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Analytical solutions of solitary waves and their collision stability in a pre-compressed one-dimensional granular crystal

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Abstract

In this paper, a pre-compressed one-dimensional granular crystal model is studied. The bright analytic single and multiple solitary wave solutions in more general forms than those obtained the KdV system in the previous studies are derived by using the homogeneous balance principle and Hirota’s bilinear method. The difference between the present solutions and those from the KdV system are investigated both analytically and numerically. By analyzing the dispersion relation and the collision process of solitary waves, we find that there are two types of double-solitary waves in the pre-compressed granular crystal model. The geometric and numerical analysis of dynamic behaviors of the solutions is presented with emphasis on the relation between the double-solitary waves and elastic collision between single-solitary waves. We find that the opposite collision between single-solitary waves may be stable and thus generate a stable double-solitary wave. It is concluded that the collision is a special stable double-solitary wave solution. We further propose a possible way to determine the stability of multiple solitary waves qualitatively. The results of this paper provide a theoretical basis for finding stable multiple solitary wave solutions.

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Acknowledgements

This piece of work is supported by the National Natural Science Foundation of China under grant numbers 11532001 and 11991031. The third author (Y. S. Wang) is also grateful for the support of National Natural Science Foundation of China under grant number 12021002.

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Authors and Affiliations

  1. School of Mathematics & Statistics, Henan University of Science & Technology, Luoyang, 471000, China

    Zhi-Guo Liu & Jinliang Zhang

  2. Department of Mechanics, Beijing Jiaotong University, Beijing, 100044, China

    Zhi-Guo Liu & Yue-Sheng Wang

  3. Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin, 300350, China

    Yue-Sheng Wang

  4. Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO, 65211, USA

    Guoliang Huang

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  1. Zhi-Guo Liu
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Correspondence to Yue-Sheng Wang.

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We would like to declare on behalf of our co-authors that the work described in this paper is original, has not been published previously, and is not considered for publication elsewhere, in whole or in part. All the authors listed have approved the submission of the manuscript. No conflict of interest exits in the submission of the manuscript.

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Liu, ZG., Zhang, J., Wang, YS. et al. Analytical solutions of solitary waves and their collision stability in a pre-compressed one-dimensional granular crystal. Nonlinear Dyn 104, 4293–4309 (2021). https://doi.org/10.1007/s11071-021-06534-9

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