  A new integrable nonlocal nonlinear Schroedinger (NLS) equation with clear
physical motivations is proposed. This equation is obtained from a special
reduction of the Manakov system, and it describes Manakov solutions whose two
components are related by a parity symmetry. Since the Manakov system governs
wave propagation in a wide variety of physical systems, this new nonlocal
equation has clear physical meanings. Solitons and multi-solitons in this
nonlocal equation are also investigated in the framework of Riemann-Hilbert
formulations. Surprisingly, symmetry relations of discrete scattering data for
this equation are found to be very complicated, where constraints between
eigenvectors in the scattering data depend on the number and locations of the
underlying discrete eigenvalues in a very complex manner. As a consequence,
general $N$-solitons are difficult to obtain in the Riemann-Hilbert framework.
However, one- and two-solitons are derived, and their dynamics investigated. It
is found that two-solitons are generally not a nonlinear superposition of
one-solitons, and they exhibit interesting dynamics such as meandering and
sudden position shifts. As a generalization, other integrable and physically
meaningful nonlocal equations are also proposed, which include NLS equations of
reverse-time and reverse-space-time types as well as nonlocal Manakov equations
of reverse-space, reverse-time and reverse-space-time types.
