Existence of Solitary Waves and Periodic Waves in a Perturbed Generalized Gardner Equation

• Weifang Yan ^[1] ; Rui Yu ^[1] ; Linlin Wang ^[1]
  1. [1] Ludong University

     Ludong University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 3, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This paper investigates a perturbed generalized Gardner equation incorporating weak backward diffusion and dissipation effects. Initially, under specific parameter conditions, we derive the exact expression of solitary wave solution for the unperturbed equation by using the dynamical system theory. Subsequently, by employing geometric singular perturbation theory and analyzing perturbations in a Hamiltonian system, we demonstrate that both solitary wave and periodic wave solutions persist for any energy parameter h within a specified interval under small perturbations. Additionally, through the analysis of Abelian integral ratios, we establish that the limiting wave speed c(h) and the period T (h) for these solutions are strictly monotonically decreasing with respect to h. Furthermore, the upper and lower bounds of the limiting wave speed and the period for these solutions are obtained. Numerical simulations are in complete agreement with the theoretical results.

• Referencias bibliográficas
  • 1. Bates, P.W., Wen, Z., Zhang, M.: Small permanent charge effects on individual fluxes via classical Poisson-Nernst-Planck systems with multiple...
  • 2. Betchewe, G., Victor, K.K., Thomas, B.B., Crepin, K.T.: New solutions of the gardner equation: analytical and numerical analysis of its...
  • 3. Chen, Y., Liu, Z.: The bifurcations of solitary and kink waves described by the Gardner equation. Discrete Cont. Dyn. Syst.-S 9, 1629–1645...
  • 4. Christopher, J.: Geometric Singular Perturbation Theory. Lecture Notes Math., vol. 1609, Springer- Verlag, (1994)
  • 5. Coffey,M.W.: On series expansions giving closed form solutions of Korteweg-de Vries like equations. SIAM J. Appl. Math. 50, 1580–1592 (1990)
  • 6. Dai, Y.,Wei, M., Han, M.: Existence of periodic waves in a perturbed generalized BBM equation. Int. J. Bifurc. Chaos 33, 2350060 (2023)
  • 7. Fu, Z., Liu, S., Liu, S.: New kinds of solutions to Gardner equation. Chaos Soliton. Fract. 20, 301–309 (2004)
  • 8. Ghanbari, B., Baleanu, D.: New solutions of Gardner’s equation using two analytical methods. Front. Phys. 7, 00202 (2019)
  • 9. Han, M.: Bifurcation Theory of Limit Cycles. Science Press, Beijing (2013)
  • 10. Han, M., Yang, J.: The maximum number of zeros of functions with parameters and application to differential equations. J. Nonlin. Model....
  • 11. Han, M., Yu, P.: Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles. Springer, NY (2012)
  • 12. Jhangeer, A., Hussain, A., Junaid-U-Rehman, M., Baleanu, D., Riaz, M.B.: Quasi-periodic, chaotic and traveling wave structures of modified...
  • 13. Konno, K., Ichikawa, Y.H.: A modified Korteweg-de Vries equation for ion acoustic waves. J. Phys. Soc. Jpn. 37, 1631–1636 (1974)
  • 14. Luo, D., Han, M., Zhu, D.: Uniqueness of limit cycles bifurcating from a singular closed orbit (I) (in chinese). Acta Math. Sin. 35, 407–417...
  • 15. Miura, R.M., Gardner, C.S., Kruskal, M.D.: Korteweg-de Vries equation and generalizations, ii. existence of conservation laws and constants...
  • 16. Mohanty, S.K., Dev, A.N., Sahoo, S.K., Emadifar, H., Arora, G.: Exact solutions of the generalized ZK and Gardner equations by extended...
  • 17. Olivier, C.P., Verheest, F., Hereman,W.A.: Collision properties of overtaking supersolitons with small amplitudes. Phys. Plasmas 25, 032309...
  • 18. Qi, Y., Tian, Y., Jiang, Y.: Existence of traveling wave solutions for the perturbed modified Gardner equation. Qua. Theor. Dyn. Sys....
  • 19. Wen, Z.: The generalized bifurcation method for deriving exact solutions of nonlinear space-time fractional partial differential equations....
  • 20. Shi, L., Wen, Z.: Bifurcations and dynamics of traveling wave solutions to a Fujimoto-Watanabe equation. Commun. Theor. Phys. 69, 631–636...
  • 21. Shi, L., Wen, Z.: Several types of periodic wave solutions and their relations of a Fujimoto-Watanabe equation. J. Appl. Anal. Comput....
  • 22. Sun, X., Huang,W., Cai, J.: Coexistence of the solitary and periodic waves in convecting shallow water fluid. Nonlinear Anal.-Real World...
  • 23. Sun, X., Zeng,Y.,Yu, P.:Analysis and simulation of periodic and solitarywaves in nonlinear dispersivedissipative solids. Commun. Nonlinear...
  • 24. Tamang, J., Saha, A.: Bifurcations of small amplitude supernonlinearwaves of the mKdV andmodified Gardner equations in a three component...
  • 25. Tang, Y., Xu, W., Shen, J., Gao, L.: Persistence of solitary wave solutions of singularly perturbed Gardner equation. Chaos Soliton. Fract....
  • 26. Wadati, M.: Wave propagation in nonlinear lattice. I. J. Phys. Soc. Jpn. 38, 673–680 (1975)
  • 27. Wadati, M.: Wave propagation in nonlinear lattice. II. J. Phys. Soc. Jpn. 38, 681–686 (1975)
  • 28. Wazwaz, A.M.: The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. Appl. Math. Comput. 187, 1131–1142...
  • 29. Wen, Z.: On existence of kink and antikink wave solutions of singularly perturbed Gardner equation. Math. Meth. Appl. Sci. 43, 4422–4427...
  • 30. Wen, Z., Bates, P.W., Zhang, M.: Effects on I-V relations from small permanent charge and channel geometry via classical Poisson-Nernst-Planck...
  • 31. Wen, Z., Shi, T.: Existence and uniqueness of a canard cycle with cyclicity at most two in a singularly perturbed Leslie-Gower Predator-Prey...
  • 32. Wen, Z., Zhang, L., Zhang, M.: Dynamics of classical Poisson-Nernst-Planck systems with multiple cations and boundary layers. J. Dyn....
  • 33. Yan, W., Liu, Z., Liang, Y.: Existence of solitary waves and periodic waves to a perturbed generalized KdV equation. Math. Model. Anal....
  • 34. Zhang, L., Han, M., Zhang, M., Khalique, C.M.: A new type of solitary wave solution of the mKdV equation under singular perturbations....