Bifurcations and Exact Peakons, Periodic Peakons, Solitary Wave, Kink Wave Solutions for the Quartic CH-Type Equation

• Xueqiong Yi ^[1] ; Yuqian Zhou ^[1] ; Qian Liu ^[1] ; Kebing Li ^[1] ; Shengning Zhang ^[1]
  1. [1] Chengdu University of Information Technology

     Chengdu University of Information Technology

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 2, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We introduce a novel quartic Camassa-Holm type equation and address their traveling wave solutions. The difficulties related to the non-analytical vector fields and complex elliptic integrals are overcome by using the theory of singular traveling wave systems. Upon analyzing the bifurcations of the traveling wave solutions, we study the phase portraits of different topologies, and obtain some exact traveling wave solutions including peakons, periodic peakons, solitary waves, and kink wave solutions. Our results are useful to better understand wave phenomena described by partial differential equations with higher-order nonlinear terms, and pave the way to obtain the traveling wave solutions of other equations with similar nonlinear terms.

• Referencias bibliográficas
  • 1. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661 (1993)
  • 2. Camassa, R., Holm, D.D., Hyman, J.M.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)
  • 3. Li, J., Chen, G.: On a class of singular nonlinear traveling wave equations. Int. J. Bifurcat. Chaos. 17(11), 4049–4065 (2007)
  • 4. Degasperis, A., Procesi, M.: Asymptotic integrability, symmetry and perturbation theory. World Sci. Publ. 1(1), 23–37 (1999)
  • 5. Wang, J., Shehzad, K., Seadawy, A.R., Arshad, M., Asmat, F.: Dynamic study of multi-peak solitons and other wave solutions of new coupled...
  • 6. Seadawy, A.R.: Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput. Math. Appl....
  • 7. Çelik, N., Seadawy, A.R., özkan, Y.S., Ya¸sar, E.: A model of solitary waves in a nonlinear elastic circular rod: abundant different type...
  • 8. Younas, U., Seadawy, A.R., Younis, M., Rizvi, S.T.R.: Optical solitons and closed form solutions to the (3+1)-dimensional resonant...
  • 9. Rizvi, S.T.R., Seadawy, A.R., Ahmed, S., Younis, M., Ali, K.: Study of multiple lump and rogue waves to the generalized unstable space...
  • 10. Shah, K., Seadawy, A.R., Arfan, M.: Evaluation of one dimensional fuzzy fractional partial differential equations. Alex. Eng. J. 59(5),...
  • 11. Seadawy, A.R., Iqbal, M., Lu, D.: Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary...
  • 12. Seadawy, A.R., Rizvi, S.T.R., Ali, I., Younis, M., Ali, K., Makhlouf, M.M., Althobaiti, A.: Conservation laws, optical molecules, modulation...
  • 13. Freire, I.L., Sales Filho, N., de Souza, L.C., Eduardo Toffoli, C.: Invariants and wave breaking analysis of a Camassa-Holm type equation...
  • 14. Liang, J., Li, J., Zhang, Y.: Bifurcations and exact solutions of an asymptotic rotation-Camassa-Holm equation. Nonlinear. Dynam. 101,...
  • 15. Raza, N., Ur Rahman, R., Seadawy, A., Jhangeer, A.: Computational and bright soliton solutions and sensitivity behavior of Camassa-Holm...
  • 16. Novikov, V.: Generalizations of the Camassa-Holm equation. J. Phys. A-Math. Theor. 42(34), 342002 (2009)
  • 17. Liu, X., Qiao, Z., Yin, Z. (2013): On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity....
  • 18. Johnson, R.S.: Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid. Mech. 455, 63–82 (2002)
