Dynamic behaviors of vector breather waves and higher-order rogue waves in the coupled Gerdjikov–Ivanov equation

We extend Darboux-dressing transformation with expansion theorem to study the coupled Gerdjikov–Ivanov (cGI) equation. We successfully find its novel vector breather wave solutions and Nth-order rouge wave solutions. By considering an expansion theory, we first construct the Darboux-dressing transformation, which could iterate with the same spectral parameters for finding interesting exact solutions of the cGI equation. Based on the resulting Darboux-dressing transformation, we derive its exact breather wave solutions by means of matrix exponential function. Moreover, the Taylor multi-series expansion method is employed to derive higher-order rouge wave solutions of the equation. Finally, some interesting dynamic behaviors of breather waves and rouge waves are analyzed graphically.