Abstract
A time-fractional Ablowitz-Ladik model is investigated by using the fractional exponential function approach, and bright and dark discrete soliton solutions, exponential discrete solutions and discrete unusual wave solutions are found. Dynamical evolutions of these fractional discrete solitons including bright and dark discrete solitons and two kinds of unusual waves are studied. When fractional order equals to 1, all fractional solutions obtained in this paper can degenerate into those in the literature.
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