A non-autonomous fractional granular model: Multi-shock, Breather, Periodic, Hybrid solutions and Soliton interactions

This paper explores a novel generalized one-dimensional fractional order Granular equation with the effect of periodic forced term. This type of equation arises in different area of mathematical physics with several rough materials in engineering applications. By taking into account of external forces in combination with Hertz constant law and the long wave approximation principle, we construct the fractional order one-dimensional crystalline chain of elastic spheres. A suitable transformation is implemented to convert the fractional order equation to a regular equation. The Hirota’s bilinear approach is used to secure solutions for kink and anti-kink types shock solutions. In order to find periodic and solitary wave solutions, the Granular model is converted to approximate KdV model. The newly developed solutions exhibit a range of interesting dynamics due to the existence of an external force and the roughness effect. Many hybrid solutions are created by taking a long wave limit of a fraction of the exponential and trigonometrical functions in the bilinear form of the granular model. The hybrid solutions show different superposed wave shapes with lumps, kinks and breathers. The dynamical interaction of these solutions are further illustrated graphically. Furthermore, the system’s stability is assessed using perturbation techniques in order to comprehend its resilience and possible uses in granular particles in real-world situations, guaranteeing their dependability in a range of circumstances.