Construction of fractional granular model and bright, dark, lump, breather types soliton solutions using Hirota bilinear method

The present article designs the granular metamaterials considering the granular structures of discrete particles which are different from elastic metamaterials consisting of continuous media. In granular metamaterials, the wave propagates through contact with neighboring particles. To identify the propagating properties of wave quantities in the rough granular medium the fractional granular equation is formulated directly in a pre-compressed spherical chain adopting Hertz law and long wave approximation theory. Using phase and group velocities, Caputo fractional derivatives are used to illustrate normal and anomalous dispersion wave dependence. To demonstrate in depth the dynamical behavior of the wave profile, various types of complex solutions like multi-shock, multi-solitons, lump, and breather solutions of the one-dimensional time fractional granular equations are explored employing Hirota's bilinear approach. Finally, the more complicated hybrid solutions such as kink with the lump, soliton with the lump, etc. are exhibited from numerical understanding. The numerical graphs and figures demonstrate the crucial role of the order of derivative (roughness parameter) in the formation of different types of soliton solutions.