  We employ the $\bar{\partial}$-steepest descent method in order to
investigate the Cauchy problem of the complex short pulse (CSP) equation with
initial conditions in weighted Sobolev space $H^{1,1}(\mathbb{R})=\{f\in
L^{2}(\mathbb{R}): f',xf\in L^{2}(\mathbb{R})\}$. The long time asymptotic
behavior of the solution $u(x,t)$ is derived in a fixed space-time cone
$S(x_{1},x_{2},v_{1},v_{2})=\{(x,t)\in\mathbb{R}^{2}: y=y_{0}+vt,
~y_{0}\in[y_{1},y_{2}], ~v\in[v_{1},v_{2}]\}$. Based on the resulting
asymptotic behavior, we prove the solution resolution conjecture of the CSP
equation which includes the soliton term confirmed by $N(I)$-soliton on
discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum
with residual error up to $O(t^{-1})$.