  • 19. Lopes, O. (2003): Stability of peakons for the generalized Camassa-Holm equation. Electron. J. Differ. Equ. Paper No. 05, 12
  • 20. Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa-Holm and DegasperisProcesi equations. Arch. Ration. Mech. An....
  • 21. Dullin, H.R., Gottwald, G.A., Holm, D.D.: An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett....
  • 22. Dullin, H.R., Gottwald, G.A., Holm, D.D.: Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow...
  • 23. Zhou, Y., Chen, G., Li, J.: Bifurcations, exact peakon, periodic peakons and solitary wave solutions of the modified camassa-holm equation....
  • 24. Li, J.: Peakon, pseudo-peakon, periodic peakon and compacton determined by exact solutions of singular nonlinear traveling wave systems....
  • 25. Liang, J., Li, J., Zhang, Y.: Bifurcations and exact solutions of an asymptotic rotation-Camassa-Holm equation. Nonlinear Dyn. 101, 2423–2439...
  • 26. Li, J.: Singular nonlinear travelling wave equations: bifurcations and exact solutions. Science Press, Beijing (2013)
  • 27. Li, K., Zhou, Y., Liu, Q.: Solving the regularized Schamel equation by the singular planar dynamical system method and the deep learning...
  • 28. Alotaibi, M.F., Raza, N., Rafiq, M.H., Soltani, A.: New solitary waves, bifurcation and chaotic patterns of Fokas system arising in monomode...
  • 29. Kazmi, S.S., Jhangeer, A., Raza, N., Alrebdi, H.I., Abdel-Aty, A.H., Eleuch, H.: The analysis of bifurcation, quasi-periodic and solitons...
  • 30. Li, R., Sinnah, Z.A.B., Shatouri, Z.M., Manafian, J., Aghdaei, M.F., Kadi, A.: Different forms of optical soliton solutions to the Kudryashov’s...
  • 31. Liu, X., Abd Alreda, B., Manafian, J., Eslami, B., Fazli Aghdaei, M., Abotaleb, M., Kadi, A.: Computational modeling of wave propagation...
  • 32. Qian, Y., Manafian, J., Asiri, M., Mahmoud, K.H., Alanssari, A.I., Alsubaie, A.S.: Nonparaxial solitons and the dynamics of solitary waves...
  • 33. Li, R., Manafian, J., Lafta, H.A., Kareem, H.A., Uktamov, K.F., Abotaleb, M.: The nonlinear vibration and dispersive wave systems with...
  • 34. Zhang, S.S.,Manafian, J., Ilhan, O.A., Jalil, A.T., Yasin, Y., Gatea,M.A.: Nonparaxial pulse propagation to the cubic-quintic nonlinear...
  • 35. Raza, N., Rani, B., Chahlaoui, Y., Shah, N.A.: A variety of new rogue wave patterns for three coupled nonlinear Maccari’s models in complex...
  • 36. Rafiq, M.H., Raza, N., Jhangeer, A.: Nonlinear dynamics of the generalized unstable nonlinear Schro¨dinger equation: a graphical perspective....
  • 37. Rafiq, M.H., Jannat, N., Rafiq, M.N.: Sensitivity analysis and analytical study of the three-component coupled NLS-type equations in fiber...
  • 38. Rafiq, M.H., Raza, N., Jhangeer, A.: Dynamic study of bifurcation, chaotic behavior and multi-soliton profiles for the system of shallow...
  • 39. Rafiq, M.H., Jhangeer, A., Raza, N.: The analysis of solitonic, supernonlinear, periodic, quasiperiodic, bifurcation and chaotic patterns...
  • 40. Kaplan, M., Akbulut, A., Raza, N.: Research on sensitivity analysis and traveling wave solutions of the (4+1)-dimensional nonlinear...
  • 41. Li, J., Zhu, W., Chen, G.: Understanding peakons, periodic peakons and compactons via a shallow water wave equation. Int. J. Bifurcat....
  • 42. Li, J., Zhou, Y., Zhuang, J., Deng, S.: Foundations of nonlinear ordinary differential equations. Science Press, Beijing (2022)
  • 43. Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, Berlin, Heidelberg (1971)