Explicit solitary wave structures for the fractional-order Sobolev-type equations and their stability analysis

The current research is concerned with solitary wave structures to the time fractional-order Sobolev-type equations. The special types of Sobolev-type equations are under consideration such as the generalized hyperelastic-rod wave (HRW) equation, and Camassa–Holm (CH) equation. These equations occur in several fields, including particularly in quantum field theory, plasma theory, ecology, consolidation of clay and fluid dynamics. The underlying models are investigated analytically by applying two techniques, such as the generalized projective Riccati equation (GPRE) and the modified auxiliary equation (MAE). The gained results are obtained from the different families of solutions such as, including a periodic wave, kink-type wave peakon, a singular wave, and dark solutions. The gained results are denoted as hyperbolic and trigonometric functions. Furthermore, we check that the underlying models are stable using the concept of linearized stability. The propagation behavior of the gained results is displayed in 3D, 2D, and contour visualizations to investigate the influence of various relevant parameters. These results will help the researchers to understand the physical situations.