Bifurcations and Exact Traveling Wave Solutions for the Model of Slightly Dispersive Quasi-Incompressible Hyperelastic Materials

• Jibin Li ^[1] ; Yanfei Dai ^[1]
  1. [1] Zhejiang Normal University

     Zhejiang Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 1, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This paper is devoted to the bifurcations and exact traveling wave solutions in a model of slightly dispersive quasi-incompressible hyperelastic materials by introducing a very general response function for the Cauchy stress tensor of dispersive hyperelastic solids, which was recently established by Saccomandi and Vergor [1]. By using the method of dynamical systems, we study the bifurcations of phase portraits and dynamical behaviors of solutions for the corresponding traveling wave system, which is a planar dynamical system. In addition to the results on the existence of solitary wave and kink wave solutions that has been proved, we not only find more traveling wave solutions, such as periodic wave and anti-kink wave solutions, but also obtain the exact explicit parametric representations and bifurcations of all possible traveling wave solutions under various parameter conditions.

• Referencias bibliográficas
  • 1. Saccomandi, G., Vergori, L.: Solitary waves in slightly dispersive quasi-incompressible hyperelastic materials. Int. J. Solids Struct....
  • 2. Li, J., Chen, G.: On a class of singular nonlinear traveling wave equations. Int. J. Bifur. Chaos 17, 4049–4065 (2007)
  • 3. Li, J.: Singular nonlinear traveling wave equations: bifurcations and exact solutions. Science Press, Beijing (2013)
  • 4. Byrd, P.F., Fridman, M.D.: Handbook of elliptic integrals for engineers and sciensists. Springer, Berlin (1971)
  • 5. Han, M.: Bifurcation theory of limit cycles. Science Press, Beijing (2013)
  • 6. Li, J., Shi, J.: Bifurcations and exact solutions of ac-driven complex Ginzburg-Landau equation. Appl. Math. Comput. 221, 102–110 (2013)
  • 7. Li, J., Qiao, Z.: Peakon, pseudo-peakon, and cuspon solutions for two generalized Camassa-Holm equations. J. Math. Phys. 54, 123501 (2013)
  • 8. Li, J., Zhu, W., Chen, G.: Understanding peakons, periodic peakons and compactons via a shallow water wave equation. Int. J. Bifurc. Chaos...
  • 9. Zhang, B., Xia, Y., Zhu, W., Bai, Y.: Explicit exact traveling wave solutions and bifurcations of the generalized combined double sinh-cosh-gordon...
  • 10. Zhang, B., Zhu, W., Xia, Y., Bai, Y.: A unified analysis of exact traveling wave solutions for the fractional-order and integer-order...
  • 11. Zhu, W., Xia, Y., Bai, Y.: Traveling wave solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity. Appl. Math. Comput....
  • 12. Zhou, Y., Zhuang, J.: Bifurcations and exact solutions of the Raman soliton model in nanoscale optical waveguides with metamaterials....
  • 13. Khorbatly, B.: Exact traveling wave solutions of a geophysical Boussinesq system. Nonlinear Anal. Real World Appl. 71, 103832 (2023)
  • 14. Zhuang, J., Zhou, Y., Li, J.: Bifurcations and exact solutions of the Gerdjikov-Ivanov equation. J. Nonlinear Model Anal. 5, 549–564 (2023)
  • 15. Liu, J., Zhao, L., He, L., Liu, W.: Bifurcation, phase portrait and traveling wave solutions of the coupled fractional Lakshmanan-Porsezian-Daniel...
  • 16. Zhou, Y., Li, J.: Bifurcations and exact solutions in the model of nonlinear elastodynamics of materials with strong ellipticity condition....
  • 17. Maugin, G., Cadet, S.: Existence of solitary waves in martensitic alloys. Int. J. Eng. Sci. 29, 243–258 (1991)
  • 18. Saccomandi, G., Vergori, L.: Waves in isotropic dispersive elastic solids. Wave Motion 116, 103066 (2023)