  In this paper, we extend $\overline\partial$ steepest descent method to study
the Cauchy problem for the nonlocal nonlinear Schr\"odinger (NNLS) equation
with 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})\cap
L^{2,1/2}(\mathbb{R})$. Based on the spectral analysis of the Lax pair, the
solution of the Cauchy problem is expressed in terms of solutions of a
Riemann-Hilbert problem, which is transformed into a solvable model after a
series of deformations. Finally, we obtain the asymptotic expansion of the
Cauchy problem for the NNLS equation in solitonic region. The leading order
term is soliton solutions, the second term is the error term is the interaction
between solitons and dispersion, the error term comes from the corresponding
$\bar{\partial}$ equation. Compared to the asymptotic results on the classical
NLS equation, the major difference is the second and third terms in asymptotic
expansion for the NNLS equation were affected by a function $ {\rm Im}\nu(\xi)$
for the stationary phase point $\xi$.
