A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques - Hossain Md Nur; Rasid Md Mamunur; Abouelfarag I.; El-Rashidy K.; Miah M. Mamun; Kanan Mohammad (Open Physics)
Soliton solutions play a crucial role in modeling stable phenomena across optical communications, fluid dynamics, and plasma physics, owing to their stability and persistence in solving nonlinear equations. This study centers on the extended Sakovich equation, emphasizing the importance of soliton solutions in predicting and controlling localized wave behaviors, which advances nonlinear dynamics and its various applications due to its integrable properties and flexible soliton characteristics. This equation is applicable across diverse fields such as fluid dynamics, nonlinear optics, and plasma physics, where it effectively models nonlinear wave phenomena, including solitons and shock waves. Additionally, it provides crucial insights into wave propagation in biological systems and acoustics, making it a valuable tool for analyzing complex wave dynamics. Additionally, we investigate bifurcation and modulation instability within this equation, employing the improved Sardar subequation method and the ℛ′ℛ,1ℛ\left(\phantom{\rule[-0.75em]{}{0ex}},\frac{{ {\mathcal R} }^{^{′} }}{ {\mathcal R} },\frac{1}{ {\mathcal R} }\right) method to derive solitary wave solutions. These methods yield a diverse range of waveforms – hyperbolic, trigonometric, and rational functions – validated rigorously using Mathematica software for accuracy. Graphical representations vividly display various soliton patterns, such as singular, multi-singular, periodic singular, kink, anti-kink, bell-shaped, Kuznetsov–Ma Breather, and parabolic-shaped, highlighting their effectiveness in revealing innovative solutions. Furthermore, a comparative analysis verified the novelty of our derived soliton solutions. This research significantly contributes to advancing soliton solutions for the Sakovich equation, promising diverse applications across scientific disciplines.