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In this letter, for the discrete parity-time-symmetric nonlocal nonlinearSchr\"{o}dinger equation, we construct the Darboux transformation, whichprovides an algebraic iterative algorithm to obtain a series of analyticsolutions from a known one. To illustrate, the breathing-soliton solutions,periodic-wave solutions and localized rational soliton solutions are derivedwith the zero and plane-wave solutions as the seeds. The properties of thosesolutions are also discussed, and particularly the asymptotic analysis revealsall possible cases of the interaction between the discrete rational dark andantidark solitons.

Darboux transformation and analytic solutions of the discrete \PT-symmetric nonlocal nonlinear Schrodinger equation