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In the present investigation, the solutions on the periodic anddouble-periodic background are successfully constructed by Darbouxtransformation using a plane wave seed solution. Firstly, the Darbouxtransformation for the reverse-space-time DNLS equation is constructed.Secondly, periodic solutions, breathers, double-periodic solutions, breatherson the periodic background and double-periodic background are studied. Thirdly,the higher-order rogue waves on the periodic and double-periodic background areconstructed by semi-degenerate Darboux transformations. In addition, thedynamic behaviors of the solutions are plotted to show some interesting newsolution structures.

Breathers and rogue waves on the double-periodic background for the reverse-space-time derivative nonlinear Schr\"odinger equation