Solitary Waves, Periodic Waves and Chaos in a Perturbed Higher-Order Gardner Equation

• El-mahdi Nafia ^[1] ; Abdelmjid Benmerrous ^[1] ; Mhamed El Omari ^[1]
  1. [1] Université Sultan Moulay Slimane

     Université Sultan Moulay Slimane

     Alkhalfia, Marruecos

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 5, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We study solitary waves, periodic waves, and chaotic dynamics in a singularly perturbed higher-order Gardner equation with reinforced convective nonlinearity, weak dissipation, and periodic modulation. The additional term μu4ux modifies the travelling-wave potential and, after normalization, leads to a slow–fast system whose reduced dynamics is governed by a sextic Hamiltonian. The critical manifold is shown to be normally hyperbolic and attracting, so Fenichel theory yields a locally invariant slow manifold for small perturbations. The unperturbed reduced system contains two symmetric centers, two homoclinic loops, and two period annuli, generating the underlying solitary and periodic profiles. Abelian integral conditions provide a wave-speed selection rule for the persistence of these waves under dissipative perturbations.

    Finally, a weak periodic modulation produces a Melnikov function with simple zeros at the selected homoclinic speed, yielding transverse homoclinic intersections and a Smale horseshoe for the Poincaré map. Hence the periodically forced reduced travelling-wave system exhibits chaotic dynamics.

• Referencias bibliográficas
  • 1. Miura, R.M., Gardner, C.S., Kruskal, M.D.: Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants...
  • 2. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)
  • 3. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equations 31, 53–98 (1979). https://doi.org/10.1016/0022-0396(79)90152-9
  • 4. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer (1983). https://doi.org/10.1007/978-1-4612-1140-2
  • 5. Drazin, P.G., Johnson, R.S.: Solitons:An Introduction. Cambridge University Press, Cambridge (1989)
  • 6. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993). https://doi.org/10.1103/PhysRevLett.71.1661
  • 7. Jones, C.K.R.T.: Geometric singular perturbation theory. In: Johnson, R. (ed.) Dynamical Systems. Lecture Notes in Mathematics, vol. 1609,...
  • 8. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer (2003). https://doi.org/10.1007/b97481
  • 9. Tang, Y., Xu, W., Shen, J., Gao, L.: Persistence of solitary wave solutions of singularly perturbed Gardner equation. Chaos, Solitons Fractals...
  • 10. Han, M.,Yu, P.:Normal Forms,Melnikov Functions and Bifurcations of Limit Cycles. Springer (2012). https://doi.org/10.1007/978-1-4471-2918-9
  • 11. Yan, W., Liu, Z., Liang, Y.: Existence of solitary waves and periodic waves to a perturbed generalized KdV equation. Math. Model. Anal....
  • 12. Kuehn, C.: Multiple Time Scale Dynamics. Springer (2015). https://doi.org/10.1007/978-3-319- 12316-5
  • 13. Chen, Y., Liu, Z.: The bifurcations of solitary and kink waves described by the Gardner equation. Discrete Contin. Dyn. Syst. Ser. S 9,...
  • 14. Du, Z., Li, J., Li, X.: Existence of solitary wave solutions of delayed Camassa-Holm equation. J. Funct. Anal. 275, 988–1007 (2018)
  • 15. Wen, Z.: On existence of kink and antikink wave solutions of singularly perturbed Gardner equation. Math. Methods Appl. Sci. 43, 4422–4427...
  • 16. Zhang, L., Han, M., Zhang, M., Khalique, C.M.: A new type of solitary wave solution of the mKdV equation under singular perturbations....
  • 17. Jhangeer, A., Hussain, A., Junaid-U-Rehman, M., Baleanu, D., Riaz, M.B.: Quasi-periodic, chaotic and travelling wave structures of modified...
  • 18. Du, Z., Li, J.: Geometric singular perturbation analysis toCamassa-Holm-Kuramoto-Sivashinsky equation. J. Differential Equations 306,...
  • 19. Dai, Y.,Wei, M., Han, M.: Existence of periodic waves in a perturbed generalized BBM equation. Int. J. Bifurcation Chaos 33, 2350060 (2023)....
  • 20. Qiao, Q., Yan, S., Zhang, X.: Periodic and solitary waves in a Korteweg-de Vries equation with delay. Proc. R. Soc. Edinb. Sect. A Math....
  • 21. Qi, Y., Tian, Y., Jiang, Y.: Existence of traveling wave solutions for the perturbed modified Gardner equation. Qual. Theory Dyn. Syst....
  • 22. Jia, G., Tian, Y., Zhang, X.: Existence of traveling wave solutions for Gardner equation and its spatiotemporal delay form. Qual. Theory...
  • 23. Akinyemi, L., Inc, M., Yusuf, A., Baleanu, D.: Novel analytical approaches and stability examination for soliton solutions of the perturbed...
  • 24. Cen, X., Zhang, C.: Periodic waves and monotonicity of limit wave speed in a perturbed generalized KdV equation with two arbitrarily high-order...
  • 25. Zhang, Z., Li, X.: Solitary waves in a singularly perturbed Camassa-Holm equation involving Atangana’s conformable derivative.Qual. Theory...
  • 26. Yan, W., Yu, R., Wang, L.: Existence of solitary waves and periodic waves in a perturbed generalized Gardner equation. Qual. Theory Dyn....
  • 27. Jawarneh, Y., Almusawa, M.Y., Haleemzai, I., Hakami, A.H., Ahmadini, A.A.H., Albaity, A.: New analytical wave solutions for the Gardner...
  • 28. Huzak, R., Rojas, D.: Abelian integrals and non-generic turning points. Qual. Theory Dyn. Syst. 21, 77 (2022). https://doi.org/10.1007/s12346-022-00609-7
  • 29. Grau,M., Mañosas, F., Villadelprat, J.: A Chebyshev criterion for Abelian integrals. Trans. Am. Math. Soc. 363, 109–129 (2011). https://doi.org/10.1090/S0002-9947-2010-05007-X