Breathers and higher order rogue waves on the double-periodic background for the nonlocal Gerdjikov-Ivanov equation

Through Darboux transformation (DT) method, the breathers and the rogue waves on the double-periodic background of the nonlocal Gerdjikov-Ivanov (GI) equation are derived. First, we use the odd-fold DT, the even-fold DT and the plane wave seed solution to obtain some novel solutions for the nonlocal GI equation. These solutions include single- and double-periodic wave, one-breather, and one-breather on the single- or the double-periodic background. Second, we construct the odd-fold semi-degenerate DT and the even-fold semi-degenerate DT to find the higher-order rogue waves on the single-periodic and the double-periodic background, respectively. Finally, the dynamics of above mentioned solutions are analysed graphically by choosing appropriate parametric values.