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A new integrable nonlocal nonlinear Schroedinger (NLS) equation with clearphysical motivations is proposed. This equation is obtained from a specialreduction of the Manakov system, and it describes Manakov solutions whose twocomponents are related by a parity symmetry. Since the Manakov system governswave propagation in a wide variety of physical systems, this new nonlocalequation has clear physical meanings. Solitons and multi-solitons in thisnonlocal equation are also investigated in the framework of Riemann-Hilbertformulations. Surprisingly, symmetry relations of discrete scattering data forthis equation are found to be very complicated, where constraints betweeneigenvectors in the scattering data depend on the number and locations of theunderlying 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. Itis found that two-solitons are generally not a nonlinear superposition ofone-solitons, and they exhibit interesting dynamics such as meandering andsudden position shifts. As a generalization, other integrable and physicallymeaningful nonlocal equations are also proposed, which include NLS equations ofreverse-time and reverse-space-time types as well as nonlocal Manakov equationsof reverse-space, reverse-time and reverse-space-time types.

A new nonlocal nonlinear Schroedinger equation and its soliton solutions