Existence of Single-Peak Solitary Waves and Double-Peaks Solitary Wave of Gardner Equation with Kuramoto–Sivashinsky Perturbation

• Keqin Zhao ^[1] ; Zhenshu Wen ^[1]
  1. [1] Huaqiao University

     Huaqiao University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 3, 2023
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This paper concerns geometric study of single-peak solitary waves and double-peaks solitary wave of Gardner equation with Kuramoto–Sivashinsky perturbation. We first reduce the high-dimensional traveling wave system of the perturbed Gardner equation to the perturbed planar system through geometric singular perturbation theory. We then show the persistence of one homoclinic orbit, and the generation of a new homoclinic orbit by the Melnikov function method. Single-peak solitary waves and double-peaks solitary wave are newly found for the perturbed Gardner equation. The numerical simulations are performed to verify the theoretical results.

• Referencias bibliográficas
  • 1. Ogawa, T.: Travelling wave solutions to a perturbed Korteweg-de Vries equation. Hiroshima Math. J. 24(2), 401–422 (1994)
  • 2. Yan, W., Liu, Z., Liang, Y.: Existence of solitary waves and periodic waves to a perturbed generalized KdV equation. Math. Modell. Anal....
  • 3. Chen, A., Guo, L., Deng, X.: Existence of solitary waves and periodic waves for a perturbed generalized BBM equation. J. Differ. Equ. 261(10),...
  • 4. Wen, Z., Zhang, L., Zhang, M.: Dynamics of classical Poisson–Nernst–Planck systems with multiple cations and boundary layers. J. Dyn. Differ....
  • 5. Ge, J., Du, Z.: The solitary wave solutions of the nonlinear perturbed shallow water wave model. Appl. Math. Lett. 103, 106202 (2020)
  • 6. Guo, L., Zhao, Y.: Existence of periodic waves for a perturbed quintic BBM equation. Discr. Contin. Dyn. Syst. 40(8), 4689 (2020)
  • 7. Wen, Z.: On existence of kink and antikink wave solutions of singularly perturbed Gardner equation. Math. Methods Appl. Sci. 43(7), 4422–4427...
  • 8. Zhang, H., Xia, Y., N’gbo, P.: Global existence and uniqueness of a periodic wave solution of the generalized Burgers-Fisher equation....
  • 9. Wen, Z., Bates, P.W., Zhang, M.: Effects on I–V relations from small permanent charge and channel geometry via classical Poisson–Nernst–Planck...
  • 10. Zhang, L., Han, M., Zhang, M., Khalique, C.M.: A new type of solitary wave solution of the mKdV equation under singular perturbations....
  • 11. Sun, X., Huang, W., Cai, J.: Coexistence of the solitary and periodic waves in convecting shallow water fluid. Nonlinear Anal. Real World...
  • 12. Huang, Z., Wen, Z.: Single-and double-peak solitary waves of two-component Drinfel’d–Sokolov–Wilson system with Kuramoto–Sivashinsky perturbation....
  • 13. Bates, P.W., Wen, Z., Zhang, M.: Small permanent charge effects on individual fluxes via PoissonNernst-Planck models with multiple cations....
  • 14. Kamchatnov, A., Kuo, Y., Lin, T., Horng, T., Gou, S., Clift, R., El, G., Grimshaw, R.: Undular bore theory for the Gardner equation. Phys....
  • 15. Miura, R.M.: Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys. 9(8), 1202–1204...
  • 16. Miura, R.M., Gardner, C.S., Kruskal, M.D.: Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants...
  • 17. Watanabe, S.: Ion acoustic soliton in plasma with negative ion. J. Phys. Soc. Jpn. 53(3), 950–956 (1984)
  • 18. Ruderman, M.S., Talipova, T., Pelinovsky, E.: Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions....
  • 19. Grimshaw, R.: Environmental Stratified Flows, no. 3. Springer Science & Business Media, Berlin (2002)
  • 20. Grimshaw, R., Pelinovsky, E., Taipova, T., Sergeeva, A.: Rogue internal waves in the ocean: long wave model. Eur. Phys. J. Special Top....
  • 21. Demler, E., Maltsev, A.: Semiclassical solitons in strongly correlated systems of ultracold bosonic atoms in optical lattices. Ann. Phys....
  • 22. Alejo, M.A.: Nonlinear stability of Gardner breathers. J. Differ. Equ. 264(2), 1192–1230 (2018)
  • 23. Grimshaw, R., Slunyaev, A., Pelinovsky, E.: Generation of solitons and breathers in the extended Korteweg-de Vries equation with positive...
  • 24. Fu, Z., Liu, S., Liu, S.: New kinds of solutions to Gardner equation. Chaos Solitons Fractals 20(2), 301–309 (2004)
  • 25. Wazwaz, A.M.: New solitons and kink solutions for the Gardner equation. Commun. Nonlinear Sci. Numer. Simul. 12(8), 1395–1404 (2007)
  • 26. Betchewe, G., Victor, K.K., Thomas, B.B., Crepin, K.T.: New solutions of the Gardner equation: analytical and numerical analysis of its...
  • 27. Saha, A., Talukdar, B., Chatterjee, S.: Dynamical systems theory for the Gardner equation. Phys. Rev. E 89(2), 023204 (2014)
  • 28. Chen, Y., Liu, Z.: The bifurcations of solitary and kink waves described by the Gardner equation. Discr. Contin. Dyn. Syst. S 9(6), 1629–1645...
  • 29. Fei, J., Cao, W., Ma, Z.: Nonlocal symmetries and explicit solutions for the Gardner equation. Appl. Math. Comput. 314, 293–298 (2017)
  • 30. Ak, T., Triki, H., Dhawan, S., Erduran, K.S.: Theoretical and numerical investigations on solitary wave solutions of Gardner equation....
  • 31. Hepson, O.E., Korkmaz, A., Dag, I.: Exponential B-spline collocation solutions to the Gardner equation. Int. J. Comput. Math. 97(4), 837–850...
  • 32. Slyunyaev, A., Pelinovski, E.: Dynamics of large-amplitude solitons. J. Exp. Theor. Phys. 89(1), 173–181 (1999)
  • 33. Tang, Y., Xu, W., Shen, J., Gao, L.: Persistence of solitary wave solutions of singularly perturbed Gardner equation. Chaos Solitons Fractals...
  • 34. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)
  • 35. Han, M.: Bifurcation Theory of Limit Cycles. Science Press, Beijing (2013)