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A nonlocal nonlinear Schr\"odinger equation with focusing nonlinearity isconsidered which has been derived as a continuum limit of theCalogero-Sutherland model in an integrable classical dynamical system. Theequation is shown to stem from the compatibility conditions of a system oflinear PDEs, assuring its complete integrability. We construct a nonsingular$N$-phase solution ($N$: positive integer) of the equation by means of a directmethod. The features of the one- and two-phase solutions are investigated incomparison with the corresponding solutions of the defocusing version of theequation. We also provide an alternative representation of the $N$-phasesolution in terms of solutions of a system of nonlinear algebraic equations.Furthermore, the eigenvalue problem associated with the $N$-phase solution isdiscussed briefly with some exact results. Subsequently, we demonstrate thatthe $N$-soliton solution can be obtained simply by taking the long-wave limitof the $N$-phase solution. The similar limiting procedure gives an alternativerepresentation of the $N$-soliton solution as well as the exact results relatedto the corresponding eigenvalue problem.

Multiphase solutions and their reductions for a nonlocal nonlinear Schr\"odinger equation with focusing nonlinearity