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In this paper, we introduce the reverse-space and reverse-space-time nonlocaldiscrete derivative nonlinear Schr\"odinger (DNLS) equations through thenonlocal symmetry reductions of the semi-discrete Gerdjikov-Ivanov equation.The muti-soliton solutions of two types of nonlocal discrete derivativenonlinear Schr\"odinger equations are derived by means of the Hirota bilinearmethod and reduction approach. We also investigate the dynamics of solitonsolutions and reveal the rich soliton structures in the reverse-space andreverse-space-time nonlocal discrete DNLS equations. Our investigation showsthat the solitons of these nonlocal equations often breathe and periodicallycollapse for some soliton parameters, but remain nonsingular for other range ofparameters.

Soliton solutions and their dynamics in reverse-space and reverse-space-time nonlocal discrete derivative nonlinear Schr\"odinger equations