On the Solitons, Shocks, and Periodic Wave Solutions to the Fractional Quintic Benney–Lin Equation for Liquid Film Dynamics

Abstract

In this study, two improved versions related to the family of $\tilde{G}$-approaches $\left(\text{for simplicity, we will use from now on}\tilde{G}\equiv \left(\frac{G^{{}^{\prime }}}{G}\right)\right)$, namely, the simple $\tilde{G}$-expansion method and the generalized $\left(r+\tilde{G}\right)$-expansion method, are applied to investigate the families of symmetric solitary wave solutions for the quintic fractional Benney–Lin equation that arises in the liquid film. The $\tilde{G}$-expansion method is a transformation-based method that has been used a lot to solve nonlinear partial differential equations and fractional partial differential equations. This method produces several solitary wave solutions to the current problem by supposing a series-form solution. The generalized $\left(r+\tilde{G}\right)$-expansion method, on the other hand, builds on the simple $\tilde{G}$-expansion method by adding more parameters $r$ to the series-form solution. This makes finding more families of solitary wave solutions possible and better shows how the system changes over time. These techniques identify various traveling waves, such as periodic, kink, $M$-shaped, bell-shaped, shock waves and others physical solutions. Some obtained solutions are graphically discussed to better visualize the wave phenomena connected to various symmetrical solitary wave solutions. The fractional Benney–Lin equation's dynamics and wave characteristics may be better understood through these graphical depictions, which makes it easier to analyze the model's behavior in detail.