Skip to main content
SpringerLink
Log in

Bifurcations and exact solutions in the model of nonlinear elastodynamics of materials with strong ellipticity condition

  • Review
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Rogers et al. (Wave Motion 56:147–164, 2015) investigated a class of deformations in nonlinear elastodynamics for an isotropic elastic solid subject to body forces corresponding to a nonlinear substrate potential. A few of exact solutions were obtained, which are descriptive of the propagation of compact waves and motions with oscillatory spatial dependence. But the bifurcations of phase portraits and dynamic behaviors of solutions have not been studied for the two dynamical systems. In fact, the two model equations are singular nonlinear wave systems of the first and second kinds, respectively. In this paper, we use the method of dynamical systems to investigate the bifurcations of phase portraits for the two systems and obtain all possible exact solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data availability

Enquiries about data availability should be directed to the authors.

References

  1. Byrd, P.F., Fridman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, Berlin (1971)

    Book  Google Scholar 

  2. Destrade, M., Saccomandi, G.: Some results on finite amplitude elastic waves propagating in rotating media. Acta Mech. 173, 19–31 (2004)

    Article  Google Scholar 

  3. Destrade, M., Saccomandi, G.: Finite amplitude elastic waves propagating in compressible solids. Phys. Rev. E 72, 016620 (2005)

    Article  MathSciNet  Google Scholar 

  4. Destrade, M., Saccomandi, G.: Nonlinear transverse waves in deformed dispersive solids. Wave Motion 45, 325–336 (2008)

    Article  MathSciNet  Google Scholar 

  5. Destrade, M., Saccomandi, G.: Transverse waves in nonlinearly elastic solids and the Milne-Pinney (or Ermakov) equations. Math. Mech. Solids 16, 586–593 (2011)

    Article  MathSciNet  Google Scholar 

  6. Li, J., Chen, G.: On a class of singular nonlinear traveling wave equations. Int. J. Bifurc. Chaos 17, 4049–4065 (2007)

    Article  MathSciNet  Google Scholar 

  7. Li, J., Zhu, W., Chen, G.: Understanding peakons, periodic peakons and compactons via a shallow water wave equation. Int. J. Bifurc. Chaos 26(12), 1650207 (2016)

    Article  MathSciNet  Google Scholar 

  8. Li, J.: Singular Nonlinear Traveling Wave Equations: Bifurcations and Exact Solutions. Science Press, Beijing (2013)

    Google Scholar 

  9. Li, J., Chen, G., Deng, S.: Smooth exact traveling wave solutions determined by singular nonlinear traveling wave systems: two models. Int. J. Bifurc. Chaos 29(4), 1950047 (2019)

    Article  MathSciNet  Google Scholar 

  10. Rogers, C., Saccomandi, G., Vergori, L.: Carroll-type deformations in nonlinear elastodynamics. J. Phys. A 47, 205204 (2014)

    Article  MathSciNet  Google Scholar 

  11. Rogers, C., Saccomandi, G., Vergori, L.: Nonlinear elastodynamics of materials with strong ellipticity condition: Carroll-type solutions. Wave Motion 56, 147–164 (2015)

Download references

Funding

This research was partially supported by the National Natural Science Foundation of China (11871231, 12071162, 11701191), Natural Science Foundation of Fujian Province (2021J01303, 2022J01303).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, Fujian Province, People’s Republic of China

    Yan Zhou & Jibin Li

Authors
  1. Yan Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jibin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jibin Li.

Ethics declarations

Conflict of interest

The authors declare that they have no relevant financial or non-financial interests to disclose.

Ethical approval

We certify that this manuscript is original and has not been published and will not be submitted elsewhere for publication while being considered by Nonlinear Dynamics. And the study is not split up into several parts to increase the quantity of submissions and submitted to various journals or to one journal over time. No data have been fabricated or manipulated (including images) to support our conclusions. No data, text, or theories by others are presented as if they were our own. And authors whose names appear on the submission have contributed sufficiently to the scientific work and therefore share collective responsibility and accountability for the results. Finally, this article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, Y., Li, J. Bifurcations and exact solutions in the model of nonlinear elastodynamics of materials with strong ellipticity condition. Nonlinear Dyn 112, 1–13 (2024). https://doi.org/10.1007/s11071-023-09042-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-09042-0

Keywords

Mathematics Subject Classification

Search

Navigation