Dark and bright soliton phenomena of the generalized time-space fractional equation with gas bubbles

The objective of this work is to provide the method of getting the closed-form solitary wave solution of the fractional $(3+1)$-generalized nonlinear wave equation that characterizes the behavior of liquids with gas bubbles. The same phenomena are evident in science, engineering, and even in the field of physics. This is done by employing the Riccati-Bernoulli sub-ode in a systematic manner as applied to the Bäcklund transformation in the study of this model. New soliton solutions, in the forms of soliton, are derived in the hyperbolic and trigonometric functions. The used software is the computational software Maple, which makes it possible to perform all the necessary calculations and the check of given solutions. The result of such calculations is graphical illustrations of the steady-state characteristics of the system and its dynamics concerning waves and the inter-relationships between the parameters. Moreover, the contour plots and the three-dimensional figures describe the essential features, helping readers understand the physical nature of the model introduced in this work.