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We employ the $\bar{\partial}$-steepest descent method in order toinvestigate the Cauchy problem of the complex short pulse (CSP) equation withinitial conditions in weighted Sobolev space $H^{1,1}(\mathbb{R})=\{f\inL^{2}(\mathbb{R}): f',xf\in L^{2}(\mathbb{R})\}$. The long time asymptoticbehavior 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 resultingasymptotic behavior, we prove the solution resolution conjecture of the CSPequation which includes the soliton term confirmed by $N(I)$-soliton ondiscrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrumwith residual error up to $O(t^{-1})$.

Soliton resolution for the complex short pulse equation with weighted Sobolev initial data