  We study the integrable nonlocal nonlinear Schr\"odinger equation proposed by
Ablowitz and Musslimani, that is considered as a particular example of
equations with parity-time ($\mathcal{PT}$) symmetric self-induced potential.
We consider dynamics (including collisions) of moving solitons. Analytically we
develop a collective coordinate approach based on variational methods and
examine its applicability in the system. We show numerically that a single
moving soliton can pass the origin and decay or be trapped at the origin and
blows up at a finite time. Using a standard soliton ansatz, the variational
approximation can capture the dynamics well, including the finite-time blow up,
even though the ansatz is relatively far from the actual blowing-up soliton
solution. In the case of two solitons moving towards each other, we show that
there can be a mass transfer between them, in addition to wave scattering. We
also demonstrate that defocusing nonlinearity can support bright solitons.
