  In this paper, we study the Cauchy problem for the focusing nonlinear
short-pluse equation by using $\overline\partial$ steepest descent method.
\begin{align} &u_{xt}=u+\frac{1}{6}(u^3)_{xx}, \nonumber\\ &u(x,0)=u_0(x)\in
H^{1,1}(R),\nonumber \end{align} where $H^{1,1}(R)$ is a weighted Sobolev
space. Because the spectral variable z is the same order in the WKI-type Lax
pair, we construct the solution of SP equation in the new scale $(y,t)$,
whereas the original scale $(x,t)$ is given in terms of functions in the new
scale and the solution of Riemann-Hilbert problem. In any fixed space-time cone
of the new scale $(y,t)$ which stratify that $v_1\leq v_1 \in R^-$ and
$\xi=\frac{y}{t}<0$, \begin{equation} C(y_1,y_2,v_1,v_2) = \left\lbrace (y,t)
\in R^2|y=y_0+vt, y_0 \in[y_1,y_2]\text{, } v\in[v_1,v_2]\right\rbrace,
\nonumber \end{equation} we compute the long time asymptotic expansion of the
solution $u(x,t)$, which prove soliton resolution conjecture consisting of
three terms: the leading order term can be characterized with an $N(I)$-soliton
whose parameters are modulated by a sum of localied soliton-soliton
interactions as one moves through the cone; the second $t^{-1/2}$ order term
coming from soliton-radiation interactions on continuous spectrum up to an
residual error order $\mathcal{O}(|t|^{-1})$ from a $\overline\partial$
equation. Our results also show that soliton solutions of short-pluse equation
are asymptotically stable.
