Solitary waves in a granular chain of elastic spheres: Multiple solitary solutions and their stabilities

A granular chain of elastic spheres via Hertzian contact incorporates a classical nonlinear force model describing dynamical elastic solitary wave propagation. In this paper, the multiple solitary waves and their dynamic behaviors and stability in such a system are considered. An approximate KdV equation with the standard form is derived under the long-wavelength approximation and small deformation. The closed-form analytical single- and multiple-soliton solutions are obtained. The construction of the multiple-soliton solutions is analyzed by using the functional analysis. It is found that the multiple-soliton solution can be excited by the single-soliton solutions. This result is confirmed by the numerical analysis. Based on the soliton solutions of the KdV equation, the analytic solutions of multiple dark solitary waves are obtained from the original dynamic equation of the granular chain in the long-wavelength approximation. The stability of the single and multiple dark solitary wave solutions are numerically analyzed by using both split-step Fourier transform method and Runge-Kutta method. The results show that the single dark solitary wave solution is stable, and the multiple dark solitary waves are unstable.

Figure 1

Schematic diagram of the granular chain of elastic spheres precompressed by a static load.

Figure 2

The profiles of soliton solutions ($\beta =1330\mu {\mathrm{m}}^2, R=2\mathrm{mm}$): (a) solution Eq. (17) with $k_1=300{\mathrm{m}}^{-1}$; (b) solution Eq. (18) with $k_1=150{\mathrm{m}}^{-1}$ and $k_2=-190{\mathrm{m}}^{-1}$; (c) solution Eq. (19) with $k_1=-180{\mathrm{m}}^{-1}, k_2=220{\mathrm{m}}^{-1}$, and $k_3=210{\mathrm{m}}^{-1}$; and (d) solution Eq. (20) with $k_1=200{\mathrm{m}}^{-1}, k_2=200{\mathrm{m}}^{-1}, k_3=-210{\mathrm{m}}^{-1}$, and $k_4=-230{\mathrm{m}}^{-1}$.

Figure 3

Intensity profiles of the double-soliton solution Eq. (18) (a) and approximate solutions Eqs. (21, 22, 23, 24) (b–e, respectively) with the highlighted areas corresponding to limiting cases (i)–(iv) where $\beta =4887\mu {\mathrm{m}}^2, R=2\mathrm{mm}, k_1=1{\mathrm{m}}^{-1}$, and $k_2=1.4{\mathrm{m}}^{-1}$.

Figure 4

Plot of the phase shift $P_s$ for ${\theta }_1$ soliton as the function of the wave number $k_2$. The other parameters are the same as described in the caption of Fig. 3.

Figure 5

Phase shifts of the ${\theta }_i(i=1,2,3,4)$-solitary waves: (a, b) intensity profiles of the double-soliton solution Eq. (18) with the areas marked with arrows showing ${\theta }_1$- and ${\theta }_2$-solitary waves for $k_1=1{\mathrm{m}}^{-1}, k_2=1.2{\mathrm{m}}^{-1}$ and $k_1=1{\mathrm{m}}^{-1},k_2=1.3{\mathrm{m}}^{-1}$, respectively; (c) intensity profiles of the triple-soliton solution Eq. (19) with the area marked with arrows showing ${\theta }_1, {\theta }_2$, and ${\theta }_3$ solitons for $k_1=1{\mathrm{m}}^{-1}, k_2=1.01{\mathrm{m}}^{-1}$, and $k_3=1.2{\mathrm{m}}^{-1}$; (d) intensity profiles of the quadruple soliton solution Eq. (20) with the area marked with arrows showing ${\theta }_1, {\theta }_2, {\theta }_3$, and ${\theta }_4$ solitons for $k_1=1{\mathrm{m}}^{-1}, k_2=1.1{\mathrm{m}}^{-1}, k_3=1.13{\mathrm{m}}^{-1}$, and $k_4=1.3{\mathrm{m}}^{-1}$. The other parameters are the same as described in the caption of Fig. 3.

