The solitary waves were found by Scott Rusell in 1834. Later, the KdV equation was obtained in 1895.^[1] An extraordinary series of events happened around 1965 when Kruskal and Zabusky discovered a pulse-like solitary wave solution of the KdV equation,^[2] for which the name “soliton” was proposed. However, the presence of solitary waves in the bead chain was first discovered and studied by Nesterenko et al.^[3] in 1984. It indicated that the interaction between spheres obeyed the Hertzian law, and the interaction forces with a form of solitary waves propagated in the bead chain. Later, Coste et al.^[4] experimentally studied the relationship between the propagation velocity of the isolated wave and the maximum amplitude and static force. In the pioneering work of Nesterenko,^[3,5,6] wave propagation in a 1D assembly of beads has a rich content in physics due to the nonlinear nature of the interaction between spheres. The most commonly used method of the study of this nonlinear wave is the Poincare–Lighthill–Kuo (PLK) method,^[7–9] which has been extensively used in many branches of physics, such as in plasma physics,^[10,11] in Boes–Einstein condensates,^[9,12] in nonlinear lattice dynamics,^[13,14] etc.

The excitation, propagation, and interaction of solitary waves are important issues in theoretical studies. Recently several studies have reported, both experimentally and numerically, a large number of phenomena associated with this type of wave in Hertzian linear chains, ranging from solitary wave interaction with boundaries^[4,15–17] and wave mitigation in tapered chains^[18–21] to energy localization in chains with mass defects^[22–24] and pulse reflection and transmission due to impurities in a granular chain.^[25–27] With the further reports on solitary wave correlation studies, the problem of interaction between solitary waves has attracted rapidly growing interest because of the significance of understanding the dynamics of granular matter. For instance, the authors in Ref. [28] studied the interaction between the solitary waves in the Toda lattice. It shows that the Toda lattice enables capturing both copropagating and counterpropagating soliton collisions and the existence of phase shift after collision. The interaction between the solitary waves in the plasma^[29–34] has been widely reported, it is the well-known feature of the presence of phase shifts as a result of the solitonic collisions in the plasma. However, the interaction of solitary waves in granular chains is little understood. Francisco Santibanez et al.^[35] studied only experimentally the interaction between two solitary waves that approach one another in a linear chain of spheres interacting via the Hertz potential. When these counterpropagating waves collide, they cross each other and a phase shift in respect to the noninteracting waves is introduced as a result of the nonlinear interaction potential. As a result, in the literature, one refers to these waves as solitary waves. Two opposite propagating solitary waves can pass through each other without suffering any interaction except for a slight shift in their positions compared to where they would have been had they not passed through one another. In this letter, by using the reductive perturbation method, we obtain a pair of KdV solitary waves described by uncoupled KdV equations, one moving rightward and the other moving leftward, to capture the evolution of general initial data, it is used to study the interaction of solitary waves in a 1D granular medium. The head on collision between two opposite propagating solitary waves is studied in the present study. The interesting result is that no phase shift is observed. It is found that the maximum amplitude in the process of the head on collision is close to the linear sum of two colliding solitary waves.

This paper is organized as follows. In Section 2 we set up the model. In Section 3 we give the numerical results. Finally, in Section 4 the conclusion is given in the last section.

We consider an alignment of N identical elastic spheres, each of mass m and radius R, placed in such a way that they are barely in contact with each other. There is no grain-grain interaction when there is no contact, but when an external perturbation is applied to the system initially, different pulses may be evolved into the system. In this process, the elastic deformations are accompanied. Under elastic deformation, the energy stored at the contact between two elastic beads under compression with the Hertz potential is , where δ is the overlap deformation between beads and , , R is the bead radius, Y is the Young modulus, and σ is the Poisson ratio of the bead material. The dynamics of the bead chain is described by the N coupled nonlinear equations,

with , where m_i is the bead mass, denotes the absolute position of bead i, and s_i is the displacement from its equilibrium position of bead i. In general, the magnitude of n depends upon the contact geometry between the beads and typically varies between n = 5/2 and 3,^[36,37] and in our work n = 5/2. If there is great enough prestress, the bead radius will be , is the initial overlap deformation between two beads. In the continuous long-wavelength limit, and , equation (1) becomes^[27,39–41]

Now we study the interaction of a pair of uncoupled KdV solitary waves in a 1D granular chain, as is shown in Fig. 1. We place a chain of N same beads from i = 1 to i = N. We assume that there are two opposite propagating KdV solitary waves labeled by L and R, respectively. One is propagating from the left to the right direction and the other is in the negative direction. Initially, both KdV solitary waves are far apart. After some time, they interact, collide, and then depart. In general, after the collision, there is a phase shift of both solitary waves.^[28–35]

Fig. 1.

