  In this paper, we are going to investigate Cauchy problem for nonlocal
nonlinear Schr\"odinger equation with the initial potential $q_0(x)$ in
weighted sobolev space $H^{1,1}(\mathbb{R})$, \begin{align*}
iq_t(x,t)&+q_{xx}(x,t)+2\sigma q^2(x,t)\bar q(-x,t)=0,\quad\sigma=\pm1,\\
q(x,0)&=q_0(x). \end{align*} We show that the solution can be represented by
the solution of a Riemann-Hilbert problem (RH problem), and assuming no
discrete spectrum, we majorly apply $\bar\partial$-steepest cescent descent
method on analyzing the long-time asymptotic behavior of it.
