Dynamical Behaviors and Abundant New Soliton Solutions of Two Nonlinear PDEs via an Efficient Expansion Method in Industrial Engineering

Ibrahim Alraddadi, M. Akher Chowdhury, M. S. Abbas, K. El-Rashidy, J. R. M. Borhan, M. Mamun Miah () and Mohammad Kanan ()
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Ibrahim Alraddadi: Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
M. Akher Chowdhury: Department of Mathematics, Bangladesh Army University of Engineering and Technology, Natore 6431, Bangladesh
M. S. Abbas: Administrative and Financial Science Department, Ranyah University College, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
K. El-Rashidy: Technology and Science Department, Ranyah University College, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
J. R. M. Borhan: Department of Mathematics, Jashore University of Science and Technology, Jashore 7408, Bangladesh
M. Mamun Miah: Division of Mathematical and Physical Sciences, Kanazawa University, Kanazawa 9201192, Japan
Mohammad Kanan: Department of Industrial Engineering, College of Engineering, University of Business and Technology, Jeddah 21448, Saudi Arabia

Mathematics, 2024, vol. 12, issue 13, 1-21

Abstract: In this study, we discuss the dynamical behaviors and extract new interesting wave soliton solutions of the two significant well-known nonlinear partial differential equations (NPDEs), namely, the Korteweg–de Vries equation (KdVE) and the Jaulent–Miodek hierarchy equation (JMHE). This investigation has applications in pattern recognition, fluid dynamics, neural networks, mechanical systems, ecological systems, control theory, economic systems, bifurcation analysis, and chaotic phenomena. In addition, bifurcation analysis and the chaotic behavior of the KdVE and JMHE are the main issues of the present research. As a result, in this study, we obtain very effective advanced exact traveling wave solutions with the aid of the proposed mathematical method, and the solutions involve rational functions, hyperbolic functions, and trigonometric functions that play a vital role in illustrating and developing the models involving the KdVE and the JMHE. These new exact wave solutions lead to utilizing real problems and give an advanced explanation of our mentioned mathematical models that we did not yet have. Some of the attained solutions of the two equations are graphically displayed with 3D, 2D, and contour panels of different shapes, like periodic, singular periodic, kink, anti-kink, bell, anti-bell, soliton, and singular soliton wave solutions. The solutions obtained in this study of our considered equations can lead to the acceptance of our proposed method, effectively utilized to investigate the solutions for the mathematical models of various important complex problems in natural science and engineering.

Keywords: the Korteweg–de Vries equation; the Jaulent–Miodek hierarchy equation; mathematical method; exact traveling wave solutions; nonlinear PDEs; dynamical behaviors (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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