  In this paper, we analyze the long-time behavior of the solution of the
initial value problem (IVP) for the short pulse (SP) equation. As the SP
equation is a complete integrable system, which posses a Wadati-Konno-Ichikawa
(WKI)-type Lax pair, we formulate a $2\times 2$ matrix Riemann-Hilbert problem
to this IVP by using the inverse scattering method. Since the spectral variable
$k$ is the same order in the WKI-type Lax pair, we construct the solution of
this IVP parametrically in the new scale $(y,t)$, whereas the original scale
$(x,t)$ is given in terms of functions in the new scale, in terms of the
solution of this Riemann-Hilbert problem. However, by employing the nonlinear
steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert
problem, we can get the explicit leading order asymptotic of the solution of
the short pulse equation in the original scale $(x,t)$ as time $t$ goes to
infinity.
