Dynamical behaviors and soliton structures in the space-time conformable Wu-Zhang system arising in coastal design

The present research work explores the space-time fractional nonlinear conformable Wu-Zhang system, that describes $(1 + 1)$ -dimensional dispersive long waves in two horizontal oriented shallow waters. We get many explicit soliton solutions for the target model with conformable derivatives by applying the $(\frac{G'}{G})$ -expansion technique. Three-dimensional, contour, and two-dimensional graphs that show the generation of kink, dark, bright, dark-bright, parabolic, bell-shaped, and breather soliton are used to illustrate the importance of these recently discovered soliton solutions. Important information about the conformable Wu-Zhang system is revealed by examining the effects of the fractional order spatial parameter on these solutions. The method proposed in this study is simple, reliable, and efficient, and it has potential for wider use with a range of nonlinear equations in various scientific fields. This study contributes to our understanding of dispersive long waves in shallow waters by unveiling new soliton dynamics and their regulating mechanisms inside dispersion and nonlinearity-managed systems. Moreover, the bifurcation and periodic behavior of the dynamical system are also investigated using phase portraits and the time series approach. The presence of periodic behaviour in the perturbed dynamical system is analyzed, and positive results regarding the dynamical behaviors of the conformable Wu-Zhang system are obtained.