The Riemann waves in two spatial dimensions are described by the fractional Calogero-Bogoyavlenskii-Schiff equation, which has been used to explain numerous physical phenomena including magneto-sound waves in plasmas, tsunamis, and flows in rivers and internal oceans. This work concerned itself with obtaining new analytic soliton solutions for the fractional Calogero-Bogoyavlenskii-Schiff model based on the fractional conformable. By solving the model equation with the Riccati-Bernoulli sub-ODE technique in association with the Bäcklund transformation, the solution was found in terms of trigonometric, hyperbolic, and rational functions. To analyze the detailed features of the wave structures as well as the pattern of dynamics of these solutions, 3D and contour diagrams were plotted by using Wolfram Mathematica. A great advantage of these types of visualizations is that they demonstrate amplitude, shape, and propagation characteristics of the selected soliton solutions. The results reveal that the proposed approach is accurate, universal, and fast for the investigation of the different aspects of the Riemann problem and the related phenomena concerning the propagation of waves.

In the present work, we consider the space-time fractional Whitham-Broer-Kaup (FWBK) equation and then find its analytical solutions under the framework of the Riccati-Bernoulli sub-ordinary differential equation method along with the Bäcklund transformation. The derived solutions are described in terms of the hyperbolic rational and trigonometric functions and resolve unique wave features. A unique feature of this work is the investigation of the impact of the fractional order on wave steepness in $2D$, $3D$, and contour plots. Also, it should be pointed that the found relationship shows that the change of the fractional order parameter really describes the traveling periodic waves, for example, Stokes waves, with potential importance for the analysis of wave propagation and stability in the nonlinear fractional structures. This work carries theoretical developments of fractional calculus and ideas for applications in fluid dynamics and wave mechanics.