Collisions between solitary waves propagating in the two-dimensional hexagonal packing granular system

Abstract

In this paper, we investigate head-on and overtaking collisions of two solitary waves propagating in two-dimensional (2D) hexagonal packing confined in a rectangular region, using the fifth-order Runge–Kutta numerical scheme. In the head-on collision, numerical results indicate that the velocity amplitude of the solitary waves and the second solitary waves is proportional to the impacting velocity of the solitary waves. The relationship between phase shift and impacting velocity satisfies the equation [Formula: see text]. For head-on collisions of the waves with an equal amplitude of velocity, this power exponent is [Formula: see text], which can be well predicted by the binary collision approximation theory. For the case of head-on collisions between the waves with unequal amplitudes of solitary waves, this power exponent becomes [Formula: see text]. During an overtaking collision, the fast solitary wave has a positive phase shift, and the slow solitary wave has a negative phase shift. The phase shift of the fast solitary wave also has a power exponential relationship with the impacting velocity. The two waves either form a single peak in transition, or remain separated throughout the encounter with a double-peak formation depending on the ratio of the impacting velocity of solitary waves. The critical condition between the two regimes is consistent with those obtained by the Korteweg–de Vries model.