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In this paper, we extend $\overline\partial$ steepest descent method to studythe Cauchy problem for the nonlocal nonlinear Schr\"odinger (NNLS) equationwith weighted Sobolev initial data %and finite density initial data\begin{align*}  &iq_{t}+q_{xx}+2\sigma q^2(x,t)\overline{q}(-x,t)=0,  & q(x,0)=q_0(x), \end{align*} where $ q_0(x)\in L^{1,1}(\mathbb{R})\capL^{2,1/2}(\mathbb{R})$. Based on the spectral analysis of the Lax pair, thesolution of the Cauchy problem is expressed in terms of solutions of aRiemann-Hilbert problem, which is transformed into a solvable model after aseries of deformations. Finally, we obtain the asymptotic expansion of theCauchy problem for the NNLS equation in solitonic region. The leading orderterm is soliton solutions, the second term is the error term is the interactionbetween solitons and dispersion, the error term comes from the corresponding$\bar{\partial}$ equation. Compared to the asymptotic results on the classicalNLS equation, the major difference is the second and third terms in asymptoticexpansion for the NNLS equation were affected by a function $ {\rm Im}\nu(\xi)$for the stationary phase point $\xi$.

Long time asymptotic behavior for the nonlocal nonlinear Schr\"odinger equation with weighted Sobolev initial data