Periodic Wave Solution of the Generalized Burgers–Fisher Equation via Abelian Integral

• Huiyang Zhang ^[1] ; Yonghui Xia ^[1]
  1. [1] Zhejiang Normal University

     Zhejiang Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 3, 2022
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • In this paper, the global existence and uniqueness of an isolated periodic wave solution is investigated for the Burgers–Fisher equation in a very general form. The method is based on employing the monotonicity of the ratio of the Abelian integrals. However, the complexity of the nonlinear term increases the difficulty to prove its monotonicity.

    The obtained results generalize and improve the previous one (Zhang et al. Appl Math Lett 121:107353, 2021), and the new results in this paper have never appeared in the previous works including (Zhang et al. 2021).

• Referencias bibliográficas
  • 1. Bratsos, A., Khaliq, A.: An exponential time differencing method of lines for Burgers–Fisher and coupled-Burgers equations. J. Comput....
  • 2. Chu, J., Ding, Q., Yang, Y.: Steady periodic waves and formal stability for fixed-depth rotational equatorial flows. J. Differ. Equ. 269,...
  • 3. Chen, H., Tang, Y.: Global dynamics of the Josephson equation in TS1. J. Differ. Equ. 269, 4884–4913 (2020)
  • 4. Chen, A., Tian, C., Huang, W.: Periodic solutions with equal period for the Friedmann–Robertson– Walker model. Appl. Math. Lett. 77, 101–107...
  • 5. Chen, A., Zhang, C., Huang, W.: Monotonicity of limit wave speed of traveling wave solutions for a perturbed generalized KdV equation....
  • 6. Hammad, D., El-Azab, M.: 2N order compact finite difference scheme with collocation method for solving the generalized Burger’s–Huxley...
  • 7. Javidi, M.: Spectral collocation method for the solution of the generalized Burger–Fisher equation. Appl. Math. Comput. 174, 345–352 (2006)
  • 8. Laurent, C., Llibre, J.: Phase portraits of cubic polynomial vector fields of Lotka–Volterra type having a rational first integral of degree...
  • 9. Liu, C., Xiao, D.: The monotonicity of the ratio of two Abelian integrals. Trans. Am. Math. Soc. 365, 5525–5544 (2013)
  • 10. Llibre, J., Zhang, X.: On the number of limit cycles for some perturbed Hamiltonian polynomial systems. Dyn. Contin. Discrete Impuls....
  • 11. Li, C., Zhang, Z.: A criterion for determining the monotonicity of the ratio of two Abelian integrals. J. Differ. Equ. 124, 407–424 (1996)
  • 12. Mohammadi, R.: Spline solution of the generalized Burgers–Fisher equation. Appl. Anal. 91, 2189– 2215 (2012)
  • 13. Macías-Díaz, J., Gallegos, A., Vargas-Rodríguez, H.: A modified Bhattacharya exponential method to approximate positive and bounded solutions...
  • 14. Moghimi, M., Hejazi, F.: Variational iteration method for solving generalized Burger–Fisher and Burger equations. Chaos Solitons Fractals...
  • 15. Mendoza, J., Murie, C.: New exact solutions for a generalised Burgers–Fisher equation. Chaos Solitons Fractals 152, 111360 (2021)
  • 16. Ruan, S., Wei, J.: Periodic solutions of planar systems with two delays. Proc. R. Soc. Edinb. Sect. A Math. 129, 1017–1032 (1999)
  • 17. Sun, X., Huang, W., Cai, J.: Coexistence of the solitary and periodic waves in convecting shallow water fluid. Nonlinear Anal. Real World...
  • 18. Sun, X., Wu, K.: The sharp bound on the number of zeros of abelian integral for a perturbation of hyper-elliptic Hamiltonian system. Sci....
  • 19. Sun, X., Yang, J.: Sharp bounds of the number of zeros of Abelian integrals with parameters. Electron. J. Differ. Equ. 2014, 1–12 (2014)
  • 20. Sun, X., Yu, P.: Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms. Discrete Contin....
  • 21. Sun, X., Yu, P., Qin, B.: Global existence and uniqueness of periodic waves in a population model with density-dependent migrations and...
  • 22. Song, Y., Jiang, H., Liu, Q., Yuan, Y.: Spatiotemporal dynamics of the diffusive Mussel–Algae model near Turing–Hopf bifurcation. SIAM...
  • 23. Song, Y., Wei, J.: Local Hopf bifurcation and global periodic solutions in a delayed predator–prey system. J. Math. Anal. Appl. 301, 1–21...
  • 24. Shi, J., Zhang, J., Zhang, X.: Stability and asymptotic profile of steady state solutions to a reaction– diffusion pelagic-benthic algae...
  • 25. Wu, J., Zhang, Y., Li, C.: On the number of zeros of Abelian integrals for a kind of quartic Hamiltonians. J. Differ. Equ. 228, 329–335...
  • 26. Xu, Z., Xian, D.: Application of Exp-function method to generalized Burgers–Fisher equation. Acta Math. Appl. Sin. 26, 669–676 (2010)
  • 27. Yi, F., Wei, J., Shi, J.: Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator– prey system. J. Differ. Equ. 246,...
  • 28. Yi, F., Wei, J., Shi, J.: Diffusion-driven instability and bifurcation in the Lengyel–Epstein system. Nonlinear Anal. Real World Appl....
  • 29. Yang, J., Zhao, L.: The cyclicity of period annuli for a class of cubic Hamiltonian systems with nilpotent singular points. J. Differ....
  • 30. Zeng, Y., Sun, X., Yu, P.: Dynamical analysis on traveling wave of a reaction–diffusion model. Appl. Math. Lett. 109, 106550 (2020)
  • 31. Zhang, H., Xia, Y., N’gbo, P.: Global existence and uniqueness of a periodic wave solution of the generalized Burgers–Fisher equation....
  • 32. Zhao, T., Li, C., Zang, Z., Wu, Y.: Chebyshev–Legendre pseudo-spectral method for the generalised Burgers–Fisher equation. Appl. Math....