  In this work, we employ the $\bar{\partial}$-steepest descent method to
investigate the Cauchy problem of the nonlocal nonlinear Schr\"{o}dinger (NNLS)
equation with finite density type initial conditions in weighted Sobolev space
$\mathcal{H}(\mathbb{R})$. Based on the Lax spectrum problem, a Riemann-Hilbert
problem corresponding to the original problem is constructed to give the
solution of the NNLS equation with the finite density type initial boundary
value condition. By developing the $\bar{\partial}$-generalization of
Deift-Zhou nonlinear steepest descent method, we derive the leading order
approximation to the solution $q(x,t)$ in soliton region of space-time,
$\left(\frac{x}{2t}\right)=\xi$ for any fixed $\xi=\in (1,K)$($K$ is a
sufficiently large real constant), and give bounds for the error decaying as
$|t|\rightarrow\infty$. Based on the resulting asymptotic behavior, the
asymptotic approximation of the NNLS equation is characterized with the soliton
term confirmed by $N(\Lambda)$-soliton on discrete spectrum and the
$t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to
$O(t^{-\frac{3}{4}})$.
