Dynamics of modulated waves in the spring-block model of earthquake with time delay

In geological fault modeling, several fragmented blocks are coupled by springs and the motion between them is not transmitted instantly, but with a delay. The dynamics of geological media is investigated by considering the time delay between blocks' deformations. Our modelization led to a complex-Landau equation, from which we derived solitary waves induced by the stick-slip process. A solitary wave propagating along the contact surface between two plates becomes stable or unstable as the time delay varies, thus producing states of solitary wave. The modulational instability of the system is performed, showing that the instability of the amplitude increases with the time delay and the nonlinear parameters. Also, the bandwidth of instability varies with the system parameters. It is shown from our results that the time delay decelerates the propagating soliton.