  We consider the Cauchy problem for the integrable nonlocal nonlinear
Schr\"odinger (NNLS) equation $ \I\partial_t
q(x,t)+\partial_{x}^2q(x,t)+2\sigma q^{2}(x,t)\overline{q(-x,t)}=0 $ with
initial data $q(x,0)\in H^{1,1}(\mathbb{R})$. It is known that the NNLS
equation is integrable and it has soliton solutions, which can have isolated
finite time blow-up points. The main aim of this work is to propose a suitable
concept for continuation of weak $H^{1,1}$ local solutions of the general
Cauchy problem (particularly, those admitting long-time soliton resolution)
beyond possible singularities. Our main tool is the inverse scattering
transform method in the form of the Riemann-Hilbert problem combined with the
PDE existence theory for nonlinear dispersive equations.
