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 Δt ∼ (VLe)P. For head-on collisions of the waves with an equal amplitude of velocity, this power exponent is P = −0.202, 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 P = −0.656. 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.

中文翻译：

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二维六角堆积颗粒系统中传播的孤波之间的碰撞

在本文中，我们使用五阶龙格-库塔数值方案研究了在二维 (2D) 六边形堆积中传播的两个孤立波的正面碰撞和超车碰撞。在正面碰撞中，数值结果表明，孤立波和二次孤立波的速度幅值与孤立波的冲击速度成正比。相移与冲击速度的关系满足方程Δ吨 ～ (五乐)磷. 对于等幅速度的波的正面碰撞，该幂指数为磷 = -0.202，这可以通过二元碰撞近似理论很好地预测。对于具有不等幅孤立波的波之间的正面碰撞的情况，该幂指数变为磷 = -0.656. 在超车碰撞过程中，快速孤立波具有正相移，慢速孤立波具有负相移。快速孤立波的相移也与冲击速度呈幂指数关系。两个波要么在过渡中形成单峰，要么在整个相遇过程中保持分离，形成双峰，具体取决于孤立波的冲击速度之比。两种制度之间的临界条件与 Korteweg-de Vries 模型获得的临界条件一致。