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.

In this paper, we explored new families of solitary wave solutions of stochastic Chaffee-Infante equation (SCIE) with Wiener process in Itô sense. SCIE is one of the most important models in mathematical physics used to describe the processes of diffusion and wave propagation. Efficient $(G^{\prime }/G)$-expansion and its version generalized $(r+G^{\prime }/G)$-expansion are applied to obtain explicit solitary wave solutions to the targeted SCIE. The strategic $(G^{\prime }/G)$-expansion method originally converts SCIE to nonlinear ordinary differential equation (NODE) by wave transformation and then converts it to a set of nonlinear algebraic equations on the assumption of finite series-form solutions. Under the analysis of the solutions of the resulting system with Maple, a number of solitary wave solutions in the form of trigonometric, hyperbolic, and rational functions were found. To verify the presence of solitary wave solutions, such as soliton, dark, bright, kink, and anti-kink solitary wave solutions in SCIEs, several solitary wave solutions were evaluated using illustrated 3D visualizations for given parameter values under zero and nonzero noise effects. The implication of our results extends widely across various fields both in stochastic phenomena and nonlinear dynamics and has a contributions to physics and nonlinear science.