A deformed reduced semi-discrete Kaup–Newell equation, the related integrable family and Darboux transformation

• We propose a deformed reduced semi-discrete Kaup–Newell equation.
• We derive the related integrable family through discrete zero curvature equation.
• We establish a Darboux transformation of Lax pair of the proposed equation.
• Three exact solutions are given.

A deformed reduced semi-discrete Kaup–Newell equation and its related integrable family are derived from discrete zero curvature equation. A Darboux transformation of Lax pair of this equation is established with the help of gauge transformation. By means of the resulting Darboux transformation, three exact solutions are given.