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In this paper, we are going to investigate Cauchy problem for nonlocalnonlinear Schr\"odinger equation with the initial potential $q_0(x)$ inweighted 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 bythe solution of a Riemann-Hilbert problem (RH problem), and assuming nodiscrete spectrum, we majorly apply $\bar\partial$-steepest cescent descentmethod on analyzing the long-time asymptotic behavior of it.

Long-time asymptotic behavior of the nonlocal nonlinear Schr\"odinger equation with initial potential in weighted sobolev space