  This study numerically investigates the nonlinear interaction of head-on
solitary waves in a granular chain (a nonintegrable system) and compares the
simulation results with the theoretical results in fluid (an integrable
system). Three stages (i.e., pre-in-phase traveling stage, central-collision
stage, and post-in-phase traveling stage) are identified to describe the
nonlinear interaction processes in the granular chain. The nonlinear scattering
effect occurs in the central-collision stage, which decreases the amplitude of
incident solitary waves. Compared with the leading-time phase in the incident
and separation collision processes, the lagging-time phase in the separation
collision process is smaller. This asymmetrical nonlinear collision results in
an occurrence of leading phase shifts of time and space in the post-in-phase
traveling stage. We next find that solitary wave amplitude does not influence
the immediate space-phase shift in the granular chain. The space$-$phase shift
of the post-in-phase traveling stage is only determined by measurement position
rather than wave amplitude. The results are reversed in the fluid. An increase
in solitary wave amplitude leads to decreased attachment, detachment and
residence times for granular chain and fluid. For the immediate time-phase
shift, leading and lagging phenomena appear in the granular chain and the
fluid, respectively. These results offer new knowledge for designing mechanical
metamaterials and energy-mitigating systems.
