Exploring the exact soliton solutions and dynamical analysis to the model in fluid dynamics and plasma physics

This research studies the modified equal-width Burgers equation, an important model in both fluid dynamics and plasma physics. This model is commonly used to describe the traffic flow, optical pulses in fibers, ion-acoustic plasma waves, shallow water waves, etc. The new P/Q-expansion method, together with the enlarged extended Sinh-Gordon equation expansion strategy, yield precise analytical solutions. These methods generate a variety of solutions, including trigonometric, hyperbolic, and mixed wave shapes. Graphs in two dimensions, three dimensions, and contour representations of the developed solutions show how fractional-order elements affect wave generation and structural performance. To investigate the basic nonlinear properties of the system, a comprehensive qualitative dynamical analysis are carried out. Furthermore, the stability of the resulting solutions is assessed by using the stability analysis. While, modulation instability analysis is used for confirming the existence of stable steady-state wave forms under appropriate parametric conditions. Bifurcation analysis is used to explain how the qualitative structures behaves with the small changing in the parameters.