^{1}

^{2}

In this paper, we reduced the governing equation describing the one-dimensional granular crystals of elastic spheres to a continuous equation by small deformation and long wave approximation. Then, the G’/G-expansion method is applied to this continuous equation, and the exact solitary wave solutions with arbitrary parameters are obtained. Compared with other papers, the solutions obtained in this paper are more extensive and contains more parameters. The simultaneous existence of exact solitary wave solutions can help us study the propagation of shock waves in one-dimensional granular crystals of elastic spheres. At the same time, it has important theoretical significance in nondestructive testing with non-linear wave.

In recent years, the study of the propagation of highly nonlinear solitary waves in granular materials has drawn considerable attention from the scientific community [

A solitary wave was shown to be an ideal method for transferring vibrational excitations [

The dynamic properties of one-dimensional granular crystals have been extensively studied, using analytical, numerical, and experimental methods. In Reference [

In the present work, we use G’/G-expansion method [

A granular crystal of elastic spheres compressed by a static force

Using the dynamic equilibrium condition, the equation describing the motion of the one-dimensional granular crystals of elastic spheres can be derived as:

where

of the beads and the radius of the contact curvature, E is the Young’s modulus,

If the force between the elastic spheres is a small nonlinear force and the static compression at the initial time is greater than the interparticle compression, we have

Then from Equation (1), we have (of Equation (2.2) in [

In the long-wave approximation, Equation (3) can be written as the continuation form:

where

Ignoring the infinitely small quantities of the fifth order, we obtain

Next, we use the G’/G-expansion method to solve Equation (5).

Firstly, the traveling wave transformation is performed

where k and

When Equation (6) is brought into Equation (5), the following ordinary differential equations are obtained.

By integrating Equation (7) once and taking the integral constant as zero, we can get the result

Assuming that the solution of Equation (8) is

where

By solving Equation (10), we can get

where

From Equation (10), it can be obtained

Substituting Equation (12) and Equation (14) into Equation (8) and applying the principle of homogeneous balance yield

It can be obtained from Equation (15) that m = 1, so Equation (9) can be written as

Substitute Equation (16) into Equation (8), merging the same power terms of (G’/G) and making the coefficients of these same power terms zero, the following equations can be obtained

or

Solving algebraic Equation (17)-(21), we can get

when

When

where

The displacement profiles of the of the exact single solitary wave solution Equation (24), are shown in

In this paper, the continuous equation of one-dimensional granular crystals of elastic spheres is derived; and the G’/G-expansion method is applied to this continuous equation, the hyperbolic function solitary wave solutions, trigonometric function periodic wave solutions and rational wave solutions with arbitrary parameters are obtained. Solitary wave solution Equation (24) is a special form of hyperbolic function solution Equation (23), and the form of the solitary wave solutions obtained in [

This work is supported by the National Natural Science Foundation of China (Grant no.51675161). The authors express their sincere thanks to the referee for valuable suggestions.

The authors declare no conflicts of interest regarding the publication of this paper.

Liu, Z.G. and Zhang, J.L. (2019) The Exact Solitary Wave Solutions in Continuity Equation of the One-Dimensional Granular Crystals of Elastic Spheres. Journal of Applied Mathematics and Physics, 7, 2760-2766. https://doi.org/10.4236/jamp.2019.711189