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	Fig. 1. (color online) Head-on collision of two KdV solitary waves in a one dimensional bead chain.

According to the reductive perturbation method,^{[7–9,30,38,39]} we introduce the stretched coordinates in the region :

We choose a 1D chain consisting of the same beads. The relevant parameters of the beads are as follows: Youngʼs modulus , Poissonʼs ratio σ = 0.3, the density , radius R = 0.001 m, and the initial overlap δ = 0.0003 m. The initial conditions of the displacements from their equilibrium positions and the corresponding velocities of all beads are chosen from the analytical results, Eqs. (7), (8) and Eqs. (9), (10), where , . In simulation, the starting point i = 1 and the ending point are fixed and the periodic boundary condition is given. All the physical quantities are the international system of unit. The bead displacement is solved by a series of equations of motion of Eq. (1) and the corresponding velocity is obtained as well.

In the following, we will study two kinds of collision between two solitary waves. One is for two same amplitude waves, the other is for two different amplitude waves.

(i) Collision between two same amplitude waves

First, according to Eqs. (7) and (8), the left solitary wave L with the initial data ε_L = 0.001, at the point of the 10⁴-th bead, propagates freely from left to right, as shown in Fig. 2. It is observed that the waveform and the propagation velocity remain unchanged during the collision process with the right solitary wave. We compare our numerical results with the analytical one of Eq. (8) and obtain a good agreement between them, as shown in Fig. 3. It indicates that the KdV solitary waves can propagate steadily in the 1D bead chain. Then according to Eqs. (9) and (10), the right solitary wave R with the same initial condition , at the point of the -th bead, propagates freely from right to left. Initially, both solitary waves are far apart. After some time, they interact, collide, and then depart. There are four different cases of colliding: (a) & , (b) & , (c) & , and (d) & , where “+” and “−” represent the propagation direction of solitary wave,^[40] shown in Fig. 4. It is clear that after collision, their wave forms still remain unchanged. However, there is an interesting phenomenon that there is no phase shift for both solitary waves compared with those without the collision, as shown in Fig. 5. Figure 5(a)–5(d) show the comparisons between the solitary waves with and without collisions corresponding to Figs. 4(a)–4(d), respectively. The black solid lines stand for solitary waves after collision and the red solid lines are the same solitary waves propagating without collisions. Figure 6 shows the corresponding collision profiles to Fig. 5 for the bead displacements. It is found that there is no phase shift after any collision. This result is interesting since it is different from that found in the nonlinear lattice, in the water waves, in the Bose–Einstein condensations.

Fig. 2.

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	Fig. 2. (color online) Simulation results of the evolution of a KdV solitary wave at different times.

Fig. 3.

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	Fig. 3. (color online) The comparisons between simulation results and the analytical ones with the same initial conditions at t = 0.0093. In the figure, the red dotted line represents the analytical solution and the blue solid line represents the simulated results.

Fig. 4.

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Fig. 4. (color online) The interaction process of head on collision between two solitary waves. Panels (a), (b), (c), and (d) correspond to the four cases. L and R represent the left and right solitary waves, where AL = 0.00582 and AR = 0.00582. The arrows stand for the propagation directions of waves. The black solid line represents the initial state of the solitary waves of L and R, the blue solid line represents the wave profile at the critical point when the maximum amplitude is reached during the collision between two solitary waves. The red solid line represents the evolution of solitary waves L and R after collision. Four different cases are shown in panels (a), (b), (c), and (d). Case (a) is for & , case (b) for & , case (c) for & , and case (d) for & . Here A L and represent amplitudes of solitary waves L and R, respectively, while “+” and “−” represent the propagation directions of solitary waves, respectively.

Fig. 5.

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	Fig. 5. (color online) The definition of the phase lags of solitary waves. The arrows stand for the propagation directions of waves, the red solid line represents the solitary wave L without collision and the black solid line represents solitary wave L after collision. “Δ” stands for the phase shift.

