Abstract
In this investigation, the analytical behavior of two prominent nonlinear wave equations, namely the doubly dispersive equation (DDE) and the Ablowitz-Kaup-Newell-Segur equation (AKNSE), have been scrutinized. Bifurcation analysis, sensitivity as well as chaotic phenomena are also performed for the earlier-mentioned dynamical systems. These analyzes have profuse applications in the field of electrical circuits and control systems, phase transitions in materials, climate patterns, chemical reaction networks, forecasting market trends, signal processing, and quantum mechanics. Using an advanced mathematical technique, the exact solutions of the mentioned two-wave equations with singular bell-shaped soliton, bell-shaped soliton, anti-bell-shaped soliton, singular soliton, and singular periodic soliton have been studied. The technique utilized in the study is dependable for solving complex nonlinear problems in various natural science and engineering disciplines employed by many researchers. This study identified ten general solutions and ten particular solutions for the two mentioned equations that are novel and precise wave solutions. The solutions obtained in our study are significant not only for understanding the mentioned field but also may be used in revealing other interesting phenomena, such as the analysis of seismology, the study of compulsive collapse, and the study of the material effects and inner construction of solids.
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