Figure 6

The double soliton solution Eq. (18) shown by the areas marked with the dashed lines [(i), (ii), (iii), and (iv)] corresponding to four conditions: (a) $e^{{\theta }_1},e^{{\theta }_2}\gg 1$; (b) $e^{{\theta }_1},e^{{\theta }_2}\ll 1$; (c) $e^{{\theta }_1}\gg 1,e^{{\theta }_2}\ll 1$; (d) $e^{{\theta }_1}\ll 1,e^{{\theta }_2}\gg 1$. The parameters and the meaning of the highlighted areas are the same as described in the caption of Fig. 3.

Figure 7

Numerical simulation of a double-soliton inspired by a perturbed excitation with $\beta =4887\mu {\mathrm{m}}^2, R=2\mathrm{mm}, k_1=1{\mathrm{m}}^{-1}$, and $k_2=1.4{\mathrm{m}}^{-1}$: (a) initial condition selected from the double-soliton solution Eq. (18) with a perturbation; (b) initial condition selected from two single-soliton solutions Eqs. (21) and (24) with a perturbation; (c) evolution of the solution with the initial condition shown in (a); and (d) evolution of the solution with the initial condition shown in (b).

Figure 8

Evolution of the double solitary wave solution Eq. (18) in the space-time domain (a) and wave-number-frequency domain (b). Panel (b) can also be viewed as the dispersion curve corresponding to panel (a); panels (c) and (d) show the energy transfer during the dynamic collision process in the time-wave-number domain and space-frequency domain, respectively.

Figure 9

The profiles of solitary wave solutions with the parameters listed in Table 1: (a) solution Eq. (30); (b) negative modulus of solution Eq. (30); (c, d) solution Eq. (31); (e, f) solution Eq. (32); and (g, h) solution Eq. (33).

Figure 10

Numerical simulation of the solitary waves inspired by a perturbed excitation at $t=0$ with the parameters listed in Table 2: (a), (c), (e), and (g) show evolution of the single, double, triple, quadruple dark solitary waves from Eq. (9) simulated by SSFT method; (b), (d), (f), and (h) show evolution of the single, double, triple, quadruple dark solitary waves from Eq. (6) simulated by Runge-Kutta method.

Figure 11

Evolution of the single (a), double (b), triple (c), and quadruple (d) dark solitary waves simulated from Eq. (4) inspired by a perturbed excitation at $t=0$ with the same parameters as in Fig. 10 by using Runge-Kutta method.

Figure 12

(a) Evolution of the rogue wave obtained from Eq. (33) with $\beta =416.65\mu {\mathrm{m}}^2, R=10.00\mathrm{mm}, {\overline{k}}_1=1.6{\mathrm{m}}^{-1}, {\overline{k}}_2=-1.5{\mathrm{m}}^{-1}, {\overline{k}}_3=-1.9{\mathrm{m}}^{-1}, {\overline{k}}_4=1.1{\mathrm{m}}^{-1}, {\overline{\omega }}_1=15.6384{\mathrm{s}}^{-1}, {\overline{\omega }}_2=7.3305{\mathrm{s}}^{-1}, {\overline{\omega }}_3=14.1137{\mathrm{s}}^{-1}$, and ${\overline{\omega }}_4=5.3757{\mathrm{s}}^{-1}$; (b) initial condition selected from Eq. (33) at $t=0$ with a perturbation; (c) evolution of the rogue wave from Eq. (6) simulated by Runge-Kutta method and corresponding to the initial conditions in panel (b); (d) evolution of the rogue wave from Eq. (9) simulated by SSFT method corresponding to the initial conditions; (e) solution in the time-wave-number domain, and (f) solution in the space-frequency domain corresponding to the evolution of the rogue wave in panel (d).

Figure 13

Numerical simulation of the single solitary wave from Eq. (4) with ${\delta }_0=0$ inspired by a perturbed excitation at $t=0$.

Figure 14

Comparison between the present approximate analytical solutions with the experimental and numerical results in Fig. 3 of Ref. [80]: forces between the 7th and 13th particles in the chain of cylindrical particles with $\alpha ={90}^{\circ }$.