  In this paper, the renowned Riemann-Hilbert method is employed to investigate
the initial value problem of Tzitz\'eica equation on the line. Initially, our
analysis focuses on elucidating the properties of two reflection coefficients,
which are determined by the initial values. Subsequently, leveraging these
reflection coefficients, we construct a Riemann-Hilbert problem that is a
powerful tool to articulate the solution of the Tzitz\'eica equation. Finally,
the nonlinear steepest descent method is applied to the oscillatory
Riemann-Hilbert problem, which enables us to delineate the long-time asymptotic
behaviors of solutions to the Tzitz\'eica equation across various regions.
Moreover, it is shown that the leading-order terms of asymptotic formulas match
well with direct numerical simulations.
