Head-on Collision of Solitary Waves Described by the Toda Lattice Model in Granular Chain

We study the head-on collision of two solitary waves in a precompressed granular chain using the discrete element method.Our study takes the Toda chain solution as the initial condition for the simulations.The simulation covers the dynamical evolution of the collision process from the start of the incident wave to the end of the collision.The interaction has a central collision region of about five-grain width in which two solitary waves merge completely and share only one peak.Four stages,i.e.,the pre-in-phase traveling stage,lag-phase collision state,lead-phase collision state,and post-in-phase traveling stage,are identified to describe the complex collision processes.Our results may be helpful for explaining the existence of long-lived solitary waves seen in the simulations by Takato and Sen [Europhys.Lett.100（2012） 24003].