to the system of collective coordinates. Analysis of the system dynamics on the SSIM reveals the two types of non-stationary beat-wave regimes. The first type is a weakly modulated beat-wave response, exhibiting super-slow amplitude modulation without amplitude relaxation. The second, more intriguing type is a strongly modulated beat-wave response, which exhibits rapid amplitude relaxations characterized by two distinct behaviors: one involving rapid amplitude decay to the trivial state, and the other manifested by the recurrent relaxation oscillations. We derive analytical approximations that describe the mechanisms of formation and the entire dynamics of these highly nonstationary beat-wave states. Remarkably, the analytical model aligns satisfactorily with numerical simulations for both weakly and strongly modulated beat-wave states.