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Peer-reviewedCC BY 4.0Royal Society Open Science

Dynamical analysis and closed-form soliton solutions for the doubly dispersive equation

  • Muhammad Zafarullah Baber1,2,3
  • Muhammad Waqas Yasin4
  • Khadeeja Arif4
  • Muhammad Qasim1
  • Baboucarr Ceesay5
  • Nauman Ahmed6,7
7 affiliations
  1. 1Department of Mathematics, Shanghai University and Newtouch Center for Mathematics of Shanghai University 1 , Shanghai, People's Republic of China
  2. 2Department of Mathematics and Statistics, The University of Lahore 2 , Sargodha campus, Pakistan
  3. 3Center for Theoretical Physics, Khazar University 3 , 41 Mehseti Str., Baku AZ1096, Azerbaijan
  4. 4Department of Mathematics, University of Narowal 4 , Narowal, Pakistan
  5. 5Mathematics Unit, The University of The Gambia 5 , Serrekunda, Gambia
  6. 6Department of Mathematics and Statistics, The University of Lahore 6 , Lahore, Pakistan
  7. 7Research Center of Applied Mathematics, Khazar University 7 , Baku, Azerbaijan
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Abstract

The purpose of this work is to examine new exact soliton solutions for the doubly dispersive equation (DDE) by utilizing a finite series in terms of Jacobi elliptic functions (JEFs). For the exact analytical solution of nonlinear partial differential equations (NLPDEs), the JEF expansion approach is frequently employed. The suggested approach yields several solutions, including hyperbolic-type solutions, singular periodic wave solutions, JEF solutions, exponential solutions, bright, dark and bright-dark combo soliton solutions and Weierstrass elliptic double periodic solutions. Experts in engineering models will be interested in the findings, which will help them comprehend waves. To this end, we use the Galilean transformation to derive a dynamical system closely related to the equation. The bifurcation behaviours observed in this derived system were then investigated using concepts from the theory of planar dynamical systems. To examine the possible existence of chaotic behaviours, we carefully evaluate the DDE and add a perturbed term to the dynamical system. Comprehensive two- and three-dimensional (2D and 3D) phase portraits are presented to further enhance this inquiry. Some solutions are selected for the graphical behaviour, and these plots are explained according to real-life applications.

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Cite

BibTeXRIS
BibTeX
@article{baber2026dynamical,
  title = {{Dynamical analysis and closed-form soliton solutions for the doubly dispersive equation}},
  author = {Muhammad Zafarullah Baber and Muhammad Waqas Yasin and Khadeeja Arif and Muhammad Qasim and Baboucarr Ceesay and Nauman Ahmed},
  journal = {Royal Society Open Science},
  year = {2026},
  month = sep,
  doi = {10.1098/rsos.260878},
  url = {https://royalsocietypublishing.org/rsos/article/13/9/260878/483417/Dynamical-analysis-and-closed-form-soliton},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Dynamical analysis and closed-form soliton solutions for the doubly dispersive equation
AU  - Muhammad Zafarullah Baber
AU  - Muhammad Waqas Yasin
AU  - Khadeeja Arif
AU  - Muhammad Qasim
AU  - Baboucarr Ceesay
AU  - Nauman Ahmed
PY  - 2026
DA  - 2026/09/30
JO  - Royal Society Open Science
DO  - 10.1098/rsos.260878
UR  - https://royalsocietypublishing.org/rsos/article/13/9/260878/483417/Dynamical-analysis-and-closed-form-soliton
AB  - The purpose of this work is to examine new exact soliton solutions for the doubly dispersive equation (DDE) by utilizing a finite series in terms of Jacobi elliptic functions (JEFs). For the exact analytical solution of nonlinear partial differential equations (NLPDEs), the JEF expansion approach is frequently employed. The suggested approach yields several solutions, including hyperbolic-type solutions, singular periodic wave solutions, JEF solutions, exponential solutions, bright, dark and bright-dark combo soliton solutions and Weierstrass elliptic double periodic solutions. Experts in engineering models will be interested in the findings, which will help them comprehend waves. To this end, we use the Galilean transformation to derive a dynamical system closely related to the equation. The bifurcation behaviours observed in this derived system were then investigated using concepts from the theory of planar dynamical systems. To examine the possible existence of chaotic behaviours, we carefully evaluate the DDE and add a perturbed term to the dynamical system. Comprehensive two- and three-dimensional (2D and 3D) phase portraits are presented to further enhance this inquiry. Some solutions are selected for the graphical behaviour, and these plots are explained according to real-life applications.
SN  - 2054-5703
ER  -