In this article, a nonlinear dispersive structure namely the modified Benjamin–Bona–Mahony (BBM) model is studied by employing three different approaches to develop some new analytical and semi-analytical solutions. The modified extended tan hyperbolic function method, the improved F-expansion method and the Adomian decomposition method are implemented to obtain various interesting wave structures. The comparison among various types of solutions is analyzed to study the corresponding absolute error. Furthermore, the stability of the results along with the modulation instability are successfully deliberated. The 3D, contour and 2D graphs are visualized for several fascinating exact solutions to comprehend their behavior.
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