Bifurcation, Quasi-Periodic, Chaotic Pattern, and Soliton Solutions to Dual-Mode Gardner Equation

Adel Elmandouh ()
Additional contact information
Adel Elmandouh: Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia

Mathematics, 2025, vol. 13, issue 5, 1-22

Abstract: This study aims to investigate various dynamical aspects of the dual-mode Gardner equation derived from an ideal fluid model. By applying a specific wave transformation, the model is reduced to a planar dynamical system, which corresponds to a conservative Hamiltonian system with one degree of freedom. Using Hamiltonian concepts, phase portraits are introduced and briefly discussed. Additionally, the conditions for the existence of periodic, super-periodic, and solitary solutions are summarized in tabular form. These solutions are explicitly constructed, with some graphically represented through their 2 D and 3 D profiles. Furthermore, the influence of specific physical parameters on these solutions is analyzed, highlighting their effects on amplitude and width. By introducing a more general periodic external influence into the model, quasi-periodic and chaotic behavior are explored. This is achieved through the presentation of 2 D and 3 D phase portraits, along with time-series analyses. To further examine chaotic patterns, the Poincaré surface of section and sensitivity analysis are employed. Numerical simulations reveal that variations in frequency and amplitude significantly alter the dynamical characteristics of the system.

Keywords: dual-mode Gardner equation; phase portrait; wave solutions; quasi-periodic; chaotic behavior (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/5/841/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/5/841/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:5:p:841-:d:1604470

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().