The Study Of Rogue Wave And Their Propagation Properties In One-dimensional Granular Chain

Granular matter is one of the most extensive substances found in the natural world,showing more abundant properties.Since Vitali F.Nesterenko gave the “solitary wave” solution in the one-dimensional bead chain for the first time in 1983,more and more scientists began to pay attention to the nonlinear wave and related problems in the one-dimensional bead chain.The rogue waves,responsible for a large number of maritime disasters,has been studied extensively for decades.It is found that the rogue wave appear not only in oceans but also in the atmosphere,in optics,in plasmas,in superfluids,in Bose-Einstein condensates,and in capillary waves.Based on the Hertz model,the rogue wave in one-dimensional bead chain is studied numerically.By using the traditional perturbation technique,a focusing nonlinear Schrodinger equation(NLSE)for the one-dimensional(1D)bead chain with the initial prestress is first obtained.In addition to the unique fundamental Peregrine soliton,NLSE can also predict an infinite hierarchy of higher order breather solutions.The study of these solutions is crucial in explaining the even higher amplitude waves that can be observed in deep water conditions.In addition to,other special solutions of the NLSE have been obtained——single breather solutions,such as the Akhmediev breather,the Kuznetsov-Ma breather,and the quasi-Akhmediev breather solutions.In our paper,we study the rogue wave in a 1D bead chain by the combination of numerical simulation and analytical analysis.Present study shows that both the higher-order rogue wave and the Peregrine soliton(rogue wave)do exist in granular materials and can be successfully simulated by directly solve the equation of the motion of 1D bead chain with Hertzian interaction.The solutions from the NLSE can correctly describe the real rogue wave as well as the super rogue wave in the limiting case of small amplitude.Both the rogue wave and the super rogue wave propagate in the granular bead chain as if they are solitary waves.