Bifurcation Analysis and Solitons Dynamics of the Fractional Biswas–Arshed Equation via Analytical Method

Asim Zafar, Waseem Razzaq, Abdullah Nazir, Mohammed Ahmed Alomair, Abdulaziz S. Al Naim and Abdulrahman Alomair ()
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Asim Zafar: Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61100, Pakistan
Waseem Razzaq: Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61100, Pakistan
Abdullah Nazir: Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61100, Pakistan
Mohammed Ahmed Alomair: Department of Quantitative Methods, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia
Abdulaziz S. Al Naim: Department of Accounting, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia
Abdulrahman Alomair: Department of Accounting, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Mathematics, 2025, vol. 13, issue 19, 1-22

Abstract: This paper investigates soliton solutions of the time-fractional Biswas–Arshed (BA) equation using the Extended Simplest Equation Method (ESEM). The model is analyzed under two distinct fractional derivative operators: the β -derivative and the M -truncated derivative. These approaches yield diverse solution types, including kink, singular, and periodic-singular forms. Also, in this work, a nonlinear second-order differential equation is reconstructed as a planar dynamical system in order to study its bifurcation structure. The stability and nature of equilibrium points are established using a conserved Hamiltonian and phase space analysis. A bifurcation parameter that determines the change from center to saddle-type behaviors is identified in the study. The findings provide insight into the fundamental dynamics of nonlinear wave propagation by showing how changes in model parameters induce qualitative changes in the phase portrait. The derived solutions are depicted via contour plots, along with two-dimensional (2D) and three-dimensional (3D) representations, utilizing Mathematica for computational validation and graphical illustration. This study is motivated by the growing role of fractional calculus in modeling nonlinear wave phenomena where memory and hereditary effects cannot be captured by classical integer-order approaches. The time-fractional Biswas–Arshed (BA) equation is investigated to obtain diverse soliton solutions using the Extended Simplest Equation Method (ESEM) under the β -derivative and M -truncated derivative operators. Beyond solution construction, a nonlinear second-order equation is reformulated as a planar dynamical system to analyze its bifurcation and stability properties. This dual approach highlights how parameter variations affect equilibrium structures and soliton behaviors, offering both theoretical insights and potential applications in physics and engineering.

Keywords: extended simplest equation method (ESEM); bifurication analysis; exact solutions; solitons; time-fractional Biswas–Arshed (BA) model; β and M-truncated fractional derivatives (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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