Fig. 6.

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	Fig. 6. (color online) Displacement diagram of solitary waves corresponding to Fig. 5. The arrows stand for the propagation directions of waves. The red solid line represents the kink L without collision and the black solid line represents the kink L after collision.

(ii) Collisions between two solitary waves with different amplitude

It is found from the above section that the waveform and the propagation velocity of two colliding solitary waves remain unchanged after collision. In this section, we consider the collision between two solitary waves with different amplitudes. The initial condition of the solitary wave R is given by a small parameter ε_R = 0.001, but the solitary wave L is given by ε_L = 0.0015. The results are shown in Fig. 7. It is noted from Fig. 7 that initial amplitudes of colliding solitary waves L and R are about A_L = 0.01309 and A_R = 0.00582, respectively. There are four cases. No phase shift is observed for a solitary wave after collision, as shown in Fig. 8. We can also obtain from Fig. 9 corresponding to Fig. 8 that the head-on collision between two opposite propagating KdV solitary waves can pass through each other as if without suffering any interaction. There is no phase shift between them.

Fig. 7.

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Fig. 7. (color online) The interaction process of the head on collision between two solitary waves L and R. There are four cases. Initial amplitudes of colliding solitary waves L and R are about AL = 0.01309 and AR = 0.00582, respectively. In the figure, the arrows stand for the propagation directions of waves. The black solid line represents the initial state of the solitary waves L and R, the blue solid line represents the collision of solitary waves L and R and the red solid line represents the evolution of solitary wave L and R at the different times after collision. Four different cases are shown in panels (a), (b), (c), and (d). Case (a) is for , case (b) for , case (c) for , and case (d) for . Here A L and A R represent amplitudes of solitary waves L and R, respectively, while “+” and “−” represent the propagation directions of solitary waves, respectively.

Fig. 8.

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	Fig. 8. (color online) The definition of the phase lags of both solitary waves. In the figure, the arrows stand for the propagation directions of waves. The red solid line represents the solitary wave L without collision and the black solid line represents solitary wave L after collision. “Δ” stands for the phase.

Fig. 9.

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	Fig. 9. (color online) Displacement diagram of solitary waves corresponding to Fig. 8. The arrows stand for the propagation directions of waves. The red solid line represents the kink L without collision and the black solid line represents the kink L after collision.

The maximum amplitude in the process of interaction between two solitary waves is also studied, as shown in Fig. 10. Comparisons are given among three results. It should be noted here that there is no obvious boundary among the numerical results, the linear result of where and , and the modified result for the different velocity solitary waves. Good agreements among them are observed. However, small differences are observed when the colliding solitary wave is large. For further investigation, Figure 11 also shows the dependence of the maximum amplitude during the colliding process on the amplitudes of two colliding solitary waves. It is obviously noted that the maximum amplitude in the colliding process increases as the two amplitudes of the colliding solitary waves increase.

Fig. 10.

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Fig. 10. (color online) The maximum amplitude in the colliding processes as a function of the amplitude of the solitary wave L. Panels (a) and (c) corresponding to , but the amplitude of the solitary wave R is fixed in panels (b) and (d). The red dots are from the numerical results, the black ones are the analytical results by further considering the higher order terms of , , the blue ones are the linear results of , where and A R are the amplitudes of solitary waves L and R, respectively.

Fig. 11.

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	Fig. 11. (color online) The dependence of the maximum amplitude in the colliding processes on both amplitudes (A L, A R) of the two colliding solitary waves L and R.

In this paper, we have studied the head on collision between two opposite propagating solitary waves. By using the PLK method, KdV equations are obtained before and after collisions. At the same time, a numerical investigation is given for the head on collision between two KdV solitary waves. The interesting result is that no phase shift is observed which is different from that found in other branches of physics, such as in plasma physics, fluid dynamics, Bose–Einstein condensation, solid physics, and optics. The maximum amplitude during the head on collision between two solitary waves in a 1D bead chain is also studied. It is found that it is almost equal to the linear sum of the amplitudes of two solitary waves. It is in good agreement between them when the amplitudes of the two colliding solitary waves are small enough, while the difference increases as the amplitudes of both colliding solitary waves increase.