Dynamics of modulated waves and localized energy in a Burridge and Knopoff model of earthquake with velocity-dependant and hydrodynamics friction forces

Abstract

We investigate both analytically and numerically the dynamics of modulated waves in a modified Burridge and Knopoff mechanical model of earthquake. We take into account the velocity-dependent and hydrodynamics friction forces between the rock blocks of the system. The theoretical framework for the analysis is presented using the multiple scale expansion method. We show that the system can be governed by the complex Ginzburg–Landau equation, which admits soliton solutions. The modulational instability of the system is performed, showing that the bandwidth of instability increases with the friction forces parameter. The planar wave solution used as the initial condition leads to more unstable waves in the system with chaotic-like behaviour, allowing energy localization. It is also found that the wave parameters, namely, the frequency, magnitude and velocity (as well as the energy of the system) strongly depend on η. In the non-friction limit, the obtained waves propagate with a coherent behaviour for a very short time, before disappearing. Our results suggest that the damages caused by an earthquake can be important as the friction forces increase and may excite the possibility of relative predictions for the dynamics of waves propagating in the earth in the field of nonlinear science.